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version 1.1, 2000/01/10 15:35:21 version 1.1.1.4, 2003/08/25 16:06:01
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 \input texinfo    @c -*-texinfo-*-  \input texinfo    @c -*-texinfo-*-
 @c %**start of header  @c %**start of header
 @setfilename gmp.info  @setfilename gmp.info
 @settitle GNU MP 2.0.2  @include version.texi
   @settitle GNU MP @value{VERSION}
 @synindex tp fn  @synindex tp fn
 @iftex  @iftex
 @afourpaper  @afourpaper
 @end iftex  @end iftex
 @comment %**end of header  @comment %**end of header
   
 @ifinfo  @copying
 @format  This manual describes how to install and use the GNU multiple precision
 START-INFO-DIR-ENTRY  arithmetic library, version @value{VERSION}.
 * gmp: (gmp.info).               GNU Multiple Precision Arithmetic Library.  
 END-INFO-DIR-ENTRY  
 @end format  
 @end ifinfo  
   
 @c smallbook  Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002
   Free Software Foundation, Inc.
   
 @iftex  Permission is granted to copy, distribute and/or modify this document under
 @finalout  the terms of the GNU Free Documentation License, Version 1.1 or any later
 @end iftex  version published by the Free Software Foundation; with no Invariant Sections,
   with the Front-Cover Texts being ``A GNU Manual'', and with the Back-Cover
   Texts being ``You have freedom to copy and modify this GNU Manual, like GNU
   software''.  A copy of the license is included in @ref{GNU Free Documentation
   License}.
   @end copying
   
 @c Note: the edition number is listed in *three* places; please update  
 @c all three.  Also, update the month and year where appropriate.  
   
 @c ==> Update edition number for settitle and subtitle, and in the  @c  Texinfo version 4.2 or up will be needed to process this into .info files.
 @c ==> following paragraph; update date, too.  @c
   @c  The supplied texinfo.tex (or newer) should be used when processing into
   @c  .dvi etc.
   @c
   @c  The version number and edition number are taken from version.texi provided
   @c  by automake (note it's regenerated only if you configure with
   @c  --enable-maintainer-mode).
   @c
   @c  Discussions about this version in relation to previous ones (for instance
   @c  in the "Compatibility" section) obviously must be looked at manually
   @c  though.
   @c
   @c  "cindex" entries have been made for function categories and programming
   @c  topics.  Minutiae like particular systems and processors mentioned in
   @c  various places have been left out so as not to bury important topics under
   @c  a lot of junk.  "mpn" functions aren't in the concept index because a
   @c  beginner looking for "GCD" or something is only going to be confused by
   @c  pointers to low level routines.
   
   
 @ifinfo  @dircategory GNU libraries
 This file documents GNU MP, a library for arbitrary-precision arithmetic.  @direntry
   * gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
   @end direntry
   
 Copyright (C) 1991, 1993, 1994, 1995, 1996 Free Software Foundation, Inc.  @c  html <meta name=description content="...">
   @documentdescription
   How to install and use the GNU multiple precision arithmetic library, version @value{VERSION}.
   @end documentdescription
   
 Permission is granted to make and distribute verbatim copies of  @c smallbook
 this manual provided the copyright notice and this permission notice  @finalout
 are preserved on all copies.  @setchapternewpage on
   
 @ignore  @ifnottex
 Permission is granted to process this file through TeX and print the  @node Top, Copying, (dir), (dir)
 results, provided the printed document carries copying permission  @top GNU MP
 notice identical to this one except for the removal of this paragraph  @end ifnottex
 (this paragraph not being relevant to the printed manual).  
   
 @end ignore  @iftex
 Permission is granted to copy and distribute modified versions of this  
 manual under the conditions for verbatim copying, provided that the entire  
 resulting derived work is distributed under the terms of a permission  
 notice identical to this one.  
   
 Permission is granted to copy and distribute translations of this manual  
 into another language, under the above conditions for modified versions,  
 except that this permission notice may be stated in a translation approved  
 by the Foundation.  
 @end ifinfo  
   
 @setchapternewpage on  
 @titlepage  @titlepage
 @c  use the new format for titles  
   
 @title GNU MP  @title GNU MP
 @subtitle The GNU Multiple Precision Arithmetic Library  @subtitle The GNU Multiple Precision Arithmetic Library
 @subtitle Edition 2.0.2  @subtitle Edition @value{EDITION}
 @subtitle June 1996  @subtitle @value{UPDATED}
   
 @author by Torbj@"orn Granlund, TMG Datakonsult  @author by Torbj@"orn Granlund, Swox AB
   @email{tege@@swox.com}
   
 @c Include the Distribution inside the titlepage so  @c Include the Distribution inside the titlepage so
 @c that headings are turned off.  @c that headings are turned off.
Line 78  by the Foundation.
Line 87  by the Foundation.
   
 @page  @page
 @vskip 0pt plus 1filll  @vskip 0pt plus 1filll
 Copyright @copyright{} 1991, 1993, 1994, 1995, 1996 Free Software Foundation, Inc.  @end iftex
   
 @sp 2  @insertcopying
   @ifnottex
   @sp 1
   @end ifnottex
   
 Published by the Free Software Foundation @*  @iftex
 59 Temple Place - Suite 330 @*  @end titlepage
 Boston, MA 02111-1307, USA @*  @headings double
   @end iftex
   
 Permission is granted to make and distribute verbatim copies of  @c  Don't bother with contents for html, the menus seem adequate.
 this manual provided the copyright notice and this permission notice  @ifnothtml
 are preserved on all copies.  @contents
   @end ifnothtml
   
 Permission is granted to copy and distribute modified versions of this  @menu
 manual under the conditions for verbatim copying, provided that the entire  * Copying::                    GMP Copying Conditions (LGPL).
 resulting derived work is distributed under the terms of a permission  * Introduction to GMP::        Brief introduction to GNU MP.
 notice identical to this one.  * Installing GMP::             How to configure and compile the GMP library.
   * GMP Basics::                 What every GMP user should know.
   * Reporting Bugs::             How to usefully report bugs.
   * Integer Functions::          Functions for arithmetic on signed integers.
   * Rational Number Functions::  Functions for arithmetic on rational numbers.
   * Floating-point Functions::   Functions for arithmetic on floats.
   * Low-level Functions::        Fast functions for natural numbers.
   * Random Number Functions::    Functions for generating random numbers.
   * Formatted Output::           @code{printf} style output.
   * Formatted Input::            @code{scanf} style input.
   * C++ Class Interface::        Class wrappers around GMP types.
   * BSD Compatible Functions::   All functions found in BSD MP.
   * Custom Allocation::          How to customize the internal allocation.
   * Language Bindings::          Using GMP from other languages.
   * Algorithms::                 What happens behind the scenes.
   * Internals::                  How values are represented behind the scenes.
   
 Permission is granted to copy and distribute translations of this manual  * Contributors::               Who brings your this library?
 into another language, under the above conditions for modified versions,  * References::                 Some useful papers and books to read.
 except that this permission notice may be stated in a translation approved  * GNU Free Documentation License::
 by the Foundation.  * Concept Index::
 @end titlepage  * Function Index::
 @headings double  @end menu
   
 @ifinfo  
 @node Top, Copying, (dir), (dir)  
   
 @top GNU MP  @c  @m{T,N} is $T$ in tex or @math{N} otherwise.  This is an easy way to give
   @c  different forms for math in tex and info.  Commas in N or T don't work,
   @c  but @C{} can be used instead.  \, works in info but not in tex.
   @iftex
   @macro m {T,N}
   @tex$\T\$@end tex
   @end macro
   @end iftex
   @ifnottex
   @macro m {T,N}
   @math{\N\}
   @end macro
   @end ifnottex
   
 This manual documents how to install and use the GNU multiple precision  @macro C {}
 arithmetic library, version 2.0.2.  ,
   @end macro
   
   @c  @ms{V,N} is $V_N$ in tex or just vn otherwise.  This suits simple
   @c  subscripts like @ms{x,0}.
   @iftex
   @macro ms {V,N}
   @tex$\V\_{\N\}$@end tex
   @end macro
   @end iftex
   @ifnottex
   @macro ms {V,N}
   \V\\N\
   @end macro
   @end ifnottex
   
   @c  @nicode{S} is plain S in info, or @code{S} elsewhere.  This can be used
   @c  when the quotes that @code{} gives in info aren't wanted, but the
   @c  fontification in tex or html is wanted.  Doesn't work as @nicode{'\\0'}
   @c  though (gives two backslashes in tex).
   @ifinfo
   @macro nicode {S}
   \S\
   @end macro
 @end ifinfo  @end ifinfo
   @ifnotinfo
   @macro nicode {S}
   @code{\S\}
   @end macro
   @end ifnotinfo
   
 @menu  @c  @nisamp{S} is plain S in info, or @samp{S} elsewhere.  This can be used
 * Copying::                   GMP Copying Conditions (LGPL).  @c  when the quotes that @samp{} gives in info aren't wanted, but the
 * Introduction to MP::        Brief introduction to GNU MP.  @c  fontification in tex or html is wanted.
 * Installing MP::             How to configure and compile the MP library.  @ifinfo
 * MP Basics::                 What every MP user should now.  @macro nisamp {S}
 * Reporting Bugs::            How to usefully report bugs.  \S\
 * Integer Functions::         Functions for arithmetic on signed integers.  @end macro
 * Rational Number Functions:: Functions for arithmetic on rational numbers.  @end ifinfo
 * Floating-point Functions::  Functions for arithmetic on floats.  @ifnotinfo
 * Low-level Functions::       Fast functions for natural numbers.  @macro nisamp {S}
 * BSD Compatible Functions::  All functions found in BSD MP.  @samp{\S\}
 * Custom Allocation::         How to customize the internal allocation.  @end macro
   @end ifnotinfo
   
 * Contributors::  @c  Usage: @GMPtimes{}
 * References::  @c  Give either \times or the word "times".
 * Concept Index::  @tex
 * Function Index::  \gdef\GMPtimes{\times}
 @end menu  @end tex
   @ifnottex
   @macro GMPtimes
   times
   @end macro
   @end ifnottex
   
 @node Copying, Introduction to MP, Top, Top  @c  Usage: @GMPmultiply{}
   @c  Give * in info, or nothing in tex.
   @tex
   \gdef\GMPmultiply{}
   @end tex
   @ifnottex
   @macro GMPmultiply
   *
   @end macro
   @end ifnottex
   
   @c  Usage: @GMPabs{x}
   @c  Give either |x| in tex, or abs(x) in info or html.
   @tex
   \gdef\GMPabs#1{|#1|}
   @end tex
   @ifnottex
   @macro GMPabs {X}
   @abs{}(\X\)
   @end macro
   @end ifnottex
   
   @c  Usage: @GMPfloor{x}
   @c  Give either \lfloor x\rfloor in tex, or floor(x) in info or html.
   @tex
   \gdef\GMPfloor#1{\lfloor #1\rfloor}
   @end tex
   @ifnottex
   @macro GMPfloor {X}
   floor(\X\)
   @end macro
   @end ifnottex
   
   @c  Usage: @GMPceil{x}
   @c  Give either \lceil x\rceil in tex, or ceil(x) in info or html.
   @tex
   \gdef\GMPceil#1{\lceil #1 \rceil}
   @end tex
   @ifnottex
   @macro GMPceil {X}
   ceil(\X\)
   @end macro
   @end ifnottex
   
   @c  Math operators already available in tex, made available in info too.
   @c  For example @bmod{} can be used in both tex and info.
   @ifnottex
   @macro bmod
   mod
   @end macro
   @macro gcd
   gcd
   @end macro
   @macro ge
   >=
   @end macro
   @macro le
   <=
   @end macro
   @macro log
   log
   @end macro
   @macro min
   min
   @end macro
   @macro rightarrow
   ->
   @end macro
   @end ifnottex
   
   @c  New math operators.
   @c  @abs{} can be used in both tex and info, or just \abs in tex.
   @tex
   \gdef\abs{\mathop{\rm abs}}
   @end tex
   @ifnottex
   @macro abs
   abs
   @end macro
   @end ifnottex
   
   @c  @cross{} is a \times symbol in tex, or an "x" in info.  In tex it works
   @c  inside or outside $ $.
   @tex
   \gdef\cross{\ifmmode\times\else$\times$\fi}
   @end tex
   @ifnottex
   @macro cross
   x
   @end macro
   @end ifnottex
   
   @c  @times{} made available as a "*" in info and html (already works in tex).
   @ifnottex
   @macro times
   *
   @end macro
   @end ifnottex
   
   @c  Usage: @W{text}
   @c  Like @w{} but working in math mode too.
   @tex
   \gdef\W#1{\ifmmode{#1}\else\w{#1}\fi}
   @end tex
   @ifnottex
   @macro W {S}
   @w{\S\}
   @end macro
   @end ifnottex
   
   @c  Usage: \GMPdisplay{text}
   @c  Put the given text in an @display style indent, but without turning off
   @c  paragraph reflow etc.
   @tex
   \gdef\GMPdisplay#1{%
   \noindent
   \advance\leftskip by \lispnarrowing
   #1\par}
   @end tex
   
   @c  Usage: \GMPhat
   @c  A new \hat that will work in math mode, unlike the texinfo redefined
   @c  version.
   @tex
   \gdef\GMPhat{\mathaccent"705E}
   @end tex
   
   @c  Usage: \GMPraise{text}
   @c  For use in a $ $ math expression as an alternative to "^".  This is good
   @c  for @code{} in an exponent, since there seems to be no superscript font
   @c  for that.
   @tex
   \gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}}
   @end tex
   
   @c  Usage: @texlinebreak{}
   @c  A line break as per @*, but only in tex.
   @iftex
   @macro texlinebreak
   @*
   @end macro
   @end iftex
   @ifnottex
   @macro texlinebreak
   @end macro
   @end ifnottex
   
   @c  Usage: @maybepagebreak
   @c  Allow tex to insert a page break, if it feels the urge.
   @c  Normally blocks of @deftypefun/funx are kept together, which can lead to
   @c  some poor page break positioning if it's a big block, like the sets of
   @c  division functions etc.
   @tex
   \gdef\maybepagebreak{\penalty0}
   @end tex
   @ifnottex
   @macro maybepagebreak
   @end macro
   @end ifnottex
   
   
   @node Copying, Introduction to GMP, Top, Top
 @comment  node-name, next, previous,  up  @comment  node-name, next, previous,  up
 @unnumbered GNU MP Copying Conditions  @unnumbered GNU MP Copying Conditions
 @cindex Copying conditions  @cindex Copying conditions
 @cindex Conditions for copying GNU MP  @cindex Conditions for copying GNU MP
   @cindex License conditions
   
 This library is @dfn{free}; this means that everyone is free to use it and  This library is @dfn{free}; this means that everyone is free to use it and
 free to redistribute it on a free basis.  The library is not in the public  free to redistribute it on a free basis.  The library is not in the public
Line 163  have is not what we distributed, so that any problems 
Line 397  have is not what we distributed, so that any problems 
 will not reflect on our reputation.@refill  will not reflect on our reputation.@refill
   
 The precise conditions of the license for the GNU MP library are found in the  The precise conditions of the license for the GNU MP library are found in the
 Library General Public License that accompany the source code.@refill  Lesser General Public License version 2.1 that accompanies the source code,
   see @file{COPYING.LIB}.  Certain demonstration programs are provided under the
   terms of the plain General Public License version 2, see @file{COPYING}.
   
 @node Introduction to MP, Installing MP, Copying, Top  
   @node Introduction to GMP, Installing GMP, Copying, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Introduction to GNU MP  @chapter Introduction to GNU MP
   @cindex Introduction
   
   
 GNU MP is a portable library written in C for arbitrary precision arithmetic  GNU MP is a portable library written in C for arbitrary precision arithmetic
 on integers, rational numbers, and floating-point numbers.  It aims to provide  on integers, rational numbers, and floating-point numbers.  It aims to provide
 the fastest possible arithmetic for all applications that need higher  the fastest possible arithmetic for all applications that need higher
 precision than is directly supported by the basic C types.  precision than is directly supported by the basic C types.
   
 Many applications use just a few hundred bits of precision; but some  Many applications use just a few hundred bits of precision; but some
 applications may need thousands or even millions of bits.  MP is designed to  applications may need thousands or even millions of bits.  GMP is designed to
 give good performance for both, by choosing algorithms based on the sizes of  give good performance for both, by choosing algorithms based on the sizes of
 the operands, and by carefully keeping the overhead at a minimum.  the operands, and by carefully keeping the overhead at a minimum.
   
 The speed of MP is achieved by using fullwords as the basic arithmetic type,  The speed of GMP is achieved by using fullwords as the basic arithmetic type,
 by using sophisticated algorithms, by including carefully optimized assembly  by using sophisticated algorithms, by including carefully optimized assembly
 code for the most common inner loops for many different CPUs, and by a general  code for the most common inner loops for many different CPUs, and by a general
 emphasis on speed (as opposed to simplicity or elegance).  emphasis on speed (as opposed to simplicity or elegance).
   
 There is carefully optimized assembly code for these CPUs: DEC Alpha, Amd  There is carefully optimized assembly code for these CPUs:
 29000, HPPA 1.0 and 1.1, Intel Pentium and generic x86, Intel i960, Motorola  @cindex CPUs supported
 MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC, National  ARM,
 NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8,  DEC Alpha 21064, 21164, and 21264,
 and DEC VAX.  Some optimizations also for ARM, Clipper, IBM ROMP (RT), and  AMD 29000,
   AMD K6, K6-2 and Athlon,
   Hitachi SuperH and SH-2,
   HPPA 1.0, 1.1 and 2.0,
   Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86,
   Intel IA-64, i960,
   Motorola MC68000, MC68020, MC88100, and MC88110,
   Motorola/IBM PowerPC 32 and 64,
   National NS32000,
   IBM POWER,
   MIPS R3000, R4000,
   SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC,
   DEC VAX,
   and
   Zilog Z8000.
   Some optimizations also for
   Cray vector systems,
   Clipper,
   IBM ROMP (RT),
   and
 Pyramid AP/XP.  Pyramid AP/XP.
   
 This version of MP is released under a more liberal license than previous  @cindex Mailing lists
 versions.  It is now permitted to link MP to non-free programs, as long as MP  There are two public mailing lists of interest.  One for general questions and
 source code is provided when distributing the non-free program.  discussions about usage of the GMP library and one for discussions about
   development of GMP.  There's more information about the mailing lists at
   @uref{http://swox.com/mailman/listinfo/}.  These lists are @strong{not} for
   bug reports.
   
   The proper place for bug reports is @email{bug-gmp@@gnu.org}.  See
   @ref{Reporting Bugs} for info about reporting bugs.
   
   @cindex Home page
   @cindex Web page
   For up-to-date information on GMP, please see the GMP web pages at
   
   @display
   @uref{http://swox.com/gmp/}
   @end display
   
   @cindex Latest version of GMP
   @cindex Anonymous FTP of latest version
   @cindex FTP of latest version
   The latest version of the library is available at
   
   @display
   @uref{ftp://ftp.gnu.org/gnu/gmp}
   @end display
   
   Many sites around the world mirror @samp{ftp.gnu.org}, please use a mirror
   near you, see @uref{http://www.gnu.org/order/ftp.html} for a full list.
   
   
 @section How to use this Manual  @section How to use this Manual
   @cindex About this manual
   
 Everyone should read @ref{MP Basics}.  If you need to install the library  Everyone should read @ref{GMP Basics}.  If you need to install the library
 yourself, you need to read @ref{Installing MP}, too.  yourself, then read @ref{Installing GMP}.  If you have a system with multiple
   ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used
   on applications.
   
 The rest of the manual can be used for later reference, although it is  The rest of the manual can be used for later reference, although it is
 probably a good idea to glance through it.  probably a good idea to glance through it.
   
   
 @node Installing MP, MP Basics, Introduction to MP, Top  @node Installing GMP, GMP Basics, Introduction to GMP, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Installing MP  @chapter Installing GMP
 @cindex Installation  @cindex Installing GMP
   @cindex Configuring GMP
   @cindex Building GMP
   
 To build MP, you first have to configure it for your CPU and operating system.  GMP has an autoconf/automake/libtool based configuration system.  On a
 You need a C compiler, preferably GCC, but any reasonable compiler should  Unix-like system a basic build can be done with
 work.  And you need a standard Unix @samp{make} program, plus some other  
 standard Unix utility programs.  
   
 (If you're on an MS-DOS machine, your can build MP using @file{make.bat}.  It  @example
 requires that djgpp is installed.  It does not require configuration, nor is  ./configure
 @samp{make} needed; @file{make.bat} both configures and builds the library.)  make
   @end example
   
 Here are the steps needed to install the library on Unix systems:  @noindent
   Some self-tests can be run with
   
 @enumerate  @example
 @item  make check
 In most cases, @samp{./configure --target=cpu-vendor-os}, should work both for  @end example
 native and cross-compilation.  If you get error messages, your machine might  
 not be supported.  
   
 If you want to compile in a separate object directory, cd to that directory,  @noindent
 and prefix the configure command with the path to the MP source directory.  And you can install (under @file{/usr/local} by default) with
 Not all @samp{make} programs have the necessary features to support this.  In  
 particular, SunOS and Slowaris @samp{make} have bugs that makes them unable to  
 build from a separate object directory.  Use GNU @samp{make} instead.  
   
 In addition to the standard cpu-vendor-os tuples, MP recognizes sparc8 and  @example
 supersparc as valid CPU names.  Specifying these CPU names for relevant  make install
 systems will improve performance significantly.  @end example
   
 In general, if you want a library that runs as fast as possible, you should  If you experience problems, please report them to @email{bug-gmp@@gnu.org}.
 make sure you configure MP for the exact CPU type your system uses.  See @ref{Reporting Bugs}, for information on what to include in useful bug
   reports.
   
 If you have @code{gcc} in your @code{PATH}, it will be used by default.  To  @menu
 override this, pass @samp{-with-gcc=no} to @file{configure}.  * Build Options::
   * ABI and ISA::
   * Notes for Package Builds::
   * Notes for Particular Systems::
   * Known Build Problems::
   @end menu
   
 @item  
 @samp{make}  
   
 This will compile MP, and create a library archive file @file{libgmp.a} in the  @node Build Options, ABI and ISA, Installing GMP, Installing GMP
 working directory.  @section Build Options
   @cindex Build options
   
 @item  All the usual autoconf configure options are available, run @samp{./configure
 @samp{make check}  --help} for a summary.  The file @file{INSTALL.autoconf} has some generic
   installation information too.
   
 This will make sure MP was built correctly.  If you get error messages, please  @table @asis
 report this to @samp{bug-gmp@@prep.ai.mit.edu}.  (@xref{Reporting Bugs}, for  @item Non-Unix Systems
 information on what to include in useful bug reports.)  
   
 @item  @samp{configure} requires various Unix-like tools.  On an MS-DOS system DJGPP
 @samp{make install}  can be used, and on MS Windows Cygwin or MINGW can be used,
   
 This will copy the file @file{gmp.h} and @file{libgmp.a}, as well as the info  @display
 files, to @file{/usr/local} (or if you passed the @samp{--prefix} option to  @uref{http://www.cygnus.com/cygwin}
 @file{configure}, to the directory given as argument to @samp{--prefix}).  @uref{http://www.delorie.com/djgpp}
 @end enumerate  @uref{http://www.mingw.org}
   @end display
   
 @noindent  Microsoft also publishes an Interix ``Services for Unix'' which can be used to
 If you wish to build and install the BSD MP compatible functions, use  build GMP on Windows (with a normal @samp{./configure}), but it's not free
 @samp{make libmp.a} and @samp{make install-bsdmp}.  software.
   
 There are some other useful make targets:  The @file{macos} directory contains an unsupported port to MacOS 9 on Power
   Macintosh, see @file{macos/README}.  Note that MacOS X ``Darwin'' should use
   the normal Unix-style @samp{./configure}.
   
   It might be possible to build without the help of @samp{configure}, certainly
   all the code is there, but unfortunately you'll be on your own.
   
   @item Build Directory
   
   To compile in a separate build directory, @command{cd} to that directory, and
   prefix the configure command with the path to the GMP source directory.  For
   example
   
   @example
   cd /my/build/dir
   /my/sources/gmp-@value{VERSION}/configure
   @end example
   
   Not all @samp{make} programs have the necessary features (@code{VPATH}) to
   support this.  In particular, SunOS and Slowaris @command{make} have bugs that
   make them unable to build in a separate directory.  Use GNU @command{make}
   instead.
   
   @item @option{--disable-shared}, @option{--disable-static}
   
   By default both shared and static libraries are built (where possible), but
   one or other can be disabled.  Shared libraries result in smaller executables
   and permit code sharing between separate running processes, but on some CPUs
   are slightly slower, having a small cost on each function call.
   
   @item Native Compilation, @option{--build=CPU-VENDOR-OS}
   
   For normal native compilation, the system can be specified with
   @samp{--build}.  By default @samp{./configure} uses the output from running
   @samp{./config.guess}.  On some systems @samp{./config.guess} can determine
   the exact CPU type, on others it will be necessary to give it explicitly.  For
   example,
   
   @example
   ./configure --build=ultrasparc-sun-solaris2.7
   @end example
   
   In all cases the @samp{OS} part is important, since it controls how libtool
   generates shared libraries.  Running @samp{./config.guess} is the simplest way
   to see what it should be, if you don't know already.
   
   @item Cross Compilation, @option{--host=CPU-VENDOR-OS}
   
   When cross-compiling, the system used for compiling is given by @samp{--build}
   and the system where the library will run is given by @samp{--host}.  For
   example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,
   
   @example
   ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
   @end example
   
   Compiler tools are sought first with the host system type as a prefix.  For
   example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain
   @command{ranlib}.  This makes it possible for a set of cross-compiling tools
   to co-exist with native tools.  The prefix is the argument to @samp{--host},
   and this can be an alias, such as @samp{m68k-linux}.  But note that tools
   don't have to be setup this way, it's enough to just have a @env{PATH} with a
   suitable cross-compiling @command{cc} etc.
   
   Compiling for a different CPU in the same family as the build system is a form
   of cross-compilation, though very possibly this would merely be special
   options on a native compiler.  In any case @samp{./configure} avoids depending
   on being able to run code on the build system, which is important when
   creating binaries for a newer CPU since they very possibly won't run on the
   build system.
   
   In all cases the compiler must be able to produce an executable (of whatever
   format) from a standard C @code{main}.  Although only object files will go to
   make up @file{libgmp}, @samp{./configure} uses linking tests for various
   purposes, such as determining what functions are available on the host system.
   
   Currently a warning is given unless an explicit @samp{--build} is used when
   cross-compiling, because it may not be possible to correctly guess the build
   system type if the @env{PATH} has only a cross-compiling @command{cc}.
   
   Note that the @samp{--target} option is not appropriate for GMP.  It's for use
   when building compiler tools, with @samp{--host} being where they will run,
   and @samp{--target} what they'll produce code for.  Ordinary programs or
   libraries like GMP are only interested in the @samp{--host} part, being where
   they'll run.  (Some past versions of GMP used @samp{--target} incorrectly.)
   
   @item CPU types
   
   In general, if you want a library that runs as fast as possible, you should
   configure GMP for the exact CPU type your system uses.  However, this may mean
   the binaries won't run on older members of the family, and might run slower on
   other members, older or newer.  The best idea is always to build GMP for the
   exact machine type you intend to run it on.
   
   The following CPUs have specific support.  See @file{configure.in} for details
   of what code and compiler options they select.
   
 @itemize @bullet  @itemize @bullet
   
   @c Keep this formatting, it's easy to read and it can be grepped to
   @c automatically test that CPUs listed get through ./config.sub
   
 @item  @item
 @samp{doc}  Alpha:
   @nisamp{alpha},
   @nisamp{alphaev5},
   @nisamp{alphaev56},
   @nisamp{alphapca56},
   @nisamp{alphapca57},
   @nisamp{alphaev6},
   @nisamp{alphaev67},
   @nisamp{alphaev68}
   
 Create a DVI version of the manual, in @file{gmp.dvi} and a set of info files,  @item
 in @file{gmp.info}, @file{gmp.info-1}, @file{gmp.info-2}, etc.  Cray:
   @nisamp{c90},
   @nisamp{j90},
   @nisamp{t90},
   @nisamp{sv1}
   
 @item  @item
 @samp{ps}  HPPA:
   @nisamp{hppa1.0},
   @nisamp{hppa1.1},
   @nisamp{hppa2.0},
   @nisamp{hppa2.0n},
   @nisamp{hppa2.0w}
   
 Create a Postscript version of the manual, in @file{gmp.ps}.  @item
   MIPS:
   @nisamp{mips},
   @nisamp{mips3},
   @nisamp{mips64}
   
 @item  @item
 @samp{html}  Motorola:
   @nisamp{m68k},
   @nisamp{m68000},
   @nisamp{m68010},
   @nisamp{m68020},
   @nisamp{m68030},
   @nisamp{m68040},
   @nisamp{m68060},
   @nisamp{m68302},
   @nisamp{m68360},
   @nisamp{m88k},
   @nisamp{m88110}
   
 Create a HTML version of the manual, in @file{gmp.html}.  @item
   POWER:
   @nisamp{power},
   @nisamp{power1},
   @nisamp{power2},
   @nisamp{power2sc}
   
 @item  @item
 @samp{clean}  PowerPC:
   @nisamp{powerpc},
   @nisamp{powerpc64},
   @nisamp{powerpc401},
   @nisamp{powerpc403},
   @nisamp{powerpc405},
   @nisamp{powerpc505},
   @nisamp{powerpc601},
   @nisamp{powerpc602},
   @nisamp{powerpc603},
   @nisamp{powerpc603e},
   @nisamp{powerpc604},
   @nisamp{powerpc604e},
   @nisamp{powerpc620},
   @nisamp{powerpc630},
   @nisamp{powerpc740},
   @nisamp{powerpc7400},
   @nisamp{powerpc7450},
   @nisamp{powerpc750},
   @nisamp{powerpc801},
   @nisamp{powerpc821},
   @nisamp{powerpc823},
   @nisamp{powerpc860},
   
 Delete all object files and archive files, but not the configuration files.  @item
   SPARC:
   @nisamp{sparc},
   @nisamp{sparcv8},
   @nisamp{microsparc},
   @nisamp{supersparc},
   @nisamp{sparcv9},
   @nisamp{ultrasparc},
   @nisamp{ultrasparc2},
   @nisamp{ultrasparc2i},
   @nisamp{ultrasparc3},
   @nisamp{sparc64}
   
 @item  @item
 @samp{distclean}  80x86 family:
   @nisamp{i386},
   @nisamp{i486},
   @nisamp{i586},
   @nisamp{pentium},
   @nisamp{pentiummmx},
   @nisamp{pentiumpro},
   @nisamp{pentium2},
   @nisamp{pentium3},
   @nisamp{pentium4},
   @nisamp{k6},
   @nisamp{k62},
   @nisamp{k63},
   @nisamp{athlon}
   
 Delete all files not included in the distribution.  @item
   Other:
   @nisamp{a29k},
   @nisamp{arm},
   @nisamp{clipper},
   @nisamp{i960},
   @nisamp{ns32k},
   @nisamp{pyramid},
   @nisamp{sh},
   @nisamp{sh2},
   @nisamp{vax},
   @nisamp{z8k}
   @end itemize
   
   CPUs not listed will use generic C code.
   
   @item Generic C Build
   
   If some of the assembly code causes problems, or if otherwise desired, the
   generic C code can be selected with CPU @samp{none}.  For example,
   
   @example
   ./configure --host=none-unknown-freebsd3.5
   @end example
   
   Note that this will run quite slowly, but it should be portable and should at
   least make it possible to get something running if all else fails.
   
   @item @option{ABI}
   
   On some systems GMP supports multiple ABIs (application binary interfaces),
   meaning data type sizes and calling conventions.  By default GMP chooses the
   best ABI available, but a particular ABI can be selected.  For example
   
   @example
   ./configure --host=mips64-sgi-irix6 ABI=n32
   @end example
   
   See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what
   applications need to do.
   
   @item @option{CC}, @option{CFLAGS}
   
   By default the C compiler used is chosen from among some likely candidates,
   with @command{gcc} normally preferred if it's present.  The usual
   @samp{CC=whatever} can be passed to @samp{./configure} to choose something
   different.
   
   For some systems, default compiler flags are set based on the CPU and
   compiler.  The usual @samp{CFLAGS="-whatever"} can be passed to
   @samp{./configure} to use something different or to set good flags for systems
   GMP doesn't otherwise know.
   
   The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure},
   and can be found in each generated @file{Makefile}.  This is the easiest way
   to check the defaults when considering changing or adding something.
   
   Note that when @samp{CC} and @samp{CFLAGS} are specified on a system
   supporting multiple ABIs it's important to give an explicit
   @samp{ABI=whatever}, since GMP can't determine the ABI just from the flags and
   won't be able to select the correct assembler code.
   
   If just @samp{CC} is selected then normal default @samp{CFLAGS} for that
   compiler will be used (if GMP recognises it).  For example @samp{CC=gcc} can
   be used to force the use of GCC, with default flags (and default ABI).
   
   @item @option{CPPFLAGS}
   
   Any flags like @samp{-D} defines or @samp{-I} includes required by the
   preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}.
   Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but
   preprocessing uses just @samp{CPPFLAGS}.  This distinction is because most
   preprocessors won't accept all the flags the compiler does.  Preprocessing is
   done separately in some configure tests, and in the @samp{ansi2knr} support
   for K&R compilers.
   
   @item C++ Support, @option{--enable-cxx}
   C++ support in GMP can be enabled with @samp{--enable-cxx}, in which case a
   C++ compiler will be required.  As a convenience @samp{--enable-cxx=detect}
   can be used to enable C++ support only if a compiler can be found.  The C++
   support consists of a library @file{libgmpxx.la} and header file
   @file{gmpxx.h}.
   
   A separate @file{libgmpxx.la} has been adopted rather than having C++ objects
   within @file{libgmp.la} in order to ensure dynamic linked C programs aren't
   bloated by a dependency on the C++ standard library, and to avoid any chance
   that the C++ compiler could be required when linking plain C programs.
   
   @file{libgmpxx.la} will use certain internals from @file{libgmp.la} and can
   only be expected to work with @file{libgmp.la} from the same GMP version.
   Future changes to the relevant internals will be accompanied by renaming, so a
   mismatch will cause unresolved symbols rather than perhaps mysterious
   misbehaviour.
   
   In general @file{libgmpxx.la} will be usable only with the C++ compiler that
   built it, since name mangling and runtime support are usually incompatible
   between different compilers.
   
   @item @option{CXX}, @option{CXXFLAGS}
   When C++ support is enabled, the C++ compiler and its flags can be set with
   variables @samp{CXX} and @samp{CXXFLAGS} in the usual way.  The default for
   @samp{CXX} is the first compiler that works from a list of likely candidates,
   with @command{g++} normally preferred when available.  The default for
   @samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then
   for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers
   @samp{-g} or nothing.  Trying @samp{CFLAGS} this way is convenient when using
   @samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will
   usually suit @samp{g++}.
   
   It's important that the C and C++ compilers match, meaning their startup and
   runtime support routines are compatible and that they generate code in the
   same ABI (if there's a choice of ABIs on the system).  @samp{./configure}
   isn't currently able to check these things very well itself, so for that
   reason @samp{--disable-cxx} is the default, to avoid a build failure due to a
   compiler mismatch.  Perhaps this will change in the future.
   
   Incidentally, it's normally not good enough to set @samp{CXX} to the same as
   @samp{CC}.  Although @command{gcc} for instance recognises @file{foo.cc} as
   C++ code, only @command{g++} will invoke the linker the right way when
   building an executable or shared library from object files.
   
   @item Temporary Memory, @option{--enable-alloca=<choice>}
   @cindex Stack overflow segfaults
   @cindex @code{alloca}
   
   GMP allocates temporary workspace using one of the following three methods,
   which can be selected with for instance
   @samp{--enable-alloca=malloc-reentrant}.
   
   @itemize @bullet
 @item  @item
 @samp{uninstall}  @samp{alloca} - C library or compiler builtin.
   @item
   @samp{malloc-reentrant} - the heap, in a re-entrant fashion.
   @item
   @samp{malloc-notreentrant} - the heap, with global variables.
   @end itemize
   
 Delete all files copied by @samp{make install}.  For convenience, the following choices are also available.
   @samp{--disable-alloca} is the same as @samp{--enable-alloca=no}.
   
   @itemize @bullet
   @item
   @samp{yes} - a synonym for @samp{alloca}.
   @item
   @samp{no} - a synonym for @samp{malloc-reentrant}.
   @item
   @samp{reentrant} - @code{alloca} if available, otherwise
   @samp{malloc-reentrant}.  This is the default.
   @item
   @samp{notreentrant} - @code{alloca} if available, otherwise
   @samp{malloc-notreentrant}.
 @end itemize  @end itemize
   
   @code{alloca} is reentrant and fast, and is recommended, but when working with
   large numbers it can overflow the available stack space, in which case one of
   the two malloc methods will need to be used.  Alternately it might be possible
   to increase available stack with @command{limit}, @command{ulimit} or
   @code{setrlimit}, or under DJGPP with @command{stubedit} or
   @code{@w{_stklen}}.  Note that depending on the system the only indication of
   stack overflow might be a segmentation violation.
   
   @samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe,
   but @samp{malloc-notreentrant} is faster and should be used if reentrancy is
   not required.
   
   The two malloc methods in fact use the memory allocation functions selected by
   @code{mp_set_memory_functions}, these being @code{malloc} and friends by
   default.  @xref{Custom Allocation}.
   
   An additional choice @samp{--enable-alloca=debug} is available, to help when
   debugging memory related problems (@pxref{Debugging}).
   
   @item FFT Multiplication, @option{--disable-fft}
   
   By default multiplications are done using Karatsuba, 3-way Toom-Cook, and
   Fermat FFT.  The FFT is only used on large to very large operands and can be
   disabled to save code size if desired.
   
   @item Berkeley MP, @option{--enable-mpbsd}
   
   The Berkeley MP compatibility library (@file{libmp}) and header file
   (@file{mp.h}) are built and installed only if @option{--enable-mpbsd} is used.
   @xref{BSD Compatible Functions}.
   
   @item MPFR, @option{--enable-mpfr}
   @cindex MPFR
   
   The optional MPFR functions are built and installed only if
   @option{--enable-mpfr} is used.  These are in a separate library
   @file{libmpfr.a} and are documented separately too (@pxref{Introduction to
   MPFR,, Introduction to MPFR, mpfr, MPFR}).
   
   @item Assertion Checking, @option{--enable-assert}
   
   This option enables some consistency checking within the library.  This can be
   of use while debugging, @pxref{Debugging}.
   
   @item Execution Profiling, @option{--enable-profiling=prof/gprof}
   
   Profiling support can be enabled either for @command{prof} or @command{gprof}.
   This adds @samp{-p} or @samp{-pg} respectively to @samp{CFLAGS}, and for some
   systems adds corresponding @code{mcount} calls to the assembler code.
   @xref{Profiling}.
   
   @item @option{MPN_PATH}
   
   Various assembler versions of each mpn subroutines are provided.  For a given
   CPU, a search is made though a path to choose a version of each.  For example
   @samp{sparcv8} has
   
   @example
   MPN_PATH="sparc32/v8 sparc32 generic"
   @end example
   
   which means look first for v8 code, then plain sparc32 (which is v7), and
   finally fall back on generic C.  Knowledgeable users with special requirements
   can specify a different path.  Normally this is completely unnecessary.
   
   @item Documentation
   
   The document you're now reading is @file{gmp.texi}.  The usual automake
   targets are available to make PostScript @file{gmp.ps} and/or DVI
   @file{gmp.dvi}.
   
   HTML can be produced with @samp{makeinfo --html}, see @ref{makeinfo
   html,Generating HTML,Generating HTML,texinfo,Texinfo}.  Or alternately
   @samp{texi2html}, see @ref{Top,Texinfo to HTML,About,texi2html,Texinfo To
   HTML}.
   
   PDF can be produced with @samp{texi2dvi --pdf} (@pxref{PDF
   Output,PDF,,texinfo,Texinfo}) or with @samp{pdftex}.
   
   Some supplementary notes can be found in the @file{doc} subdirectory.
   
   @end table
   
   
   @need 2000
   @node ABI and ISA, Notes for Package Builds, Build Options, Installing GMP
   @section ABI and ISA
   @cindex ABI
   @cindex Application Binary Interface
   @cindex ISA
   @cindex Instruction Set Architecture
   
   ABI (Application Binary Interface) refers to the calling conventions between
   functions, meaning what registers are used and what sizes the various C data
   types are.  ISA (Instruction Set Architecture) refers to the instructions and
   registers a CPU has available.
   
   Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
   latter for compatibility with older CPUs in the family.  GMP supports some
   CPUs like this in both ABIs.  In fact within GMP @samp{ABI} means a
   combination of chip ABI, plus how GMP chooses to use it.  For example in some
   32-bit ABIs, GMP may support a limb as either a 32-bit @code{long} or a 64-bit
   @code{long long}.
   
   By default GMP chooses the best ABI available for a given system, and this
   generally gives significantly greater speed.  But an ABI can be chosen
   explicitly to make GMP compatible with other libraries, or particular
   application requirements.  For example,
   
   @example
   ./configure ABI=32
   @end example
   
   In all cases it's vital that all object code used in a given program is
   compiled for the same ABI.
   
   Usually a limb is implemented as a @code{long}.  When a @code{long long} limb
   is used this is encoded in the generated @file{gmp.h}.  This is convenient for
   applications, but it does mean that @file{gmp.h} will vary, and can't be just
   copied around.  @file{gmp.h} remains compiler independent though, since all
   compilers for a particular ABI will be expected to use the same limb type.
   
   Currently no attempt is made to follow whatever conventions a system has for
   installing library or header files built for a particular ABI.  This will
   probably only matter when installing multiple builds of GMP, and it might be
   as simple as configuring with a special @samp{libdir}, or it might require
   more than that.  Note that builds for different ABIs need to done separately,
   with a fresh @command{./configure} and @command{make} each.
   
   @table @asis
   @sp 1
   @need 1000
   @item HPPA 2.0 (@samp{hppa2.0*})
   
   @table @asis
   @item @samp{ABI=2.0w}
   
   The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up
   when using @command{cc}.  @command{gcc} support for this is in progress.
   Applications must be compiled with
   
   @example
   cc  +DD64
   @end example
   
   @item @samp{ABI=2.0n}
   
   The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit limb using
   @code{long long}.  This is available on HP-UX 10 or up when using
   @command{cc}.  No @command{gcc} support is planned for this.  Applications
   must be compiled with
   
   @example
   cc  +DA2.0 +e
   @end example
   
   @item @samp{ABI=1.0}
   
   HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI.
   No special compiler options are needed for applications.
   @end table
   
   All three ABIs are available for CPUs @samp{hppa2.0w} and @samp{hppa2.0}, but
   for CPU @samp{hppa2.0n} only 2.0n or 1.0 are allowed.
   
   @sp 1
   @need 1000
   @item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]})
   
   IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3 or better
   CPU.  In both these ABIs GMP uses a 64-bit limb.  A new enough @command{gcc}
   is required (2.95 for instance).
   
   @table @asis
   @item @samp{ABI=n32}
   
   The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a
   @code{long long}.  Applications must be compiled with
   
   @example
   gcc  -mabi=n32
   cc   -n32
   @end example
   
   @item @samp{ABI=64}
   
   The 64-bit ABI is 64-bit pointers and integers.  Applications must be compiled
   with
   
   @example
   gcc  -mabi=64
   cc   -64
   @end example
   @end table
   
   Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary
   support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.
   
   @sp 1
   @need 1000
   @item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630})
   
   @table @asis
   @item @samp{ABI=aix64}
   
   The AIX 64 ABI uses 64-bit limbs and pointers and is available on systems
   @samp{*-*-aix*}.  Applications must be compiled (and linked) with
   
   @example
   gcc  -maix64
   xlc  -q64
   @end example
   
   @item @samp{ABI=32}
   
   This is the basic 32-bit PowerPC ABI.  No special compiler options are needed
   for applications.
   @end table
   
   @sp 1
   @need 1000
   @item Sparc V9 (@samp{sparcv9} and @samp{ultrasparc*})
   
   @table @asis
   @item @samp{ABI=64}
   
   The 64-bit V9 ABI is available on Solaris 2.7 and up and GNU/Linux.  GCC 2.95
   or up, or Sun @command{cc} is required.  Applications must be compiled with
   
   @example
   gcc  -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
   cc   -xarch=v9
   @end example
   
   @item @samp{ABI=32}
   
   On Solaris 2.6 and earlier, and on Solaris 2.7 with the kernel in 32-bit mode,
   only the plain V8 32-bit ABI can be used, since the kernel doesn't save all
   registers.  GMP still uses as much of the V9 ISA as it can in these
   circumstances.  No special compiler options are required for applications,
   though using something like the following requesting V9 code within the V8 ABI
   is recommended.
   
   @example
   gcc  -mv8plus
   cc   -xarch=v8plus
   @end example
   
   @command{gcc} 2.8 and earlier only supports @samp{-mv8} though.
   @end table
   
   Don't be confused by the names of these sparc @samp{-m} and @samp{-x} options,
   they're called @samp{arch} but they effectively control the ABI.
   
   On Solaris 2.7 with the kernel in 32-bit-mode, a normal native build will
   reject @samp{ABI=64} because the resulting executables won't run.
   @samp{ABI=64} can still be built if desired by making it look like a
   cross-compile, for example
   
   @example
   ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
   @end example
   @end table
   
   
   @need 2000
   @node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing GMP
   @section Notes for Package Builds
   @cindex Build notes for binary packaging
   @cindex Packaged builds
   
   GMP should present no great difficulties for packaging in a binary
   distribution.
   
   @cindex Libtool versioning
   @cindex Shared library versioning
   Libtool is used to build the library and @samp{-version-info} is set
   appropriately, having started from @samp{3:0:0} in GMP 3.0.  The GMP 4 series
   will be upwardly binary compatible in each release and will be upwardly binary
   compatible with all of the GMP 3 series.  Additional function interfaces may
   be added in each release, so on systems where libtool versioning is not fully
   checked by the loader an auxiliary mechanism may be needed to express that a
   dynamic linked application depends on a new enough GMP.
   
   An auxiliary mechanism may also be needed to express that @file{libgmpxx.la}
   (from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libgmp.la}
   from the same GMP version, since this is not done by the libtool versioning,
   nor otherwise.  A mismatch will result in unresolved symbols from the linker,
   or perhaps the loader.
   
   Using @samp{DESTDIR} or a @samp{prefix} override with @samp{make install} and
   a shared @file{libgmpxx} may run into a libtool relinking problem, see
   @ref{Known Build Problems}.
   
   When building a package for a CPU family, care should be taken to use
   @samp{--host} (or @samp{--build}) to choose the least common denominator among
   the CPUs which might use the package.  For example this might necessitate
   @samp{i386} for x86s, or plain @samp{sparc} (meaning V7) for SPARCs.
   
   Users who care about speed will want GMP built for their exact CPU type, to
   make use of the available optimizations.  Providing a way to suitably rebuild
   a package may be useful.  This could be as simple as making it possible for a
   user to omit @samp{--build} (and @samp{--host}) so @samp{./config.guess} will
   detect the CPU.  But a way to manually specify a @samp{--build} will be wanted
   for systems where @samp{./config.guess} is inexact.
   
   Note that @file{gmp.h} is a generated file, and will be architecture and ABI
   dependent.
   
   
   @need 2000
   @node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing GMP
   @section Notes for Particular Systems
   @cindex Build notes for particular systems
   @cindex Particular systems
   @cindex Systems
   @table @asis
   
   @c This section is more or less meant for notes about performance or about
   @c build problems that have been worked around but might leave a user
   @c scratching their head.  Fun with different ABIs on a system belongs in the
   @c above section.
   
   @item AIX 3 and 4
   
   On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since
   some versions of the native @command{ar} fail on the convenience libraries
   used.  A shared build can be attempted with
   
   @example
   ./configure --enable-shared --disable-static
   @end example
   
   Note that the @samp{--disable-static} is necessary because in a shared build
   libtool makes @file{libgmp.a} a symlink to @file{libgmp.so}, apparently for
   the benefit of old versions of @command{ld} which only recognise @file{.a},
   but unfortunately this is done even if a fully functional @command{ld} is
   available.
   
   @item ARM
   
   On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a
   bug in unsigned division, giving wrong results for some operands.  GMP
   @samp{./configure} will demand GCC 2.95.4 or later.
   
   @item Compaq C++
   Compaq C++ on OSF 5.1 has two flavours of @code{iostream}, a standard one and
   an old pre-standard one (see @samp{man iostream_intro}).  GMP can only use the
   standard one, which unfortunately is not the default but must be selected by
   defining @code{__USE_STD_IOSTREAM}.  Configure with for instance
   
   @example
   ./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
   @end example
   
   @item Microsoft Windows
   On systems @samp{*-*-cygwin*}, @samp{*-*-mingw*} and @samp{*-*-pw32*} by
   default GMP builds only a static library, but a DLL can be built instead using
   
   @example
   ./configure --disable-static --enable-shared
   @end example
   
   Static and DLL libraries can't both be built, since certain export directives
   in @file{gmp.h} must be different.  @samp{--enable-cxx} cannot be used when
   building a DLL, since libtool doesn't currently support C++ DLLs.  This might
   change in the future.
   
   @item Microsoft C
   A MINGW DLL build of GMP can be used with Microsoft C.  Libtool doesn't
   install @file{.lib} and @file{.exp} files, but they can be created with the
   following commands, where @file{/my/inst/dir} is the install directory (with a
   @file{lib} subdirectory).
   
   @example
   lib /machine:IX86 /def:_libs/libgmp-3.dll-def
   cp libgmp-3.lib /my/inst/dir/lib
   cp _libs/libgmp-3.dll-exp /my/inst/dir/lib/libgmp-3.exp
   @end example
   
   MINGW uses @samp{msvcrt.dll} for I/O, so applications wanting to use the GMP
   I/O routines must be compiled with @samp{cl /MD} to do the same.  If one of
   the other I/O choices provided by MS C is desired then the suggestion is to
   use the GMP string functions and confine I/O to the application.
   
   @item Motorola 68k CPU Types
   
   @samp{m68k} is taken to mean 68000.  @samp{m68020} or higher will give a
   performance boost on applicable CPUs.  @samp{m68360} can be used for CPU32
   series chips.  @samp{m68302} can be used for ``Dragonball'' series chips,
   though this is merely a synonym for @samp{m68000}.
   
   @item OpenBSD 2.6
   
   @command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it
   unsuitable for @file{.asm} file processing.  @samp{./configure} will detect
   the problem and either abort or choose another m4 in the @env{PATH}.  The bug
   is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
   
   @item Power CPU Types
   
   In GMP, CPU types @samp{power*} and @samp{powerpc*} will each use instructions
   not available on the other, so it's important to choose the right one for the
   CPU that will be used.  Currently GMP has no assembler code support for using
   just the common instruction subset.  To get executables that run on both, the
   current suggestion is to use the generic C code (CPU @samp{none}), possibly
   with appropriate compiler options (like @samp{-mcpu=common} for
   @command{gcc}).  CPU @samp{rs6000} (which is not a CPU but a family of
   workstations) is accepted by @file{config.sub}, but is currently equivalent to
   @samp{none}.
   
   @item Sparc CPU Types
   
   @samp{sparcv8} or @samp{supersparc} on relevant systems will give a
   significant performance increase over the V7 code.
   
   @item Sparc App Regs
   @cindex Sparc
   The GMP assembler code for both 32-bit and 64-bit Sparc clobbers the
   ``application registers'' @code{g2}, @code{g3} and @code{g4}, the same way
   that the GCC default @samp{-mapp-regs} does (@pxref{SPARC Options,,, gcc,
   Using the GNU Compiler Collection (GCC)}).
   
   This makes that code unsuitable for use with the special V9
   @samp{-mcmodel=embmedany} (which uses @code{g4} as a data segment pointer),
   and for applications wanting to use those registers for special purposes.  In
   these cases the only suggestion currently is to build GMP with CPU @samp{none}
   to avoid the assembler code.
   
   @item SunOS 4
   
   @command{/usr/bin/m4} lacks various features needed to process @file{.asm}
   files, and instead @samp{./configure} will automatically use
   @command{/usr/5bin/m4}, which we believe is always available (if not then use
   GNU m4).
   
   @item x86 CPU Types
   
   @samp{i386} selects generic code which will run reasonably well on all x86
   chips.
   
   @samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for the intended
   P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
   P-III)@.  @samp{i386} is a better choice when making binaries that must run on
   both.
   
   @samp{pentium4} and an SSE2 capable assembler are important for best results
   on Pentium 4.  The specific code is for instance roughly a 2@cross{} to
   3@cross{} speedup over the generic @samp{i386} code.
   
   @item x86 MMX and SSE2 Code
   
   If the CPU selected has MMX code but the assembler doesn't support it, a
   warning is given and non-MMX code is used instead.  This will be an inferior
   build, since the MMX code that's present is there because it's faster than the
   corresponding plain integer code.  The same applies to SSE2.
   
   Old versions of @samp{gas} don't support MMX instructions, in particular
   version 1.92.3 that comes with FreeBSD 2.2.8 doesn't (and unfortunately
   there's no newer assembler for that system).
   
   Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register
   to register @code{movq} instructions, and so can't be used for MMX code.
   Install a recent @command{gas} if MMX code is wanted on these systems.
   @end table
   
   
   @need 2000
   @node Known Build Problems,  , Notes for Particular Systems, Installing GMP
 @section Known Build Problems  @section Known Build Problems
   @cindex Build problems known
   
 GCC 2.7.2 (as well as 2.6.3) for the RS/6000 and PowerPC can not be used to  @c This section is more or less meant for known build problems that are not
 compile MP, due to a bug in GCC.  If you want to use GCC for these machines,  @c otherwise worked around and require some sort of manual intervention.
 you need to apply the patch below to GCC, or use a later version of the  
 compiler.  
   
 If you are on a Sequent Symmetry, use the GNU assembler instead of the  You might find more up-to-date information at @uref{http://swox.com/gmp/}.
 system's assembler, since the latter has serious bugs.  
   
 The system compiler on NeXT is a massacred and old gcc, even if the compiler  @table @asis
 calls itself @file{cc}.  This compiler cannot be used to build MP.  You need  @item Compiler link options
 to get a real gcc, and install that before you compile MP.  (NeXT might have  The version of libtool currently in use rather aggressively strips compiler
 fixed this in newer releases of their system.)  options when linking a shared library.  This will hopefully be relaxed in the
   future, but for now if this is a problem the suggestion is to create a little
   script to hide them, and for instance configure with
   
 The system C compiler under SunOS 4 has a bug that makes it miscompile  @example
 mpq/get_d.c.  This will make @samp{make check} fail.  ./configure CC=gcc-with-my-options
   @end example
   
 Please report other problems to @samp{bug-gmp@@prep.ai.mit.edu}.  @item DJGPP
 @xref{Reporting Bugs}.  The DJGPP port of @command{bash} 2.03 is unable to run the @samp{configure}
   script, it exits silently, having died writing a preamble to
   @file{config.log}.  Use @command{bash} 2.04 or higher.
   
   @samp{make all} was found to run out of memory during the final
   @file{libgmp.la} link on one system tested, despite having 64Mb available.  A
   separate @samp{make libgmp.la} helped, perhaps recursing into the various
   subdirectories uses up memory.
   
 Patch to apply to GCC 2.6.3 and 2.7.2:  @item @samp{DESTDIR} and shared @file{libgmpxx}
   @cindex @samp{DESTDIR}
   @samp{make install DESTDIR=/my/staging/area}, or the same with a @samp{prefix}
   override, to install to a temporary directory is not fully supported by
   current versions of libtool when building a shared version of a library which
   depends on another being built at the same time, like @file{libgmpxx} and
   @file{libgmp}.
   
   The problem is that @file{libgmpxx} is relinked at the install stage to ensure
   that if the system puts a hard-coded path to @file{libgmp} within
   @file{libgmpxx} then that path will be correct.  Naturally the linker is
   directed to look only at the final location, not the staging area, so if
   @file{libgmp} is not already in that final location then the link will fail.
   
   A workaround for this on SVR4 style systems, such as GNU/Linux, where paths
   are not hard-coded, is to include the staging area in the linker's search
   using @code{LD_LIBRARY_PATH}.  For example with @samp{--prefix=/usr} but
   installing under @samp{/my/staging/area},
   
 @example  @example
 *** config/rs6000/rs6000.md     Sun Feb 11 08:22:11 1996  LD_LIBRARY_PATH=/my/staging/area/usr/lib \
 --- config/rs6000/rs6000.md.new Sun Feb 18 03:33:37 1996    make install DESTDIR=/my/staging/area
 ***************  
 *** 920,926 ****  
      (set (match_operand:SI 0 "gpc_reg_operand" "=r")  
         (not:SI (match_dup 1)))]  
     ""  
 !   "nor. %0,%2,%1"  
     [(set_attr "type" "compare")])  
   
   (define_insn ""  
 --- 920,926 ----  
      (set (match_operand:SI 0 "gpc_reg_operand" "=r")  
         (not:SI (match_dup 1)))]  
     ""  
 !   "nor. %0,%1,%1"  
     [(set_attr "type" "compare")])  
   
   (define_insn ""  
 @end example  @end example
   
 @node MP Basics, Reporting Bugs, Installing MP, Top  @item GNU binutils @command{strip} prior to 2.12
   @cindex Stripped libraries
   
   @command{strip} from GNU binutils 2.11 and earlier should not be used on the
   static libraries @file{libgmp.a} and @file{libmp.a} since it will discard all
   but the last of multiple archive members with the same name, like the three
   versions of @file{init.o} in @file{libgmp.a}.  Binutils 2.12 or higher can be
   used successfully.
   
   The shared libraries @file{libgmp.so} and @file{libmp.so} are not affected by
   this and any version of @command{strip} can be used on them.
   
   @item @command{make} syntax error
   
   On certain versions of SCO OpenServer 5 and IRIX 6.5 the native @command{make}
   is unable to handle the long dependencies list for @file{libgmp.la}.  The
   symptom is a ``syntax error'' on the following line of the top-level
   @file{Makefile}.
   
   @example
   libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
   @end example
   
   Either use GNU Make, or as a workaround remove
   @code{$(libgmp_la_DEPENDENCIES)} from that line (which will make the initial
   build work, but if any recompiling is done @file{libgmp.la} might not be
   rebuilt).
   
   @item MacOS X and GCC
   Libtool currently only knows how to create shared libraries on MacOS X using
   the native @command{cc} (which is a modified GCC), not a plain GCC.  A
   static-only build should work though (@samp{--disable-shared}).
   
   Also, libtool currently cannot build C++ shared libraries on MacOS X, so if
   @samp{--enable-cxx} is desired then @samp{--disable-shared} must be used.
   Hopefully this will be fixed in the future.
   
   @item NeXT prior to 3.3
   
   The system compiler on old versions of NeXT was a massacred and old GCC, even
   if it called itself @file{cc}.  This compiler cannot be used to build GMP, you
   need to get a real GCC, and install that.  (NeXT may have fixed this in
   release 3.3 of their system.)
   
   @item POWER and PowerPC
   
   Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or
   PowerPC.  If you want to use GCC for these machines, get GCC 2.7.2.1 (or
   later).
   
   @item Sequent Symmetry
   
   Use the GNU assembler instead of the system assembler, since the latter has
   serious bugs.
   
   @item Solaris 2.6
   
   The system @command{sed} prints an error ``Output line too long'' when libtool
   builds @file{libgmp.la}.  This doesn't seem to cause any obvious ill effects,
   but GNU @command{sed} is recommended, to avoid any doubt.
   
   @item Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32
   
   A shared library build of GMP seems to fail in this combination, it builds but
   then fails the tests, apparently due to some incorrect data relocations within
   @code{gmp_randinit_lc_2exp_size}.  The exact cause is unknown,
   @samp{--disable-shared} is recommended.
   
   @item Windows DLL test programs
   
   When creating a DLL version of @file{libgmp}, libtool creates wrapper scripts
   like @file{t-mul} for programs that would normally be @file{t-mul.exe}, in
   order to setup the right library paths etc.  This works fine, but the absence
   of @file{t-mul.exe} etc causes @command{make} to think they need recompiling
   every time, which is an annoyance when re-running a @samp{make check}.
   @end table
   
   
   @node GMP Basics, Reporting Bugs, Installing GMP, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter MP Basics  @chapter GMP Basics
   @cindex Basics
   
   @strong{Using functions, macros, data types, etc.@: not documented in this
   manual is strongly discouraged.  If you do so your application is guaranteed
   to be incompatible with future versions of GMP.}
   
   @menu
   * Headers and Libraries::
   * Nomenclature and Types::
   * Function Classes::
   * Variable Conventions::
   * Parameter Conventions::
   * Memory Management::
   * Reentrancy::
   * Useful Macros and Constants::
   * Compatibility with older versions::
   * Demonstration Programs::
   * Efficiency::
   * Debugging::
   * Profiling::
   * Autoconf::
   * Emacs::
   @end menu
   
   @node Headers and Libraries, Nomenclature and Types, GMP Basics, GMP Basics
   @section Headers and Libraries
   @cindex Headers
   
 @cindex @file{gmp.h}  @cindex @file{gmp.h}
 All declarations needed to use MP are collected in the include file  All declarations needed to use GMP are collected in the include file
 @file{gmp.h}.  It is designed to work with both C and C++ compilers.  @file{gmp.h}.  It is designed to work with both C and C++ compilers.
   
   @example
   #include <gmp.h>
   @end example
   
   Note however that prototypes for GMP functions with @code{FILE *} parameters
   are only provided if @code{<stdio.h>} is included too.
   
   @example
   #include <stdio.h>
   #include <gmp.h>
   @end example
   
   Likewise @code{<stdarg.h>} (or @code{<varargs.h>}) is required for prototypes
   with @code{va_list} parameters, such as @code{gmp_vprintf}.  And
   @code{<obstack.h>} for prototypes with @code{struct obstack} parameters, such
   as @code{gmp_obstack_printf}, when available.
   
   @cindex Libraries
   @cindex Linking
   All programs using GMP must link against the @file{libgmp} library.  On a
   typical Unix-like system this can be done with @samp{-lgmp}, for example
   
   @example
   gcc myprogram.c -lgmp
   @end example
   
   GMP C++ functions are in a separate @file{libgmpxx} library.  This is built
   and installed if C++ support has been enabled (@pxref{Build Options}).  For
   example,
   
   @example
   g++ mycxxprog.cc -lgmpxx -lgmp
   @end example
   
   GMP is built using Libtool and an application can use that to link if desired,
   @pxref{Top,Shared library support for GNU,Introduction,libtool,GNU Libtool}
   
   If GMP has been installed to a non-standard location then it may be necessary
   to use @samp{-I} and @samp{-L} compiler options to point to the right
   directories, and some sort of run-time path for a shared library.  Consult
   your compiler documentation, for instance @ref{Top,,Introduction,gcc,Using and
   Porting the GNU Compiler Collection}.
   
   
   @node Nomenclature and Types, Function Classes, Headers and Libraries, GMP Basics
 @section Nomenclature and Types  @section Nomenclature and Types
   @cindex Nomenclature
   @cindex Types
   
 @cindex Integer  @cindex Integer
 @tindex @code{mpz_t}  @tindex @code{mpz_t}
 @noindent  @noindent
 In this manual, @dfn{integer} usually means a multiple precision integer, as  In this manual, @dfn{integer} usually means a multiple precision integer, as
 defined by the MP library.  The C data type for such integers is @code{mpz_t}.  defined by the GMP library.  The C data type for such integers is @code{mpz_t}.
 Here are some examples of how to declare such integers:  Here are some examples of how to declare such integers:
   
 @example  @example
Line 392  mpq_t quotient;
Line 1633  mpq_t quotient;
 @tindex @code{mpf_t}  @tindex @code{mpf_t}
 @noindent  @noindent
 @dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision  @dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision
 mantissa with an limited precision exponent.  The C data type for such objects  mantissa with a limited precision exponent.  The C data type for such objects
 is @code{mpf_t}.  is @code{mpf_t}.
   
 @cindex Limb  @cindex Limb
 @tindex @code{mp_limb_t}  @tindex @code{mp_limb_t}
 @noindent  @noindent
 A @dfn{limb} means the part of a multi-precision number that fits in a single  A @dfn{limb} means the part of a multi-precision number that fits in a single
 word.  (We chose this word because a limb of the human body is analogous to a  machine word.  (We chose this word because a limb of the human body is
 digit, only larger, and containing several digits.)  Normally a limb contains  analogous to a digit, only larger, and containing several digits.)  Normally a
 32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.  limb is 32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.
   
   
   @node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics
 @section Function Classes  @section Function Classes
   @cindex Function classes
   
 There are six classes of functions in the MP library:  There are six classes of functions in the GMP library:
   
 @enumerate  @enumerate
 @item  @item
 Functions for signed integer arithmetic, with names beginning with  Functions for signed integer arithmetic, with names beginning with
 @code{mpz_}.  The associated type is @code{mpz_t}.  There are about 100  @code{mpz_}.  The associated type is @code{mpz_t}.  There are about 150
 functions in this class.  functions in this class.
   
 @item  @item
 Functions for rational number arithmetic, with names beginning with  Functions for rational number arithmetic, with names beginning with
 @code{mpq_}.  The associated type is @code{mpq_t}.  There are about 20  @code{mpq_}.  The associated type is @code{mpq_t}.  There are about 40
 functions in this class, but the functions in the previous class can be used  functions in this class, but the integer functions can be used for arithmetic
 for performing arithmetic on the numerator and denominator separately.  on the numerator and denominator separately.
   
 @item  @item
 Functions for floating-point arithmetic, with names beginning with  Functions for floating-point arithmetic, with names beginning with
 @code{mpf_}.  The associated type is @code{mpf_t}.  There are about 50  @code{mpf_}.  The associated type is @code{mpf_t}.  There are about 60
 functions is this class.  functions is this class.
   
 @item  @item
Line 433  Functions compatible with Berkeley MP, such as @code{i
Line 1676  Functions compatible with Berkeley MP, such as @code{i
 Fast low-level functions that operate on natural numbers.  These are used by  Fast low-level functions that operate on natural numbers.  These are used by
 the functions in the preceding groups, and you can also call them directly  the functions in the preceding groups, and you can also call them directly
 from very time-critical user programs.  These functions' names begin with  from very time-critical user programs.  These functions' names begin with
 @code{mpn_}.  There are about 30 (hard-to-use) functions in this class.  @code{mpn_}.  The associated type is array of @code{mp_limb_t}.  There are
   about 30 (hard-to-use) functions in this class.
   
 The associated type is array of @code{mp_limb_t}.  
   
 @item  @item
 Miscellaneous functions.  Functions for setting up custom allocation.  Miscellaneous functions.  Functions for setting up custom allocation and
   functions for generating random numbers.
 @end enumerate  @end enumerate
   
   
 @section MP Variable Conventions  @node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics
   @section Variable Conventions
   @cindex Variable conventions
   @cindex Conventions for variables
   
 As a general rule, all MP functions expect output arguments before input  GMP functions generally have output arguments before input arguments.  This
 arguments.  This notation is based on an analogy with the assignment operator.  notation is by analogy with the assignment operator.  The BSD MP compatibility
 (The BSD MP compatibility functions disobey this rule, having the output  functions are exceptions, having the output arguments last.
 argument(s) last.)  
   
 MP allows you to use the same variable for both input and output in the same  GMP lets you use the same variable for both input and output in one call.  For
 expression.  For example, the main function for integer multiplication,  example, the main function for integer multiplication, @code{mpz_mul}, can be
 @code{mpz_mul}, can be used like this: @code{mpz_mul (x, x, x)}.  This  used to square @code{x} and put the result back in @code{x} with
 computes the square of @var{x} and puts the result back in @var{x}.  
   
 Before you can assign to an MP variable, you need to initialize it by calling  @example
   mpz_mul (x, x, x);
   @end example
   
   Before you can assign to a GMP variable, you need to initialize it by calling
 one of the special initialization functions.  When you're done with a  one of the special initialization functions.  When you're done with a
 variable, you need to clear it out, using one of the functions for that  variable, you need to clear it out, using one of the functions for that
 purpose.  Which function to use depends on the type of variable.  See the  purpose.  Which function to use depends on the type of variable.  See the
 chapters on integer functions, rational number functions, and floating-point  chapters on integer functions, rational number functions, and floating-point
 functions for details.  functions for details.
   
 A variable should only be initialized once, or at least cleared out between  A variable should only be initialized once, or at least cleared between each
 each initialization.  After a variable has been initialized, it may be  initialization.  After a variable has been initialized, it may be assigned to
 assigned to any number of times.  any number of times.
   
 For efficiency reasons, avoid to initialize and clear out a variable in loops.  For efficiency reasons, avoid excessive initializing and clearing.  In
 Instead, initialize it before entering the loop, and clear it out after the  general, initialize near the start of a function and clear near the end.  For
 loop has exited.  example,
   
 You don't need to be concerned about allocating additional space for MP  @example
 variables.  All functions in MP automatically allocate additional space when a  void
 variable does not already have enough space.  They do not, however, reduce the  foo (void)
 space when a smaller number is stored in the object.  Most of the time, this  @{
 policy is best, since it avoids frequent re-allocation.    mpz_t  n;
     int    i;
     mpz_init (n);
     for (i = 1; i < 100; i++)
       @{
         mpz_mul (n, @dots{});
         mpz_fdiv_q (n, @dots{});
         @dots{}
       @}
     mpz_clear (n);
   @}
   @end example
   
   
   @node Parameter Conventions, Memory Management, Variable Conventions, GMP Basics
   @section Parameter Conventions
   @cindex Parameter conventions
   @cindex Conventions for parameters
   
   When a GMP variable is used as a function parameter, it's effectively a
   call-by-reference, meaning if the function stores a value there it will change
   the original in the caller.  Parameters which are input-only can be designated
   @code{const} to provoke a compiler error or warning on attempting to modify
   them.
   
   When a function is going to return a GMP result, it should designate a
   parameter that it sets, like the library functions do.  More than one value
   can be returned by having more than one output parameter, again like the
   library functions.  A @code{return} of an @code{mpz_t} etc doesn't return the
   object, only a pointer, and this is almost certainly not what's wanted.
   
   Here's an example accepting an @code{mpz_t} parameter, doing a calculation,
   and storing the result to the indicated parameter.
   
   @example
   void
   foo (mpz_t result, const mpz_t param, unsigned long n)
   @{
     unsigned long  i;
     mpz_mul_ui (result, param, n);
     for (i = 1; i < n; i++)
       mpz_add_ui (result, result, i*7);
   @}
   
   int
   main (void)
   @{
     mpz_t  r, n;
     mpz_init (r);
     mpz_init_set_str (n, "123456", 0);
     foo (r, n, 20L);
     gmp_printf ("%Zd\n", r);
     return 0;
   @}
   @end example
   
   @code{foo} works even if the mainline passes the same variable for
   @code{param} and @code{result}, just like the library functions.  But
   sometimes it's tricky to make that work, and an application might not want to
   bother supporting that sort of thing.
   
   For interest, the GMP types @code{mpz_t} etc are implemented as one-element
   arrays of certain structures.  This is why declaring a variable creates an
   object with the fields GMP needs, but then using it as a parameter passes a
   pointer to the object.  Note that the actual fields in each @code{mpz_t} etc
   are for internal use only and should not be accessed directly by code that
   expects to be compatible with future GMP releases.
   
   
   @need 1000
   @node Memory Management, Reentrancy, Parameter Conventions, GMP Basics
   @section Memory Management
   @cindex Memory Management
   
   The GMP types like @code{mpz_t} are small, containing only a couple of sizes,
   and pointers to allocated data.  Once a variable is initialized, GMP takes
   care of all space allocation.  Additional space is allocated whenever a
   variable doesn't have enough.
   
   @code{mpz_t} and @code{mpq_t} variables never reduce their allocated space.
   Normally this is the best policy, since it avoids frequent reallocation.
   Applications that need to return memory to the heap at some particular point
   can use @code{mpz_realloc2}, or clear variables no longer needed.
   
   @code{mpf_t} variables, in the current implementation, use a fixed amount of
   space, determined by the chosen precision and allocated at initialization, so
   their size doesn't change.
   
   All memory is allocated using @code{malloc} and friends by default, but this
   can be changed, see @ref{Custom Allocation}.  Temporary memory on the stack is
   also used (via @code{alloca}), but this can be changed at build-time if
   desired, see @ref{Build Options}.
   
   
   @node Reentrancy, Useful Macros and Constants, Memory Management, GMP Basics
   @section Reentrancy
   @cindex Reentrancy
   @cindex Thread safety
   @cindex Multi-threading
   
   GMP is reentrant and thread-safe, with some exceptions:
   
   @itemize @bullet
   @item
   If configured with @option{--enable-alloca=malloc-notreentrant} (or with
   @option{--enable-alloca=notreentrant} when @code{alloca} is not available),
   then naturally GMP is not reentrant.
   
   @item
   @code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the
   selected precision.  @code{mpf_init2} can be used instead.
   
   @item
   @code{mpz_random} and the other old random number functions use a global
   random state and are hence not reentrant.  The newer random number functions
   that accept a @code{gmp_randstate_t} parameter can be used instead.
   
   @item
   @code{mp_set_memory_functions} uses global variables to store the selected
   memory allocation functions.
   
   @item
   If the memory allocation functions set by a call to
   @code{mp_set_memory_functions} (or @code{malloc} and friends by default) are
   not reentrant, then GMP will not be reentrant either.
   
   @item
   If the standard I/O functions such as @code{fwrite} are not reentrant then the
   GMP I/O functions using them will not be reentrant either.
   
   @item
   It's safe for two threads to read from the same GMP variable simultaneously,
   but it's not safe for one to read while the another might be writing, nor for
   two threads to write simultaneously.  It's not safe for two threads to
   generate a random number from the same @code{gmp_randstate_t} simultaneously,
   since this involves an update of that variable.
   
   @item
   On SCO systems the default @code{<ctype.h>} macros use per-file static
   variables and may not be reentrant, depending whether the compiler optimizes
   away fetches from them.  The GMP text-based input functions are affected.
   @end itemize
   
   
   @need 2000
   @node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics
 @section Useful Macros and Constants  @section Useful Macros and Constants
   @cindex Useful macros and constants
   @cindex Constants
   
 @deftypevr {Global Constant} {const int} mp_bits_per_limb  @deftypevr {Global Constant} {const int} mp_bits_per_limb
   @findex mp_bits_per_limb
   @cindex Bits per limb
   @cindex Limb size
 The number of bits per limb.  The number of bits per limb.
 @end deftypevr  @end deftypevr
   
 @defmac __GNU_MP_VERSION  @defmac __GNU_MP_VERSION
 @defmacx __GNU_MP_VERSION_MINOR  @defmacx __GNU_MP_VERSION_MINOR
 The major and minor MP version, respectively, as integers.  @defmacx __GNU_MP_VERSION_PATCHLEVEL
   @cindex Version number
   @cindex GMP version number
   The major and minor GMP version, and patch level, respectively, as integers.
   For GMP i.j, these numbers will be i, j, and 0, respectively.
   For GMP i.j.k, these numbers will be i, j, and k, respectively.
 @end defmac  @end defmac
   
 @section Compatibility with Version 1.x  @deftypevr {Global Constant} {const char * const} gmp_version
   @findex gmp_version
   The GMP version number, as a null-terminated string, in the form ``i.j'' or
   ``i.j.k''.  This release is @nicode{"@value{VERSION}"}.
   @end deftypevr
   
 This version of MP is upward compatible with previous versions of MP, with a  
 few exceptions.  
   
 @enumerate  @node Compatibility with older versions, Demonstration Programs, Useful Macros and Constants, GMP Basics
 @item Integer division functions round the result differently.  The old  @section Compatibility with older versions
 functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv},  @cindex Compatibility with older versions
 @code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the  @cindex Upward compatibility
 quotient to @minus{}infinity).  There are a lot of new functions for integer  
 division, giving the user better control over the rounding.  
   
 @item The function @code{mpz_mod} now compute the true @strong{mod} function.  This version of GMP is upwardly binary compatible with all 4.x and 3.x
   versions, and upwardly compatible at the source level with all 2.x versions,
   with the following exceptions.
   
 @item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use  @itemize @bullet
 @strong{mod} for reduction.  @item
   @code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency
   with other @code{mpn} functions.
   
 @item The assignment functions for rational numbers do no longer canonicalize  @item
 their results.  In the case a non-canonical result could arise from an  @code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and
 assignment, the user need to insert an explicit call to  3.0.1, but in 3.1 reverted to the 2.x style.
 @code{mpq_canonicalize}.  This change was made for efficiency.  @end itemize
   
 @item Output generated by @code{mpz_out_raw} in this release cannot be read  There are a number of compatibility issues between GMP 1 and GMP 2 that of
 by @code{mpz_inp_raw} in previous releases.  This change was made for making  course also apply when porting applications from GMP 1 to GMP 4.  Please
 the file format truly portable between machines with different word sizes.  see the GMP 2 manual for details.
   
 @item Several @code{mpn} functions have changed.  But they were intentionally  The Berkeley MP compatibility library (@pxref{BSD Compatible Functions}) is
 undocumented in previous releases.  source and binary compatible with the standard @file{libmp}.
   
 @item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui}  @c @enumerate
 are now implementated as macros, and thereby sometimes evaluate their  @c @item Integer division functions round the result differently.  The obsolete
 arguments multiple times.  @c functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv},
   @c @code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the
   @c quotient towards
   @c @ifinfo
   @c @minus{}infinity).
   @c @end ifinfo
   @c @iftex
   @c @tex
   @c $-\infty$).
   @c @end tex
   @c @end iftex
   @c There are a lot of functions for integer division, giving the user better
   @c control over the rounding.
   
 @item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1  @c @item The function @code{mpz_mod} now compute the true @strong{mod} function.
 for 0^0.  (In version 1, they yielded 0.)  
   
 @end enumerate  @c @item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use
   @c @strong{mod} for reduction.
   
   @c @item The assignment functions for rational numbers do no longer canonicalize
   @c their results.  In the case a non-canonical result could arise from an
   @c assignment, the user need to insert an explicit call to
   @c @code{mpq_canonicalize}.  This change was made for efficiency.
   
 @section Getting the Latest Version of MP  @c @item Output generated by @code{mpz_out_raw} in this release cannot be read
   @c by @code{mpz_inp_raw} in previous releases.  This change was made for making
   @c the file format truly portable between machines with different word sizes.
   
 The latest version of the MP library is available by anonymous ftp from  @c @item Several @code{mpn} functions have changed.  But they were intentionally
 from @samp{prep.ai.mit.edu}.  The file name is  @c undocumented in previous releases.
 @file{/pub/gnu/gmp-M.N.tar.gz}.  Many sites around the world mirror  
 @samp{prep}; please use a mirror site near you.  
   
 @node Reporting Bugs, Integer Functions, MP Basics, Top  @c @item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui}
   @c are now implemented as macros, and thereby sometimes evaluate their
   @c arguments multiple times.
   
   @c @item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1
   @c for 0^0.  (In version 1, they yielded 0.)
   
   @c In version 1 of the library, @code{mpq_set_den} handled negative
   @c denominators by copying the sign to the numerator.  That is no longer done.
   
   @c Pure assignment functions do not canonicalize the assigned variable.  It is
   @c the responsibility of the user to canonicalize the assigned variable before
   @c any arithmetic operations are performed on that variable.
   @c Note that this is an incompatible change from version 1 of the library.
   
   @c @end enumerate
   
   
   @need 1000
   @node Demonstration Programs, Efficiency, Compatibility with older versions, GMP Basics
   @section Demonstration programs
   @cindex Demonstration programs
   @cindex Example programs
   @cindex Sample programs
   The @file{demos} subdirectory has some sample programs using GMP.  These
   aren't built or installed, but there's a @file{Makefile} with rules for them.
   For instance,
   
   @example
   make pexpr
   ./pexpr 68^975+10
   @end example
   
   @noindent
   The following programs are provided
   
   @itemize @bullet
   @item
   @samp{pexpr} is an expression evaluator, the program used on the GMP web page.
   @item
   The @samp{calc} subdirectory has a similar but simpler evaluator using
   @command{lex} and @command{yacc}.
   @item
   The @samp{expr} subdirectory is yet another expression evaluator, a library
   designed for ease of use within a C program.  See @file{demos/expr/README} for
   more information.
   @item
   @samp{factorize} is a Pollard-Rho factorization program.
   @item
   @samp{isprime} is a command-line interface to the @code{mpz_probab_prime_p}
   function.
   @item
   @samp{primes} counts or lists primes in an interval, using a sieve.
   @item
   @samp{qcn} is an example use of @code{mpz_kronecker_ui} to estimate quadratic
   class numbers.
   @item
   @cindex @code{perl}
   The @samp{perl} subdirectory is a comprehensive perl interface to GMP.  See
   @file{demos/perl/INSTALL} for more information.  Documentation is in POD
   format in @file{demos/perl/GMP.pm}.
   @end itemize
   
   
   @need 1000
   @node Efficiency, Debugging, Demonstration Programs, GMP Basics
   @section Efficiency
   @cindex Efficiency
   
   @table @asis
   @item Small operands
   On small operands, the time for function call overheads and memory allocation
   can be significant in comparison to actual calculation.  This is unavoidable
   in a general purpose variable precision library, although GMP attempts to be
   as efficient as it can on both large and small operands.
   
   @item Static Linking
   On some CPUs, in particular the x86s, the static @file{libgmp.a} should be
   used for maximum speed, since the PIC code in the shared @file{libgmp.so} will
   have a small overhead on each function call and global data address.  For many
   programs this will be insignificant, but for long calculations there's a gain
   to be had.
   
   @item Initializing and clearing
   Avoid excessive initializing and clearing of variables, since this can be
   quite time consuming, especially in comparison to otherwise fast operations
   like addition.
   
   A language interpreter might want to keep a free list or stack of
   initialized variables ready for use.  It should be possible to integrate
   something like that with a garbage collector too.
   
   @item Reallocations
   An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing
   values will have its memory repeatedly @code{realloc}ed, which could be quite
   slow or could fragment memory, depending on the C library.  If an application
   can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can
   be called to allocate the necessary space from the beginning
   (@pxref{Initializing Integers}).
   
   It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2}
   is too small, since all functions will do a further reallocation if necessary.
   Badly overestimating memory required will waste space though.
   
   @item @code{2exp} functions
   It's up to an application to call functions like @code{mpz_mul_2exp} when
   appropriate.  General purpose functions like @code{mpz_mul} make no attempt to
   identify powers of two or other special forms, because such inputs will
   usually be very rare and testing every time would be wasteful.
   
   @item @code{ui} and @code{si} functions
   The @code{ui} functions and the small number of @code{si} functions exist for
   convenience and should be used where applicable.  But if for example an
   @code{mpz_t} contains a value that fits in an @code{unsigned long} there's no
   need extract it and call a @code{ui} function, just use the regular @code{mpz}
   function.
   
   @item In-Place Operations
   @code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg}
   and @code{mpf_neg} are fast when used for in-place operations like
   @code{mpz_abs(x,x)}, since in the current implementation only a single field
   of @code{x} needs changing.  On suitable compilers (GCC for instance) this is
   inlined too.
   
   @code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui}
   benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since
   usually only one or two limbs of @code{x} will need to be changed.  The same
   applies to the full precision @code{mpz_add} etc if @code{y} is small.  If
   @code{y} is big then cache locality may be helped, but that's all.
   
   @code{mpz_mul} is currently the opposite, a separate destination is slightly
   better.  A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one
   limb, make a temporary copy of @code{x} before forming the result.  Normally
   that copying will only be a tiny fraction of the time for the multiply, so
   this is not a particularly important consideration.
   
   @code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make
   no attempt to recognise a copy of something to itself, so a call like
   @code{mpz_set(x,x)} will be wasteful.  Naturally that would never be written
   deliberately, but if it might arise from two pointers to the same object then
   a test to avoid it might be desirable.
   
   @example
   if (x != y)
     mpz_set (x, y);
   @end example
   
   Note that it's never worth introducing extra @code{mpz_set} calls just to get
   in-place operations.  If a result should go to a particular variable then just
   direct it there and let GMP take care of data movement.
   
   @item Divisibility Testing (Small Integers)
   
   @code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions
   for testing whether an @code{mpz_t} is divisible by an individual small
   integer.  They use an algorithm which is faster than @code{mpz_tdiv_ui}, but
   which gives no useful information about the actual remainder, only whether
   it's zero (or a particular value).
   
   However when testing divisibility by several small integers, it's best to take
   a remainder modulo their product, to save multi-precision operations.  For
   instance to test whether a number is divisible by any of 23, 29 or 31 take a
   remainder modulo @math{23@times{}29@times{}31 = 20677} and then test that.
   
   The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well
   as a remainder are generally a little slower than the remainder-only functions
   like @code{mpz_tdiv_ui}.  If the quotient is only rarely wanted then it's
   probably best to just take a remainder and then go back and calculate the
   quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the
   remainder is zero).
   
   @item Rational Arithmetic
   The @code{mpq} functions operate on @code{mpq_t} values with no common factors
   in the numerator and denominator.  Common factors are checked-for and cast out
   as necessary.  In general, cancelling factors every time is the best approach
   since it minimizes the sizes for subsequent operations.
   
   However, applications that know something about the factorization of the
   values they're working with might be able to avoid some of the GCDs used for
   canonicalization, or swap them for divisions.  For example when multiplying by
   a prime it's enough to check for factors of it in the denominator instead of
   doing a full GCD.  Or when forming a big product it might be known that very
   little cancellation will be possible, and so canonicalization can be left to
   the end.
   
   The @code{mpq_numref} and @code{mpq_denref} macros give access to the
   numerator and denominator to do things outside the scope of the supplied
   @code{mpq} functions.  @xref{Applying Integer Functions}.
   
   The canonical form for rationals allows mixed-type @code{mpq_t} and integer
   additions or subtractions to be done directly with multiples of the
   denominator.  This will be somewhat faster than @code{mpq_add}.  For example,
   
   @example
   /* mpq increment */
   mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
   
   /* mpq += unsigned long */
   mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
   
   /* mpq -= mpz */
   mpz_submul (mpq_numref(q), mpq_denref(q), z);
   @end example
   
   @item Number Sequences
   Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui}
   are designed for calculating isolated values.  If a range of values is wanted
   it's probably best to call to get a starting point and iterate from there.
   
   @item Text Input/Output
   Hexadecimal or octal are suggested for input or output in text form.
   Power-of-2 bases like these can be converted much more efficiently than other
   bases, like decimal.  For big numbers there's usually nothing of particular
   interest to be seen in the digits, so the base doesn't matter much.
   
   Maybe we can hope octal will one day become the normal base for everyday use,
   as proposed by King Charles XII of Sweden and later reformers.
   @c Reference: Knuth volume 2 section 4.1, page 184 of second edition.  :-)
   @end table
   
   
   @node Debugging, Profiling, Efficiency, GMP Basics
   @section Debugging
   @cindex Debugging
   
   @table @asis
   @item Stack Overflow
   Depending on the system, a segmentation violation or bus error might be the
   only indication of stack overflow.  See @samp{--enable-alloca} choices in
   @ref{Build Options}, for how to address this.
   
   In new enough versions of GCC, @samp{-fstack-check} may be able to ensure an
   overflow is recognised by the system before too much damage is done, or
   @samp{-fstack-limit-symbol} or @samp{-fstack-limit-register} may be able to
   add checking if the system itself doesn't do any (@pxref{Code Gen Options,,
   Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}).
   These options must be added to the @samp{CFLAGS} used in the GMP build
   (@pxref{Build Options}), adding them just to an application will have no
   effect.  Note also they're a slowdown, adding overhead to each function call
   and each stack allocation.
   
   @item Heap Problems
   The most likely cause of application problems with GMP is heap corruption.
   Failing to @code{init} GMP variables will have unpredictable effects, and
   corruption arising elsewhere in a program may well affect GMP.  Initializing
   GMP variables more than once or failing to clear them will cause memory leaks.
   
   In all such cases a malloc debugger is recommended.  On a GNU or BSD system
   the standard C library @code{malloc} has some diagnostic facilities, see
   @ref{Allocation Debugging,,,libc,The GNU C Library Reference Manual}, or
   @samp{man 3 malloc}.  Other possibilities, in no particular order, include
   
   @display
   @uref{http://www.inf.ethz.ch/personal/biere/projects/ccmalloc}
   @uref{http://quorum.tamu.edu/jon/gnu} @ (debauch)
   @uref{http://dmalloc.com}
   @uref{http://www.perens.com/FreeSoftware} @ (electric fence)
   @uref{http://packages.debian.org/fda}
   @uref{http://www.gnupdate.org/components/leakbug}
   @uref{http://people.redhat.com/~otaylor/memprof}
   @uref{http://www.cbmamiga.demon.co.uk/mpatrol}
   @end display
   
   The GMP default allocation routines in @file{memory.c} also have a simple
   sentinel scheme which can be enabled with @code{#define DEBUG} in that file.
   This is mainly designed for detecting buffer overruns during GMP development,
   but might find other uses.
   
   @item Stack Backtraces
   On some systems the compiler options GMP uses by default can interfere with
   debugging.  In particular on x86 and 68k systems @samp{-fomit-frame-pointer}
   is used and this generally inhibits stack backtracing.  Recompiling without
   such options may help while debugging, though the usual caveats about it
   potentially moving a memory problem or hiding a compiler bug will apply.
   
   @item GNU Debugger
   A sample @file{.gdbinit} is included in the distribution, showing how to call
   some undocumented dump functions to print GMP variables from within GDB.  Note
   that these functions shouldn't be used in final application code since they're
   undocumented and may be subject to incompatible changes in future versions of
   GMP.
   
   @item Source File Paths
   GMP has multiple source files with the same name, in different directories.
   For example @file{mpz}, @file{mpq}, @file{mpf} and @file{mpfr} each have an
   @file{init.c}.  If the debugger can't already determine the right one it may
   help to build with absolute paths on each C file.  One way to do that is to
   use a separate object directory with an absolute path to the source directory.
   
   @example
   cd /my/build/dir
   /my/source/dir/gmp-@value{VERSION}/configure
   @end example
   
   This works via @code{VPATH}, and might require GNU @command{make}.
   Alternately it might be possible to change the @code{.c.lo} rules
   appropriately.
   
   @item Assertion Checking
   The build option @option{--enable-assert} is available to add some consistency
   checks to the library (see @ref{Build Options}).  These are likely to be of
   limited value to most applications.  Assertion failures are just as likely to
   indicate memory corruption as a library or compiler bug.
   
   Applications using the low-level @code{mpn} functions, however, will benefit
   from @option{--enable-assert} since it adds checks on the parameters of most
   such functions, many of which have subtle restrictions on their usage.  Note
   however that only the generic C code has checks, not the assembler code, so
   CPU @samp{none} should be used for maximum checking.
   
   @item Temporary Memory Checking
   The build option @option{--enable-alloca=debug} arranges that each block of
   temporary memory in GMP is allocated with a separate call to @code{malloc} (or
   the allocation function set with @code{mp_set_memory_functions}).
   
   This can help a malloc debugger detect accesses outside the intended bounds,
   or detect memory not released.  In a normal build, on the other hand,
   temporary memory is allocated in blocks which GMP divides up for its own use,
   or may be allocated with a compiler builtin @code{alloca} which will go
   nowhere near any malloc debugger hooks.
   
   @item Maximum Debuggability
   To summarize the above, a GMP build for maximum debuggability would be
   
   @example
   ./configure --disable-shared --enable-assert \
     --enable-alloca=debug --host=none CFLAGS=-g
   @end example
   
   For C++, add @samp{--enable-cxx CXXFLAGS=-g}.
   
   @item Checker
   The checker program (@uref{http://savannah.gnu.org/projects/checker}) can be
   used with GMP.  It contains a stub library which means GMP applications
   compiled with checker can use a normal GMP build.
   
   A build of GMP with checking within GMP itself can be made.  This will run
   very very slowly.  Configure with
   
   @example
   ./configure --host=none-pc-linux-gnu CC=checkergcc
   @end example
   
   @samp{--host=none} must be used, since the GMP assembler code doesn't support
   the checking scheme.  The GMP C++ features cannot be used, since current
   versions of checker (0.9.9.1) don't yet support the standard C++ library.
   
   @item Valgrind
   The valgrind program (@uref{http://devel-home.kde.org/~sewardj}) is a memory
   checker for x86s.  It translates and emulates machine instructions to do
   strong checks for uninitialized data (at the level of individual bits), memory
   accesses through bad pointers, and memory leaks.
   
   Current versions (20020226 snapshot) don't support MMX or SSE, so GMP must be
   configured for an x86 without those (eg. plain @samp{i386}), or with a special
   @code{MPN_PATH} that excludes those subdirectories (@pxref{Build Options}).
   
   @item Other Problems
   Any suspected bug in GMP itself should be isolated to make sure it's not an
   application problem, see @ref{Reporting Bugs}.
   @end table
   
   
   @node Profiling, Autoconf, Debugging, GMP Basics
   @section Profiling
   @cindex Profiling
   
   Running a program under a profiler is a good way to find where it's spending
   most time and where improvements can be best sought.
   
   Depending on the system, it may be possible to get a flat profile, meaning
   simple timer sampling of the program counter, with no special GMP build
   options, just a @samp{-p} when compiling the mainline.  This is a good way to
   ensure minimum interference with normal operation.  The necessary symbol type
   and size information exists in most of the GMP assembler code.
   
   The @samp{--enable-profiling} build option can be used to add suitable
   compiler flags, either for @command{prof} (@samp{-p}) or @command{gprof}
   (@samp{-pg}), see @ref{Build Options}.  Which of the two is available and what
   they do will depend on the system, and possibly on support available in
   @file{libc}.  For some systems appropriate corresponding @code{mcount} calls
   are added to the assembler code too.
   
   On x86 systems @command{prof} gives call counting, so that average time spent
   in a function can be determined.  @command{gprof}, where supported, adds call
   graph construction, so for instance calls to @code{mpn_add_n} from
   @code{mpz_add} and from @code{mpz_mul} can be differentiated.
   
   On x86 and 68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are
   incompatible, so the latter is not used when @command{gprof} profiling is
   selected, which may result in poorer code generation.  If @command{prof}
   profiling is selected instead it should still be possible to use
   @command{gprof}, but only the @samp{gprof -p} flat profile and call counts can
   be expected to be valid, not the @samp{gprof -q} call graph.
   
   
   @node Autoconf, Emacs, Profiling, GMP Basics
   @section Autoconf
   @cindex Autoconf detections
   
   Autoconf based applications can easily check whether GMP is installed.  The
   only thing to be noted is that GMP library symbols from version 3 onwards have
   prefixes like @code{__gmpz}.  The following therefore would be a simple test,
   
   @example
   AC_CHECK_LIB(gmp, __gmpz_init)
   @end example
   
   This just uses the default @code{AC_CHECK_LIB} actions for found or not found,
   but an application that must have GMP would want to generate an error if not
   found.  For example,
   
   @example
   AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR(
   [GNU MP not found, see http://swox.com/gmp])])
   @end example
   
   If functions added in some particular version of GMP are required, then one of
   those can be used when checking.  For example @code{mpz_mul_si} was added in
   GMP 3.1,
   
   @example
   AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR(
   [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
   @end example
   
   An alternative would be to test the version number in @file{gmp.h} using say
   @code{AC_EGREP_CPP}.  That would make it possible to test the exact version,
   if some particular sub-minor release is known to be necessary.
   
   An application that can use either GMP 2 or 3 will need to test for
   @code{__gmpz_init} (GMP 3 and up) or @code{mpz_init} (GMP 2), and it's also
   worth checking for @file{libgmp2} since Debian GNU/Linux systems used that
   name in the past.  For example,
   
   @example
   AC_CHECK_LIB(gmp, __gmpz_init, ,
     [AC_CHECK_LIB(gmp, mpz_init, ,
       [AC_CHECK_LIB(gmp2, mpz_init)])])
   @end example
   
   In general it's suggested that applications should simply demand a new enough
   GMP rather than trying to provide supplements for features not available in
   past versions.
   
   Occasionally an application will need or want to know the size of a type at
   configuration or preprocessing time, not just with @code{sizeof} in the code.
   This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or
   up is best for this, since prior versions needed certain @samp{-D} defines on
   systems using a @code{long long} limb.  The following would suit Autoconf 2.50
   or up,
   
   @example
   AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
   @end example
   
   The optional @code{mpfr} functions are provided in a separate
   @file{libmpfr.a}, and this might be from GMP with @option{--enable-mpfr} or
   from MPFR installed separately.  Either way @file{libmpfr} depends on
   @file{libgmp}, it doesn't stand alone.  Currently only a static
   @file{libmpfr.a} will be available, not a shared library, since upward binary
   compatibility is not guaranteed.
   
   @example
   AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR(
   [Need MPFR either from GNU MP 4 or separate MPFR package.
   See http://www.mpfr.org or http://swox.com/gmp])
   @end example
   
   
   @node Emacs,  , Autoconf, GMP Basics
   @section Emacs
   @cindex Emacs
   
   @key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation
   on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup,
   emacs, The Emacs Editor}).
   
   The GMP manual can be included in such lookups by putting the following in
   your @file{.emacs},
   
   @c  This isn't pretty, but there doesn't seem to be a better way (in emacs
   @c  21.2 at least).  info-lookup->mode-value could be used for the "assoc"s,
   @c  but that function isn't documented, whereas info-lookup-alist is.
   @c
   @example
   (eval-after-load "info-look"
     '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
        (setcar (nthcdr 3 mode-value)
                (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
                      (nth 3 mode-value)))))
   @end example
   
   The same can be done for MPFR, with @code{(mpfr)} in place of @code{(gmp)}.
   
   
   @node Reporting Bugs, Integer Functions, GMP Basics, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Reporting Bugs  @chapter Reporting Bugs
 @cindex Reporting bugs  @cindex Reporting bugs
   @cindex Bug reporting
   
 If you think you have found a bug in the MP library, please investigate it and  If you think you have found a bug in the GMP library, please investigate it
 report it.  We have made this library available to you, and it is not to ask  and report it.  We have made this library available to you, and it is not too
 too much from you, to ask you to report the bugs that you find.  much to ask you to report the bugs you find.
   
 There are a few things you should think about when you put your bug report  Before you report a bug, check it's not already addressed in @ref{Known Build
 together.  Problems}, or perhaps @ref{Notes for Particular Systems}.  You may also want
   to check @uref{http://swox.com/gmp/} for patches for this release.
   
 You have to send us a test case that makes it possible for us to reproduce the  Please include the following in any report,
 bug.  Include instructions on how to run the test case.  
   
 You also have to explain what is wrong; if you get a crash, or if the results  @itemize @bullet
 printed are incorrect and in that case, in what way.  @item
   The GMP version number, and if pre-packaged or patched then say so.
   
   @item
   A test program that makes it possible for us to reproduce the bug.  Include
   instructions on how to run the program.
   
   @item
   A description of what is wrong.  If the results are incorrect, in what way.
   If you get a crash, say so.
   
   @item
   If you get a crash, include a stack backtrace from the debugger if it's
   informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).
   
   @item
   Please do not send core dumps, executables or @command{strace}s.
   
   @item
   The configuration options you used when building GMP, if any.
   
   @item
   The name of the compiler and its version.  For @command{gcc}, get the version
   with @samp{gcc -v}, otherwise perhaps @samp{what `which cc`}, or similar.
   
   @item
   The output from running @samp{uname -a}.
   
   @item
   The output from running @samp{./config.guess}, and from running
   @samp{./configfsf.guess} (might be the same).
   
   @item
   If the bug is related to @samp{configure}, then the contents of
   @file{config.log}.
   
   @item
   If the bug is related to an @file{asm} file not assembling, then the contents
   of @file{config.m4} and the offending line or lines from the temporary
   @file{mpn/tmp-<file>.s}.
   @end itemize
   
   Please make an effort to produce a self-contained report, with something
   definite that can be tested or debugged.  Vague queries or piecemeal messages
   are difficult to act on and don't help the development effort.
   
 It is not uncommon that an observed problem is actually due to a bug in the  It is not uncommon that an observed problem is actually due to a bug in the
 compiler used when building MP; the MP code tends to explore interesting  compiler; the GMP code tends to explore interesting corners in compilers.
 corners in compilers.  Therefore, please include compiler version information  
 in your bug report.  This can be extracted using @samp{what `which cc`}, or,  
 if you're using gcc, @samp{gcc -v}.  Also, include the output from @samp{uname  
 -a}.  
   
 If your bug report is good, we will do our best to help you to get a corrected  If your bug report is good, we will do our best to help you get a corrected
 version of the library; if the bug report is poor, we won't do anything about  version of the library; if the bug report is poor, we won't do anything about
 it (aside of chiding you to send better bug reports).  it (except maybe ask you to send a better report).
   
 Send your bug report to: @samp{bug-gmp@@prep.ai.mit.edu}.  Send your report to: @email{bug-gmp@@gnu.org}.
   
 If you think something in this manual is unclear, or downright incorrect, or if  If you think something in this manual is unclear, or downright incorrect, or if
 the language needs to be improved, please send a note to the same address.  the language needs to be improved, please send a note to the same address.
Line 573  the language needs to be improved, please send a note 
Line 2531  the language needs to be improved, please send a note 
 @chapter Integer Functions  @chapter Integer Functions
 @cindex Integer functions  @cindex Integer functions
   
 This chapter describes the MP functions for performing integer arithmetic.  This chapter describes the GMP functions for performing integer arithmetic.
 These functions start with the prefix @code{mpz_}.  These functions start with the prefix @code{mpz_}.
   
 Arbitrary precision integers are stored in objects of type @code{mpz_t}.  GMP integers are stored in objects of type @code{mpz_t}.
   
 @menu  @menu
 * Initializing Integers::  * Initializing Integers::
 * Assigning Integers::  * Assigning Integers::
 * Simultaneous Integer Init & Assign::  * Simultaneous Integer Init & Assign::
 * Converting Integers::  * Converting Integers::
 * Integer Arithmetic::  * Integer Arithmetic::
 * Comparison Functions::  * Integer Division::
 * Integer Logic and Bit Fiddling::  * Integer Exponentiation::
 * I/O of Integers::  * Integer Roots::
 * Miscellaneous Integer Functions::  * Number Theoretic Functions::
   * Integer Comparisons::
   * Integer Logic and Bit Fiddling::
   * I/O of Integers::
   * Integer Random Numbers::
   * Integer Import and Export::
   * Miscellaneous Integer Functions::
 @end menu  @end menu
   
 @node Initializing Integers, Assigning Integers, , Integer Functions  @node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Initialization and Assignment Functions  @section Initialization Functions
   @cindex Integer initialization functions
   @cindex Initialization functions
   
 The functions for integer arithmetic assume that all integer objects are  The functions for integer arithmetic assume that all integer objects are
 initialized.  You do that by calling the function @code{mpz_init}.  initialized.  You do that by calling the function @code{mpz_init}.  For
   example,
   
 @deftypefun void mpz_init (mpz_t @var{integer})  
 Initialize @var{integer} with limb space and set the initial numeric value to  
 0.  Each variable should normally only be initialized once, or at least cleared  
 out (using @code{mpz_clear}) between each initialization.  
 @end deftypefun  
   
 Here is an example of using @code{mpz_init}:  
   
 @example  @example
 @{  @{
   mpz_t integ;    mpz_t integ;
Line 619  Here is an example of using @code{mpz_init}:
Line 2578  Here is an example of using @code{mpz_init}:
 @}  @}
 @end example  @end example
   
 @noindent  
 As you can see, you can store new values any number of times, once an  As you can see, you can store new values any number of times, once an
 object is initialized.  object is initialized.
   
   @deftypefun void mpz_init (mpz_t @var{integer})
   Initialize @var{integer}, and set its value to 0.
   @end deftypefun
   
   @deftypefun void mpz_init2 (mpz_t @var{integer}, unsigned long @var{n})
   Initialize @var{integer}, with space for @var{n} bits, and set its value to 0.
   
   @var{n} is only the initial space, @var{integer} will grow automatically in
   the normal way, if necessary, for subsequent values stored.  @code{mpz_init2}
   makes it possible to avoid such reallocations if a maximum size is known in
   advance.
   @end deftypefun
   
 @deftypefun void mpz_clear (mpz_t @var{integer})  @deftypefun void mpz_clear (mpz_t @var{integer})
 Free the limb space occupied by @var{integer}.  Make sure to call this  Free the space occupied by @var{integer}.  Call this function for all
 function for all @code{mpz_t} variables when you are done with them.  @code{mpz_t} variables when you are done with them.
 @end deftypefun  @end deftypefun
   
 @deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})  @deftypefun void mpz_realloc2 (mpz_t @var{integer}, unsigned long @var{n})
 Change the limb space allocation to @var{new_alloc} limbs.  This function is  Change the space allocated for @var{integer} to @var{n} bits.  The value in
 not normally called from user code, but it can be used to give memory back to  @var{integer} is preserved if it fits, or is set to 0 if not.
 the heap, or to increase the space of a variable to avoid repeated automatic  
 re-allocation.  This function can be used to increase the space for a variable in order to
   avoid repeated automatic reallocations, or to decrease it to give memory back
   to the heap.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, mp_size_t @var{fixed_num_bits})  @deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
 Allocate @strong{fixed} limb space for all @var{array_size} integers in  This is a special type of initialization.  @strong{Fixed} space of
 @var{integer_array}.  The fixed allocation for each integer in the array is  @var{fixed_num_bits} bits is allocated to each of the @var{array_size}
 enough to store @var{fixed_num_bits}.  If the fixed space will be insufficient  integers in @var{integer_array}.
 for storing the result of a subsequent calculation, the result is  
 unpredictable.  
   
 This function is useful for decreasing the working set for some algorithms  The space will not be automatically increased, unlike the normal
 that use large integer arrays.  @code{mpz_init}, but instead an application must ensure it's sufficient for
   any value stored.  The following space requirements apply to various
   functions,
   
 There is no way to de-allocate the storage allocated by this function.  @itemize @bullet
 Don't call @code{mpz_clear}!  @item
   @code{mpz_abs}, @code{mpz_neg}, @code{mpz_set}, @code{mpz_set_si} and
   @code{mpz_set_ui} need room for the value they store.
   
   @item
   @code{mpz_add}, @code{mpz_add_ui}, @code{mpz_sub} and @code{mpz_sub_ui} need
   room for the larger of the two operands, plus an extra
   @code{mp_bits_per_limb}.
   
   @item
   @code{mpz_mul}, @code{mpz_mul_ui} and @code{mpz_mul_ui} need room for the sum
   of the number of bits in their operands, but each rounded up to a multiple of
   @code{mp_bits_per_limb}.
   
   @item
   @code{mpz_swap} can be used between two array variables, but not between an
   array and a normal variable.
   @end itemize
   
   For other functions, or if in doubt, the suggestion is to calculate in a
   regular @code{mpz_init} variable and copy the result to an array variable with
   @code{mpz_set}.
   
   @code{mpz_array_init} can reduce memory usage in algorithms that need large
   arrays of integers, since it avoids allocating and reallocating lots of small
   memory blocks.  There is no way to free the storage allocated by this
   function.  Don't call @code{mpz_clear}!
 @end deftypefun  @end deftypefun
   
   @deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
   Change the space for @var{integer} to @var{new_alloc} limbs.  The value in
   @var{integer} is preserved if it fits, or is set to 0 if not.  The return
   value is not useful to applications and should be ignored.
   
   @code{mpz_realloc2} is the preferred way to accomplish allocation changes like
   this.  @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that
   @code{_mpz_realloc} takes the new size in limbs.
   @end deftypefun
   
   
 @node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions  @node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @subsection Assignment Functions  @section Assignment Functions
 @cindex Integer assignment functions  @cindex Integer assignment functions
   @cindex Assignment functions
   
 These functions assign new values to already initialized integers  These functions assign new values to already initialized integers
 (@pxref{Initializing Integers}).  (@pxref{Initializing Integers}).
Line 665  These functions assign new values to already initializ
Line 2675  These functions assign new values to already initializ
 @deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op})  @deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op})
 @deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op})  @deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op})
 Set the value of @var{rop} from @var{op}.  Set the value of @var{rop} from @var{op}.
   
   @code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to
   make it an integer.
 @end deftypefun  @end deftypefun
   
 @deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})  @deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
 Set the value of @var{rop} from @var{str}, a '\0'-terminated C string in base  Set the value of @var{rop} from @var{str}, a null-terminated C string in base
 @var{base}.  White space is allowed in the string, and is simply ignored.  The  @var{base}.  White space is allowed in the string, and is simply ignored.  The
 base may vary from 2 to 36.  If @var{base} is 0, the actual base is determined  base may vary from 2 to 36.  If @var{base} is 0, the actual base is determined
 from the leading characters: if the first two characters are `0x' or `0X',  from the leading characters: if the first two characters are ``0x'' or ``0X'',
 hexadecimal is assumed, otherwise if the first character is `0', octal is  hexadecimal is assumed, otherwise if the first character is ``0'', octal is
 assumed, otherwise decimal is assumed.  assumed, otherwise decimal is assumed.
   
 This function returns 0 if the entire string up to the '\0' is a valid  This function returns 0 if the entire string is a valid number in base
 number in base @var{base}.  Otherwise it returns @minus{}1.  @var{base}.  Otherwise it returns @minus{}1.
   
   [It turns out that it is not entirely true that this function ignores
   white-space.  It does ignore it between digits, but not after a minus sign or
   within or after ``0x''.  We are considering changing the definition of this
   function, making it fail when there is any white-space in the input, since
   that makes a lot of sense.  Send your opinion of this change to
   @email{bug-gmp@@gnu.org}.  Do you really want it to accept @nicode{"3 14"} as
   meaning 314 as it does now?]
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
   Swap the values @var{rop1} and @var{rop2} efficiently.
   @end deftypefun
   
   
 @node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions  @node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @subsection Combined Initialization and Assignment Functions  @section Combined Initialization and Assignment Functions
 @cindex Initialization and assignment functions  @cindex Initialization and assignment functions
   @cindex Integer init and assign
   
 For convenience, MP provides a parallel series of initialize-and-set functions  For convenience, GMP provides a parallel series of initialize-and-set functions
 which initialize the output and then store the value there.  These functions'  which initialize the output and then store the value there.  These functions'
 names have the form @code{mpz_init_set@dots{}}  names have the form @code{mpz_init_set@dots{}}
   
Line 726  an error occurs.  (I.e., you have to call @code{mpz_cl
Line 2752  an error occurs.  (I.e., you have to call @code{mpz_cl
 @end deftypefun  @end deftypefun
   
   
 @node Converting Integers,  Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions  @node Converting Integers, Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Conversion Functions  @section Conversion Functions
 @cindex Integer conversion functions  @cindex Integer conversion functions
 @cindex Conversion functions  @cindex Conversion functions
   
 This section describes functions for converting arbitrary precision integers  This section describes functions for converting GMP integers to standard C
 to standard C types.  Functions for converting @emph{to} arbitrary  types.  Functions for converting @emph{to} GMP integers are described in
 precision integers are described in @ref{Assigning Integers} and @ref{I/O of  @ref{Assigning Integers} and @ref{I/O of Integers}.
 Integers}.  
   
 @deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})  @deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})
 Return the least significant part from @var{op}.  This function combined  Return the value of @var{op} as an @code{unsigned long}.
 with @* @code{mpz_tdiv_q_2exp(@dots{}, @var{op}, CHAR_BIT*sizeof(unsigned  
 long int))} can be used to extract the limbs of an integer.  If @var{op} is too big to fit an @code{unsigned long} then just the least
   significant bits that do fit are returned.  The sign of @var{op} is ignored,
   only the absolute value is used.
 @end deftypefun  @end deftypefun
   
 @deftypefun {signed long int} mpz_get_si (mpz_t @var{op})  @deftypefun {signed long int} mpz_get_si (mpz_t @var{op})
Line 748  If @var{op} fits into a @code{signed long int} return 
Line 2775  If @var{op} fits into a @code{signed long int} return 
 Otherwise return the least significant part of @var{op}, with the same sign  Otherwise return the least significant part of @var{op}, with the same sign
 as @var{op}.  as @var{op}.
   
 If @var{op} is too large to fit in a @code{signed long int}, the returned  If @var{op} is too big to fit in a @code{signed long int}, the returned
 result is probably not very useful.  @c To find out if the value will fit, use  result is probably not very useful.  To find out if the value will fit, use
 @c the function @code{mpz_fits_si}.  the function @code{mpz_fits_slong_p}.
 @end deftypefun  @end deftypefun
   
 @deftypefun double mpz_get_d (mpz_t @var{op})  @deftypefun double mpz_get_d (mpz_t @var{op})
 Convert @var{op} to a double.  Convert @var{op} to a @code{double}.
 @end deftypefun  @end deftypefun
   
   @deftypefun double mpz_get_d_2exp (signed long int *@var{exp}, mpz_t @var{op})
   Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
   raised to @var{exp}}, with @math{0.5@le{}@GMPabs{@var{d}}<1}, is a good
   approximation to @var{op}.
   @end deftypefun
   
 @deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op})  @deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op})
 Convert @var{op} to a string of digits in base @var{base}.  The base may vary  Convert @var{op} to a string of digits in base @var{base}.  The base may vary
 from 2 to 36.  from 2 to 36.
   
 If @var{str} is NULL, space for the result string is allocated using the  If @var{str} is @code{NULL}, the result string is allocated using the current
 default allocation function, and a pointer to the string is returned.  allocation function (@pxref{Custom Allocation}).  The block will be
   @code{strlen(str)+1} bytes, that being exactly enough for the string and
   null-terminator.
   
 If @var{str} is not NULL, it should point to a block of storage enough large  If @var{str} is not @code{NULL}, it should point to a block of storage large
 for the result.  To find out the right amount of space to provide for  enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
 @var{str}, use @code{mpz_sizeinbase (@var{op}, @var{base}) + 2}.  The two  + 2}.  The two extra bytes are for a possible minus sign, and the
 extra bytes are for a possible minus sign, and for the terminating null  null-terminator.
 character.  
   A pointer to the result string is returned, being either the allocated block,
   or the given @var{str}.
 @end deftypefun  @end deftypefun
   
   @deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
   Return limb number @var{n} from @var{op}.  The sign of @var{op} is ignored,
   just the absolute value is used.  The least significant limb is number 0.
   
 @node Integer Arithmetic, Comparison Functions, Converting Integers, Integer Functions  @code{mpz_size} can be used to find how many limbs make up @var{op}.
   @code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to
   @code{mpz_size(@var{op})-1}.
   @end deftypefun
   
   
   @need 2000
   @node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Arithmetic Functions  @section Arithmetic Functions
 @cindex Integer arithmetic functions  @cindex Integer arithmetic functions
Line 780  character.
Line 2827  character.
   
 @deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
 @deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @math{@var{op1} + @var{op2}}.
 Set @var{rop} to @var{op1} + @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} + @var{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
 @deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
   @deftypefunx void mpz_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, mpz_t @var{op2})
 Set @var{rop} to @var{op1} @minus{} @var{op2}.  Set @var{rop} to @var{op1} @minus{} @var{op2}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefunx void mpz_mul_si (mpz_t @var{rop}, mpz_t @var{op1}, long int @var{op2})
 @deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
 Set @var{rop} to @var{op1} times @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} \times @var{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpz_addmul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
   Set @var{rop} to @math{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}.
   @end deftypefun
   
   @deftypefun void mpz_submul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefunx void mpz_submul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
   Set @var{rop} to @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
   @end deftypefun
   
 @deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  @cindex Bit shift left
 Set @var{rop} to @var{op1} times 2 raised to @var{op2}.  This operation can  Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
 also be defined as a left shift, @var{op2} steps.  @var{op2}}.  This operation can also be defined as a left shift by @var{op2}
 @end ifinfo  bits.
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} \times 2^{op2}$.  This operation can also be  
 defined as a left shift, @var{op2} steps.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})  @deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
Line 828  Set @var{rop} to @minus{}@var{op}.
Line 2867  Set @var{rop} to @minus{}@var{op}.
 Set @var{rop} to the absolute value of @var{op}.  Set @var{rop} to the absolute value of @var{op}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{op})  
 Set @var{rop} to @var{op}!, the factorial of @var{op}.  
 @end deftypefun  
   
 @subsection Division functions  @need 2000
   @node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions
   @section Division Functions
   @cindex Integer division functions
   @cindex Division functions
   
 Division is undefined if the divisor is zero, and passing a zero divisor to  Division is undefined if the divisor is zero.  Passing a zero divisor to the
 the divide or modulo functions, as well passing a zero mod argument to the  division or modulo functions (including the modular powering functions
 @code{mpz_powm} and @code{mpz_powm_ui} functions, will make these functions  @code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
 intentionally divide by zero.  This gives the user the possibility to handle  zero.  This lets a program handle arithmetic exceptions in these functions the
 arithmetic exceptions in these functions in the same manner as other  same way as for normal C @code{int} arithmetic.
 arithmetic exceptions.  
   
 There are three main groups of division functions:  @c  Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
 @itemize @bullet  @c  between each, and seem to let tex do a better job of page breaks than an
 @item  @c  @sp 1 in the middle of one big set.
 Functions that truncate the quotient towards 0.  The names of these  
 functions start with @code{mpz_tdiv}.  The @samp{t} in the name is short for  
 @samp{truncate}.  
 @item  
 Functions that round the quotient towards @minus{}infinity.  The names of  
 these routines start with @code{mpz_fdiv}.  The @samp{f} in the name is  
 short for @samp{floor}.  
 @item  
 Functions that round the quotient towards +infinity.  The names of  
 these routines start with @code{mpz_cdiv}.  The @samp{c} in the name is  
 short for @samp{ceil}.  
 @end itemize  
   
 For each rounding mode, there are a couple of variants.  Here @samp{q} means  @deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
 that the quotient is computed, while @samp{r} means that the remainder is  @deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 computed.  Functions that compute both the quotient and remainder have  @deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 @samp{qr} in the name.  @maybepagebreak
   @deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
 @deftypefun void mpz_tdiv_q (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
 @deftypefunx void mpz_tdiv_q_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
 Set @var{rop} to [@var{op1}/@var{op2}].  The quotient is truncated towards  @deftypefunx {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
 0.  @maybepagebreak
   @deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
   @deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_tdiv_r (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
 @deftypefunx void mpz_tdiv_r_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 Set @var{rop} to (@var{op1} - [@var{op1}/@var{op2}] * @var{op2}).  @deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 Unless the remainder is zero, it has the same sign as the dividend.  @maybepagebreak
   @deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
   @maybepagebreak
   @deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
   @deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_tdiv_qr (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
 @deftypefunx void mpz_tdiv_qr_ui (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 Divide @var{op1} by @var{op2} and put the quotient in @var{rop1} and the  @deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 remainder in @var{rop2}.  The quotient is rounded towards 0.  Unless the  @maybepagebreak
 remainder is zero, it has the same sign as the dividend.  @deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
   @deftypefunx {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
   @maybepagebreak
   @deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
   @deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
   @cindex Bit shift right
   
 If @var{rop1} and @var{rop2} are the same variable, the results are  @sp 1
 undefined.  Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
 @end deftypefun  @var{r}.  For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}.
   The rounding is in three styles, each suiting different applications.
   
 @deftypefun void mpz_fdiv_q (mpz_t @var{rop1}, mpz_t @var{op1}, mpz_t @var{op2})  @itemize @bullet
 @deftypefunx void mpz_fdiv_q_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @item
 @ifinfo  @code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
 Set @var{rop} to @var{op1}/@var{op2}.  The quotient is rounded towards  have the opposite sign to @var{d}.  The @code{c} stands for ``ceil''.
 @minus{}infinity.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $\lfloor@var{op1}/@var{op2}\rfloor$.  (I.e., round  
 the quotient towards $-\infty$.)  
 @end tex  
 @end iftex  
 @end deftypefun  
   
 @deftypefun void mpz_fdiv_r (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @item
 @deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
 Divide @var{op1} by @var{op2} and put the remainder in @var{rop}.  Unless  @var{r} will have the same sign as @var{d}.  The @code{f} stands for
 the remainder is zero, it has the same sign as the divisor.  ``floor''.
   
 For @code{mpz_fdiv_r_ui} the remainder is small enough to fit in an  @item
 @code{unsigned long int}, and is therefore returned.  @code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
 @end deftypefun  as @var{n}.  The @code{t} stands for ``truncate''.
   @end itemize
   
 @deftypefun void mpz_fdiv_qr (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, mpz_t @var{op2})  In all cases @var{q} and @var{r} will satisfy
 @deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, unsigned long int @var{op2})  @m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and
 Divide @var{op1} by @var{op2} and put the quotient in @var{rop1} and the  @var{r} will satisfy @math{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.
 remainder in @var{rop2}.  The quotient is rounded towards @minus{}infinity.  
 Unless the remainder is zero, it has the same sign as the divisor.  
   
 For @code{mpz_fdiv_qr_ui} the remainder is small enough to fit in an  The @code{q} functions calculate only the quotient, the @code{r} functions
 @code{unsigned long int}, and is therefore returned.  only the remainder, and the @code{qr} functions calculate both.  Note that for
   @code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or
   results will be unpredictable.
   
 If @var{rop1} and @var{rop2} are the same variable, the results are  For the @code{ui} variants the return value is the remainder, and in fact
 undefined.  returning the remainder is all the @code{div_ui} functions do.  For
 @end deftypefun  @code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
   return value is the absolute value of the remainder.
   
 @deftypefun {unsigned long int} mpz_fdiv_ui (mpz_t @var{op1}, unsigned long int @var{op2})  The @code{2exp} functions are right shifts and bit masks, but of course
 This function is similar to @code{mpz_fdiv_r_ui}, but the remainder is only  rounding the same as the other functions.  For positive @var{n} both
 returned; it is not stored anywhere.  @code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp} are simple bitwise right
   shifts.  For negative @var{n}, @code{mpz_fdiv_q_2exp} is effectively an
   arithmetic right shift treating @var{n} as twos complement the same as the
   bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp} effectively
   treats @var{n} as sign and magnitude.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_cdiv_q (mpz_t @var{rop1}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
 @deftypefunx void mpz_cdiv_q_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
 @ifinfo  Set @var{r} to @var{n} @code{mod} @var{d}.  The sign of the divisor is
 Set @var{rop} to @var{op1}/@var{op2}.  The quotient is rounded towards  ignored; the result is always non-negative.
 +infinity.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $\lceil@var{op1}/@var{op2}\rceil$.  (I.e., round the  
 quotient towards $+\infty$.)  
 @end tex  
 @end iftex  
 @end deftypefun  
   
 @deftypefun void mpz_cdiv_r (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
 @deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  remainder as well as setting @var{r}.  See @code{mpz_fdiv_ui} above if only
 Divide @var{op1} by @var{op2} and put the remainder in @var{rop}.  Unless  the return value is wanted.
 the remainder is zero, it has the opposite sign as the divisor.  
   
 For @code{mpz_cdiv_r_ui} the negated remainder is small enough to fit in an  
 @code{unsigned long int}, and it is therefore returned.  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_cdiv_qr (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
 @deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long @var{d})
 Divide @var{op1} by @var{op2} and put the quotient in @var{rop1} and the  @cindex Exact division functions
 remainder in @var{rop2}.  The quotient is rounded towards +infinity.  Unless  Set @var{q} to @var{n}/@var{d}.  These functions produce correct results only
 the remainder is zero, it has the opposite sign as the divisor.  when it is known in advance that @var{d} divides @var{n}.
   
 For @code{mpz_cdiv_qr_ui} the negated remainder is small enough to fit in an  These routines are much faster than the other division functions, and are the
 @code{unsigned long int}, and it is therefore returned.  best choice when exact division is known to occur, for example reducing a
   
 If @var{rop1} and @var{rop2} are the same variable, the results are  
 undefined.  
 @end deftypefun  
   
 @deftypefun {unsigned long int} mpz_cdiv_ui (mpz_t @var{op1}, unsigned long int @var{op2})  
 Return the negated remainder, similar to @code{mpz_cdiv_r_ui}.  (The  
 difference is that this function doesn't store the remainder anywhere.)  
 @end deftypefun  
   
 @deftypefun void mpz_mod (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  
 @deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  
 Set @var{rop} to @var{op1} @code{mod} @var{op2}.  The sign of the divisor is  
 ignored, and the result is always non-negative.  
   
 For @code{mpz_mod_ui} the remainder is small enough to fit in an  
 @code{unsigned long int}, and is therefore returned.  
 @end deftypefun  
   
 @deftypefun void mpz_divexact (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  
 Set @var{rop} to @var{op1}/@var{op2}.  This function produces correct  
 results only when it is known in advance that @var{op2} divides  
 @var{op1}.  
   
 Since mpz_divexact is much faster than any of the other routines that produce  
 the quotient (@pxref{References} Jebelean), it is the best choice for  
 instances in which exact division is known to occur, such as reducing a  
 rational to lowest terms.  rational to lowest terms.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_tdiv_q_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d})
 @ifinfo  @deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, unsigned long int @var{d})
 Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.  The quotient is  @deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, unsigned long int @var{b})
 rounded towards 0.  Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
 @end ifinfo  @code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1}/2^{op2}$.  The quotient is rounded towards 0.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_tdiv_r_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d})
 @ifinfo  @deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d})
 Divide @var{op1} by (2 raised to @var{op2}) and put the remainder in  @deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, unsigned long int @var{b})
 @var{rop}.  Unless it is zero, @var{rop} will have the same sign as @var{op1}.  Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
 @end ifinfo  case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.
 @iftex  
 @tex  
 Divide @var{op1} by $2^{op2}$ and put the remainder in @var{rop}.  Unless it is  
 zero, @var{rop} will have the same sign as @var{op1}.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_fdiv_q_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  
 @ifinfo  
 Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.  The quotient is  
 rounded towards @minus{}infinity.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $\lfloor@var{op1}/2^{op2}\rfloor$.  The quotient is rounded  
 towards $-\infty$.  
 @end tex  
 @end iftex  
 @end deftypefun  
   
 @deftypefun void mpz_fdiv_r_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @need 2000
 @ifinfo  @node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
 Divide @var{op1} by (2 raised to @var{op2}) and put the remainder in  @section Exponentiation Functions
 @var{rop}.  The sign of @var{rop} will always be positive.  @cindex Integer exponentiation functions
 @end ifinfo  @cindex Exponentiation functions
 @iftex  @cindex Powering functions
 @tex  
 Divide @var{op1} by $2^{op2}$ and put the remainder in @var{rop}.  The sign of  
 @var{rop} will always be positive.  
 @end tex  
 @end iftex  
   
 This operation can also be defined as masking of the @var{op2} least  
 significant bits.  
 @end deftypefun  
   
 @subsection Exponentialization Functions  
   
 @deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})  @deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
 @deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod})  @deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod})
 Set @var{rop} to (@var{base} raised to @var{exp}) @code{mod} @var{mod}.  If  Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
 @var{exp} is negative, the result is undefined.  modulo @var{mod}}.
   
   Negative @var{exp} is supported if an inverse @math{@var{base}^@W{-1} @bmod
   @var{mod}} exists (see @code{mpz_invert} in @ref{Number Theoretic Functions}).
   If an inverse doesn't exist then a divide by zero is raised.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp})  @deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp})
 @deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp})  @deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp})
 Set @var{rop} to @var{base} raised to @var{exp}.  Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}.  The case
 @ifinfo  @math{0^0} yields 1.
 The case of 0^0 yields 1.  
 @end ifinfo  
 @iftex  
 @tex  
 The case of $0^0$ yields 1.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @subsection Square Root Functions  
   
   @need 2000
   @node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions
   @section Root Extraction Functions
   @cindex Integer root functions
   @cindex Root extraction functions
   
   @deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n})
   Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor@C{},} the truncated integer
   part of the @var{n}th root of @var{op}.  Return non-zero if the computation
   was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power.
   @end deftypefun
   
 @deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})  @deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})
 @ifinfo  Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor@C{},} the truncated
 Set @var{rop} to the truncated integer part of the square root of  integer part of the square root of @var{op}.
 @var{op}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $\lfloor\sqrt{@var{op}}\rfloor$, the truncated integer  
 part of the square root of @var{op}.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})  @deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
 @ifinfo  Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
 Set @var{rop1} to the truncated integer part of the square root of @var{op},  of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{rop2} to the
 like @code{mpz_sqrt}.  Set @var{rop2} to  remainder @m{(@var{op} - @var{rop1}^2),
 @var{op}@minus{}@var{rop1}*@var{rop1},  @var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
 @end ifinfo  perfect square.
 @iftex  
 @tex  
 Set @var{rop1} to $\lfloor\sqrt{@var{op}}\rfloor$, like @code{mpz_sqrt}.  
 Set @var{rop2} to $(@var{op} - @var{rop1}^2)$,  
 @end tex  
 @end iftex  
 (i.e., zero if @var{op} is a perfect square).  
   
 If @var{rop1} and @var{rop2} are the same variable, the results are  If @var{rop1} and @var{rop2} are the same variable, the results are
 undefined.  undefined.
 @end deftypefun  @end deftypefun
   
   @deftypefun int mpz_perfect_power_p (mpz_t @var{op})
   Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
   @m{a,@var{a}} and @m{b,@var{b}}, with @m{b>1, @var{b}>1}, such that
   @m{@var{op}=a^b, @var{op} equals @var{a} raised to the power @var{b}}.
   
   Under this definition both 0 and 1 are considered to be perfect powers.
   Negative values of @var{op} are accepted, but of course can only be odd
   perfect powers.
   @end deftypefun
   
 @deftypefun int mpz_perfect_square_p (mpz_t @var{op})  @deftypefun int mpz_perfect_square_p (mpz_t @var{op})
 Return non-zero if @var{op} is a perfect square, i.e., if the square root of  Return non-zero if @var{op} is a perfect square, i.e., if the square root of
 @var{op} is an integer.  Return zero otherwise.  @var{op} is an integer.  Under this definition both 0 and 1 are considered to
   be perfect squares.
 @end deftypefun  @end deftypefun
   
 @subsection Number Theoretic Functions  
   
 @deftypefun int mpz_probab_prime_p (mpz_t @var{op}, int @var{reps})  @need 2000
 @ifinfo  @node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions
 If this function returns 0, @var{op} is definitely not prime.  If it returns  @section Number Theoretic Functions
 1, then @var{op} is `probably' prime.  The probability of a false positive is  @cindex Number theoretic functions
 (1/4)**@var{reps}.  
 @end ifinfo  
 @iftex  
 @tex  
 If this function returns 0, @var{op} is definitely not prime.  If it returns  
 1, then @var{op} is `probably' prime.  The probability of a false positive is  
 $(1/4)^{{reps}}$.  
 @end tex  
 @end iftex  
 A reasonable value of reps is 25.  
   
 An implementation of the probabilistic primality test found in Seminumerical  @deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
 Algorithms (@pxref{References} Knuth).  @cindex Prime testing functions
   Determine whether @var{n} is prime.  Return 2 if @var{n} is definitely prime,
   return 1 if @var{n} is probably prime (without being certain), or return 0 if
   @var{n} is definitely composite.
   
   This function does some trial divisions, then some Miller-Rabin probabilistic
   primality tests.  @var{reps} controls how many such tests are done, 5 to 10 is
   a reasonable number, more will reduce the chances of a composite being
   returned as ``probably prime''.
   
   Miller-Rabin and similar tests can be more properly called compositeness
   tests.  Numbers which fail are known to be composite but those which pass
   might be prime or might be composite.  Only a few composites pass, hence those
   which pass are considered probably prime.
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op})
   Set @var{rop} to the next prime greater than @var{op}.
   
   This function uses a probabilistic algorithm to identify primes.  For
   practical purposes it's adequate, the chance of a composite passing will be
   extremely small.
   @end deftypefun
   
   @c mpz_prime_p not implemented as of gmp 3.0.
   
   @c @deftypefun int mpz_prime_p (mpz_t @var{n})
   @c Return non-zero if @var{n} is prime and zero if @var{n} is a non-prime.
   @c This function is far slower than @code{mpz_probab_prime_p}, but then it
   @c never returns non-zero for composite numbers.
   
   @c (For practical purposes, using @code{mpz_probab_prime_p} is adequate.
   @c The likelihood of a programming error or hardware malfunction is orders
   @c of magnitudes greater than the likelihood for a composite to pass as a
   @c prime, if the @var{reps} argument is in the suggested range.)
   @c @end deftypefun
   
 @deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @cindex Greatest common divisor functions
 Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.  Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.
   The result is always positive even if one or both input operands
   are negative.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
 Compute the greatest common divisor of @var{op1} and @var{op2}.  If  Compute the greatest common divisor of @var{op1} and @var{op2}.  If
 @var{rop} is not NULL, store the result there.  @var{rop} is not @code{NULL}, store the result there.
   
 If the result is small enough to fit in an @code{unsigned long int}, it is  If the result is small enough to fit in an @code{unsigned long int}, it is
 returned.  If the result does not fit, 0 is returned, and the result is equal  returned.  If the result does not fit, 0 is returned, and the result is equal
Line 1135  is non-zero.
Line 3132  is non-zero.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})  @deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})
 Compute @var{g}, @var{s}, and @var{t}, such that @var{a}@var{s} +  @cindex Extended GCD
 @var{b}@var{t} = @var{g} = @code{gcd} (@var{a}, @var{b}).  If @var{t} is  Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
 NULL, that argument is not computed.  addition set @var{s} and @var{t} to coefficients satisfying
   @math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}.
   @var{g} is always positive, even if one or both of @var{a} and @var{b} are
   negative.
   
   If @var{t} is @code{NULL} then that value is not computed.
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long @var{op2})
   @cindex Least common multiple functions
   Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
   @var{rop} is always positive, irrespective of the signs of @var{op1} and
   @var{op2}.  @var{rop} will be zero if either @var{op1} or @var{op2} is zero.
   @end deftypefun
   
 @deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
   @cindex Modular inverse functions
 Compute the inverse of @var{op1} modulo @var{op2} and put the result in  Compute the inverse of @var{op1} modulo @var{op2} and put the result in
 @var{rop}.  Return non-zero if an inverse exist, zero otherwise.  When the  @var{rop}.  If the inverse exists, the return value is non-zero and @var{rop}
 function returns zero, do not assume anything about the value in @var{rop}.  will satisfy @math{0 @le{} @var{rop} < @var{op2}}.  If an inverse doesn't exist
   the return value is zero and @var{rop} is undefined.
 @end deftypefun  @end deftypefun
   
 @deftypefun int mpz_jacobi (mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b})
 @deftypefunx int mpz_legendre (mpz_t @var{op1}, mpz_t @var{op2})  @cindex Jacobi symbol functions
 Compute the Jacobi and Legendre symbols, respectively.  Calculate the Jacobi symbol @m{\left(a \over b\right),
   (@var{a}/@var{b})}.  This is defined only for @var{b} odd.
 @end deftypefun  @end deftypefun
   
 @need 2000  @deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p})
 @node Comparison Functions, Integer Logic and Bit Fiddling, Integer Arithmetic, Integer Functions  Calculate the Legendre symbol @m{\left(a \over p\right),
 @comment  node-name,  next,  previous,  up  (@var{a}/@var{p})}.  This is defined only for @var{p} an odd positive
 @section Comparison Functions  prime, and for such @var{p} it's identical to the Jacobi symbol.
   @end deftypefun
   
 @deftypefun int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b})
 @ifinfo  @deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b})
 Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >  @deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b})
 @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <  @deftypefunx int mpz_si_kronecker (long @var{a}, mpz_t @var{b})
 @var{op2}.  @deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b})
 @end ifinfo  @cindex Kronecker symbol functions
 @iftex  Calculate the Jacobi symbol @m{\left(a \over b\right),
 @tex  (@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
 Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >  2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or
 @var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}  @m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even.
 < @var{op2}$.  
 @end tex  When @var{b} is odd the Jacobi symbol and Kronecker symbol are
 @end iftex  identical, so @code{mpz_kronecker_ui} etc can be used for mixed
   precision Jacobi symbols too.
   
   For more information see Henri Cohen section 1.4.2 (@pxref{References}),
   or any number theory textbook.  See also the example program
   @file{demos/qcn.c} which uses @code{mpz_kronecker_ui}.
 @end deftypefun  @end deftypefun
   
 @deftypefn Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})  @deftypefun {unsigned long int} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f})
   Remove all occurrences of the factor @var{f} from @var{op} and store the
   result in @var{rop}.  The return value is how many such occurrences were
   removed.
   @end deftypefun
   
   @deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{op})
   @cindex Factorial functions
   Set @var{rop} to @var{op}!, the factorial of @var{op}.
   @end deftypefun
   
   @deftypefun void mpz_bin_ui (mpz_t @var{rop}, mpz_t @var{n}, unsigned long int @var{k})
   @deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, unsigned long int @var{n}, @w{unsigned long int @var{k}})
   @cindex Binomial coefficient functions
   Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over
   @var{k}} and store the result in @var{rop}.  Negative values of @var{n} are
   supported by @code{mpz_bin_ui}, using the identity
   @m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
   bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6
   part G.
   @end deftypefun
   
   @deftypefun void mpz_fib_ui (mpz_t @var{fn}, unsigned long int @var{n})
   @deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, unsigned long int @var{n})
   @cindex Fibonacci sequence functions
   @code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci
   number.  @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to
   @m{F_{n-1},F[n-1]}.
   
   These functions are designed for calculating isolated Fibonacci numbers.  When
   a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and
   iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
   similar.
   @end deftypefun
   
   @deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, unsigned long int @var{n})
   @deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, unsigned long int @var{n})
   @cindex Lucas number functions
   @code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas
   number.  @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1}
   to @m{L_{n-1},L[n-1]}.
   
   These functions are designed for calculating isolated Lucas numbers.  When a
   sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and
   iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
   similar.
   
   The Fibonacci numbers and Lucas numbers are related sequences, so it's never
   necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}.  The
   formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers
   Algorithm}, the reverse is straightforward too.
   @end deftypefun
   
   
   @node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions
   @comment  node-name,  next,  previous,  up
   @section Comparison Functions
   @cindex Integer comparison functions
   @cindex Comparison functions
   
   @deftypefn Function int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefnx Function int mpz_cmp_d (mpz_t @var{op1}, double @var{op2})
 @deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2})  @deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2})
 @ifinfo  @deftypefnx Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})
 Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >  Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
 @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <  @var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
 @var{op2}.  @math{@var{op1} < @var{op2}}.
 @end ifinfo  
 @iftex  
 @tex  
 Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >  
 @var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}  
 < @var{op2}$.  
 @end tex  
 @end iftex  
   
 These functions are actually implemented as macros.  They evaluate their  Note that @code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate
 arguments multiple times.  their arguments more than once.
 @end deftypefn  @end deftypefn
   
   @deftypefn Function int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2})
   @deftypefnx Function int mpz_cmpabs_d (mpz_t @var{op1}, double @var{op2})
   @deftypefnx Function int mpz_cmpabs_ui (mpz_t @var{op1}, unsigned long int @var{op2})
   Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
   value if @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
   @math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
   @math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.
   
   Note that @code{mpz_cmpabs_si} is a macro and will evaluate its arguments more
   than once.
   @end deftypefn
   
 @deftypefn Macro int mpz_sgn (mpz_t @var{op})  @deftypefn Macro int mpz_sgn (mpz_t @var{op})
 @ifinfo  @cindex Sign tests
 Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.  @cindex Integer sign tests
 @end ifinfo  Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
 @iftex  @math{-1} if @math{@var{op} < 0}.
 @tex  
 Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.  
 @end tex  
 @end iftex  
   
 This function is actually implemented as a macro.  It evaluates its  This function is actually implemented as a macro.  It evaluates its argument
 arguments multiple times.  multiple times.
 @end deftypefn  @end deftypefn
   
 @node Integer Logic and Bit Fiddling, I/O of Integers, Comparison Functions, Integer Functions  
   @node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Logical and Bit Manipulation Functions  @section Logical and Bit Manipulation Functions
 @cindex Logical functions  @cindex Logical functions
 @cindex Bit manipulation functions  @cindex Bit manipulation functions
   @cindex Integer bit manipulation functions
   
 These functions behave as if two's complement arithmetic were used (although  These functions behave as if twos complement arithmetic were used (although
 sign-magnitude is used by the actual implementation).  sign-magnitude is the actual implementation).  The least significant bit is
   number 0.
   
 @deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
 Set @var{rop} to @var{op1} logical-and @var{op2}.  Set @var{rop} to @var{op1} logical-and @var{op2}.
Line 1221  Set @var{rop} to @var{op1} logical-and @var{op2}.
Line 3304  Set @var{rop} to @var{op1} logical-and @var{op2}.
 Set @var{rop} to @var{op1} inclusive-or @var{op2}.  Set @var{rop} to @var{op1} inclusive-or @var{op2}.
 @end deftypefun  @end deftypefun
   
 @c @deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
 @c Set @var{rop} to @var{op1} exclusive-or @var{op2}.  Set @var{rop} to @var{op1} exclusive-or @var{op2}.
 @c @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op})  @deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op})
 Set @var{rop} to the one's complement of @var{op}.  Set @var{rop} to the one's complement of @var{op}.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpz_popcount (mpz_t @var{op})  @deftypefun {unsigned long int} mpz_popcount (mpz_t @var{op})
 For non-negative numbers, return the population count of @var{op}.  For  If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is
 negative numbers, return the largest possible value (@var{MAX_ULONG}).  the number of 1 bits in the binary representation.  If @math{@var{op}<0}, the
   number of 1s is infinite, and the return value is @var{MAX_ULONG}, the largest
   possible @code{unsigned long}.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})  @deftypefun {unsigned long int} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
 If @var{op1} and @var{op2} are both non-negative, return the hamming distance  If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return
 between the two operands.  Otherwise, return the largest possible value  the hamming distance between the two operands, which is the number of bit
 (@var{MAX_ULONG}).  positions where @var{op1} and @var{op2} have different bit values.  If one
   operand is @math{@ge{}0} and the other @math{<0} then the number of bits
 It is possible to extend this function to return a useful value when the  different is infinite, and the return value is @var{MAX_ULONG}, the largest
 operands are both negative, but the current implementation returns  possible @code{unsigned long}.
 @var{MAX_ULONG} in this case.  @strong{Do not depend on this behavior, since  
 it will change in future versions of the library.}  
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpz_scan0 (mpz_t @var{op}, unsigned long int @var{starting_bit})  @deftypefun {unsigned long int} mpz_scan0 (mpz_t @var{op}, unsigned long int @var{starting_bit})
 Scan @var{op}, starting with bit @var{starting_bit}, towards more significant  @deftypefunx {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})
 bits, until the first clear bit is found.  Return the index of the found bit.  Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
 @end deftypefun  bits, until the first 0 or 1 bit (respectively) is found.  Return the index of
   the found bit.
   
 @deftypefun {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})  If the bit at @var{starting_bit} is already what's sought, then
 Scan @var{op}, starting with bit @var{starting_bit}, towards more significant  @var{starting_bit} is returned.
 bits, until the first set bit is found.  Return the index of the found bit.  
   If there's no bit found, then @var{MAX_ULONG} is returned.  This will happen
   in @code{mpz_scan0} past the end of a positive number, or @code{mpz_scan1}
   past the end of a negative.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_setbit (mpz_t @var{rop}, unsigned long int @var{bit_index})  @deftypefun void mpz_setbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
 Set bit @var{bit_index} in @var{op1}.  Set bit @var{bit_index} in @var{rop}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_clrbit (mpz_t @var{rop}, unsigned long int @var{bit_index})  @deftypefun void mpz_clrbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
 Clear bit @var{bit_index} in @var{op1}.  Clear bit @var{bit_index} in @var{rop}.
 @end deftypefun  @end deftypefun
   
 @node I/O of Integers, Miscellaneous Integer Functions, Integer Logic and Bit Fiddling, Integer Functions  @deftypefun int mpz_tstbit (mpz_t @var{op}, unsigned long int @var{bit_index})
   Test bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
   @end deftypefun
   
   @node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Input and Output Functions  @section Input and Output Functions
 @cindex Integer input and output functions  @cindex Integer input and output functions
Line 1272  Clear bit @var{bit_index} in @var{op1}.
Line 3363  Clear bit @var{bit_index} in @var{op1}.
 @cindex I/O functions  @cindex I/O functions
   
 Functions that perform input from a stdio stream, and functions that output to  Functions that perform input from a stdio stream, and functions that output to
 a stdio stream.  Passing a NULL pointer for a @var{stream} argument to any of  a stdio stream.  Passing a @code{NULL} pointer for a @var{stream} argument to any of
 these functions will make them read from @code{stdin} and write to  these functions will make them read from @code{stdin} and write to
 @code{stdout}, respectively.  @code{stdout}, respectively.
   
Line 1324  machines.
Line 3415  machines.
   
   
 @need 2000  @need 2000
 @node Miscellaneous Integer Functions,, I/O of Integers, Integer Functions  @node Integer Random Numbers, Integer Import and Export, I/O of Integers, Integer Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Miscellaneous Functions  @section Random Number Functions
 @cindex Miscellaneous integer functions  @cindex Integer random number functions
   @cindex Random number functions
   
   The random number functions of GMP come in two groups; older function
   that rely on a global state, and newer functions that accept a state
   parameter that is read and modified.  Please see the @ref{Random Number
   Functions} for more information on how to use and not to use random
   number functions.
   
   @deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
   Generate a uniformly distributed random integer in the range 0 to @m{2^n-1,
   2^@var{n}@minus{}1}, inclusive.
   
   The variable @var{state} must be initialized by calling one of the
   @code{gmp_randinit} functions (@ref{Random State Initialization}) before
   invoking this function.
   @end deftypefun
   
   @deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, mpz_t @var{n})
   Generate a uniform random integer in the range 0 to @math{@var{n}-1},
   inclusive.
   
   The variable @var{state} must be initialized by calling one of the
   @code{gmp_randinit} functions (@ref{Random State Initialization})
   before invoking this function.
   @end deftypefun
   
   @deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
   Generate a random integer with long strings of zeros and ones in the
   binary representation.  Useful for testing functions and algorithms,
   since this kind of random numbers have proven to be more likely to
   trigger corner-case bugs.  The random number will be in the range
   0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive.
   
   The variable @var{state} must be initialized by calling one of the
   @code{gmp_randinit} functions (@ref{Random State Initialization})
   before invoking this function.
   @end deftypefun
   
 @deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size})  @deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size})
 Generate a random integer of at most @var{max_size} limbs.  The generated  Generate a random integer of at most @var{max_size} limbs.  The generated
 random number doesn't satisfy any particular requirements of randomness.  random number doesn't satisfy any particular requirements of randomness.
 Negative random numbers are generated when @var{max_size} is negative.  Negative random numbers are generated when @var{max_size} is negative.
   
   This function is obsolete.  Use @code{mpz_urandomb} or
   @code{mpz_urandomm} instead.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size})  @deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size})
Line 1341  of zeros and ones in the binary representation.  Usefu
Line 3472  of zeros and ones in the binary representation.  Usefu
 and algorithms, since this kind of random numbers have proven to be more  and algorithms, since this kind of random numbers have proven to be more
 likely to trigger corner-case bugs.  Negative random numbers are generated  likely to trigger corner-case bugs.  Negative random numbers are generated
 when @var{max_size} is negative.  when @var{max_size} is negative.
   
   This function is obsolete.  Use @code{mpz_rrandomb} instead.
 @end deftypefun  @end deftypefun
   
   
   @node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions
   @section Integer Import and Export
   
   @code{mpz_t} variables can be converted to and from arbitrary words of binary
   data with the following functions.
   
   @deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, int @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op})
   @cindex Integer import
   @cindex Import
   Set @var{rop} from an array of word data at @var{op}.
   
   The parameters specify the format of the data.  @var{count} many words are
   read, each @var{size} bytes.  @var{order} can be 1 for most significant word
   first or -1 for least significant first.  Within each word @var{endian} can be
   1 for most significant byte first, -1 for least significant first, or 0 for
   the native endianness of the host CPU.  The most significant @var{nails} bits
   of each word are skipped, this can be 0 to use the full words.
   
   There are no data alignment restrictions on @var{op}, any address is allowed.
   
   Here's an example converting an array of @code{unsigned long} data, most
   significant element first and host byte order within each value.
   
   @example
   unsigned long  a[20];
   mpz_t          z;
   mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
   @end example
   
   This example assumes the full @code{sizeof} bytes are used for data in the
   given type, which is usually true, and certainly true for @code{unsigned long}
   everywhere we know of.  However on Cray vector systems it may be noted that
   @code{short} and @code{int} are always stored in 8 bytes (and with
   @code{sizeof} indicating that) but use only 32 or 46 bits.  The @var{nails}
   feature can account for this, by passing for instance
   @code{8*sizeof(int)-INT_BIT}.
   @end deftypefun
   
   @deftypefun void *mpz_export (void *@var{rop}, size_t *@var{count}, int @var{order}, int @var{size}, int @var{endian}, size_t @var{nails}, mpz_t @var{op})
   @cindex Integer export
   @cindex Export
   Fill @var{rop} with word data from @var{op}.
   
   The parameters specify the format of the data produced.  Each word will be
   @var{size} bytes and @var{order} can be 1 for most significant word first or
   -1 for least significant first.  Within each word @var{endian} can be 1 for
   most significant byte first, -1 for least significant first, or 0 for the
   native endianness of the host CPU.  The most significant @var{nails} bits of
   each word are unused and set to zero, this can be 0 to produce full words.
   
   The number of words produced is written to @code{*@var{count}}.  @var{rop}
   must have enough space for the data, or if @var{rop} is @code{NULL} then a
   result array of the necessary size is allocated using the current GMP
   allocation function (@pxref{Custom Allocation}).  In either case the return
   value is the destination used, @var{rop} or the allocated block.
   
   If @var{op} is non-zero then the most significant word produced will be
   non-zero.  If @var{op} is zero then the count returned will be zero and
   nothing written to @var{rop}.  If @var{rop} is @code{NULL} in this case, no
   block is allocated, just @code{NULL} is returned.
   
   There are no data alignment restrictions on @var{rop}, any address is allowed.
   The sign of @var{op} is ignored, just the absolute value is used.
   
   When an application is allocating space itself the required size can be
   determined with a calculation like the following.  Since @code{mpz_sizeinbase}
   always returns at least 1, @code{count} here will be at least one, which
   avoids any portability problems with @code{malloc(0)}, though if @code{z} is
   zero no space at all is actually needed.
   
   @example
   numb = 8*size - nail;
   count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
   p = malloc (count * size);
   @end example
   @end deftypefun
   
   
   @need 2000
   @node Miscellaneous Integer Functions,  , Integer Import and Export, Integer Functions
   @comment  node-name,  next,  previous,  up
   @section Miscellaneous Functions
   @cindex Miscellaneous integer functions
   @cindex Integer miscellaneous functions
   
   @deftypefun int mpz_fits_ulong_p (mpz_t @var{op})
   @deftypefunx int mpz_fits_slong_p (mpz_t @var{op})
   @deftypefunx int mpz_fits_uint_p (mpz_t @var{op})
   @deftypefunx int mpz_fits_sint_p (mpz_t @var{op})
   @deftypefunx int mpz_fits_ushort_p (mpz_t @var{op})
   @deftypefunx int mpz_fits_sshort_p (mpz_t @var{op})
   Return non-zero iff the value of @var{op} fits in an @code{unsigned long int},
   @code{signed long int}, @code{unsigned int}, @code{signed int}, @code{unsigned
   short int}, or @code{signed short int}, respectively.  Otherwise, return zero.
   @end deftypefun
   
   @deftypefn Macro int mpz_odd_p (mpz_t @var{op})
   @deftypefnx Macro int mpz_even_p (mpz_t @var{op})
   Determine whether @var{op} is odd or even, respectively.  Return non-zero if
   yes, zero if no.  These macros evaluate their argument more than once.
   @end deftypefn
   
 @deftypefun size_t mpz_size (mpz_t @var{op})  @deftypefun size_t mpz_size (mpz_t @var{op})
 Return the size of @var{op} measured in number of limbs.  If @var{op} is zero,  Return the size of @var{op} measured in number of limbs.  If @var{op} is zero,
 the returned value will be zero.  the returned value will be zero.
 @c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)  @c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)
   
 @strong{This function is obsolete.  It will disappear from future MP  
 releases.}  
 @end deftypefun  @end deftypefun
   
 @deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base})  @deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base})
 Return the size of @var{op} measured in number of digits in base @var{base}.  Return the size of @var{op} measured in number of digits in base @var{base}.
 The base may vary from 2 to 36.  The returned value will be exact or 1 too  The base may vary from 2 to 36.  The sign of @var{op} is ignored, just the
 big.  If @var{base} is a power of 2, the returned value will always be exact.  absolute value is used.  The result will be exact or 1 too big.  If @var{base}
   is a power of 2, the result will always be exact.  If @var{op} is zero the
   return value is always 1.
   
 This function is useful in order to allocate the right amount of space before  This function is useful in order to allocate the right amount of space before
 converting @var{op} to a string.  The right amount of allocation is normally  converting @var{op} to a string.  The right amount of allocation is normally
 two more than the value returned by @code{mpz_sizeinbase} (one extra for a  two more than the value returned by @code{mpz_sizeinbase} (one extra for a
 minus sign and one for the terminating '\0').  minus sign and one for the null-terminator).
 @end deftypefun  @end deftypefun
   
   
Line 1369  minus sign and one for the terminating '\0').
Line 3604  minus sign and one for the terminating '\0').
 @chapter Rational Number Functions  @chapter Rational Number Functions
 @cindex Rational number functions  @cindex Rational number functions
   
 This chapter describes the MP functions for performing arithmetic on rational  This chapter describes the GMP functions for performing arithmetic on rational
 numbers.  These functions start with the prefix @code{mpq_}.  numbers.  These functions start with the prefix @code{mpq_}.
   
 Rational numbers are stored in objects of type @code{mpq_t}.  Rational numbers are stored in objects of type @code{mpq_t}.
Line 1381  Zero has the unique representation 0/1.
Line 3616  Zero has the unique representation 0/1.
   
 Pure assignment functions do not canonicalize the assigned variable.  It is  Pure assignment functions do not canonicalize the assigned variable.  It is
 the responsibility of the user to canonicalize the assigned variable before  the responsibility of the user to canonicalize the assigned variable before
 any arithmetic operations are performed on that variable.  @strong{Note that  any arithmetic operations are performed on that variable.
 this is an incompatible change from version 1 of the library.}  
   
 @deftypefun void mpq_canonicalize (mpq_t @var{op})  @deftypefun void mpq_canonicalize (mpq_t @var{op})
 Remove any factors that are common to the numerator and denominator of  Remove any factors that are common to the numerator and denominator of
Line 1390  Remove any factors that are common to the numerator an
Line 3624  Remove any factors that are common to the numerator an
 @end deftypefun  @end deftypefun
   
 @menu  @menu
 * Initializing Rationals::  * Initializing Rationals::
 * Assigning Rationals::  * Rational Conversions::
 * Simultaneous Integer Init & Assign::  * Rational Arithmetic::
 * Comparing Rationals::  * Comparing Rationals::
 * Applying Integer Functions::  * Applying Integer Functions::
 * Miscellaneous Rational Functions::  * I/O of Rationals::
 @end menu  @end menu
   
 @node Initializing Rationals, Assigning Rationals, Rational Number Functions, Rational Number Functions  @node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Initialization and Assignment Functions  @section Initialization and Assignment Functions
   @cindex Initialization and assignment functions
   @cindex Rational init and assign
   
 @deftypefun void mpq_init (mpq_t @var{dest_rational})  @deftypefun void mpq_init (mpq_t @var{dest_rational})
 Initialize @var{dest_rational} and set it to 0/1.  Each variable should  Initialize @var{dest_rational} and set it to 0/1.  Each variable should
Line 1425  Set the value of @var{rop} to @var{op1}/@var{op2}.  No
Line 3661  Set the value of @var{rop} to @var{op1}/@var{op2}.  No
 @code{mpq_canonicalize} before any operations are performed on @var{rop}.  @code{mpq_canonicalize} before any operations are performed on @var{rop}.
 @end deftypefun  @end deftypefun
   
 @node Assigning Rationals, Comparing Rationals, Initializing Rationals, Rational Number Functions  @deftypefun int mpq_set_str (mpq_t @var{rop}, char *@var{str}, int @var{base})
   Set @var{rop} from a null-terminated string @var{str} in the given @var{base}.
   
   The string can be an integer like ``41'' or a fraction like ``41/152''.  The
   fraction must be in canonical form (@pxref{Rational Number Functions}), or if
   not then @code{mpq_canonicalize} must be called.
   
   The numerator and optional denominator are parsed the same as in
   @code{mpz_set_str} (@pxref{Assigning Integers}).  White space is allowed in
   the string, and is simply ignored.  The @var{base} can vary from 2 to 36, or
   if @var{base} is 0 then the leading characters are used: @code{0x} for hex,
   @code{0} for octal, or decimal otherwise.  Note that this is done separately
   for the numerator and denominator, so for instance @code{0xEF/100} is 239/100,
   whereas @code{0xEF/0x100} is 239/256.
   
   The return value is 0 if the entire string is a valid number, or @minus{}1 if
   not.
   @end deftypefun
   
   @deftypefun void mpq_swap (mpq_t @var{rop1}, mpq_t @var{rop2})
   Swap the values @var{rop1} and @var{rop2} efficiently.
   @end deftypefun
   
   
   @need 2000
   @node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
   @section Conversion Functions
   @cindex Rational conversion functions
   @cindex Conversion functions
   
   @deftypefun double mpq_get_d (mpq_t @var{op})
   Convert @var{op} to a @code{double}.
   @end deftypefun
   
   @deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op})
   @deftypefunx void mpq_set_f (mpq_t @var{rop}, mpf_t @var{op})
   Set @var{rop} to the value of @var{op}, without rounding.
   @end deftypefun
   
   @deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, mpq_t @var{op})
   Convert @var{op} to a string of digits in base @var{base}.  The base may vary
   from 2 to 36.  The string will be of the form @samp{num/den}, or if the
   denominator is 1 then just @samp{num}.
   
   If @var{str} is @code{NULL}, the result string is allocated using the current
   allocation function (@pxref{Custom Allocation}).  The block will be
   @code{strlen(str)+1} bytes, that being exactly enough for the string and
   null-terminator.
   
   If @var{str} is not @code{NULL}, it should point to a block of storage large
   enough for the result, that being
   
   @example
   mpz_sizeinbase (mpq_numref(@var{op}), @var{base})
   + mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3
   @end example
   
   The three extra bytes are for a possible minus sign, possible slash, and the
   null-terminator.
   
   A pointer to the result string is returned, being either the allocated block,
   or the given @var{str}.
   @end deftypefun
   
   
   @node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions
   @comment  node-name,  next,  previous,  up
 @section Arithmetic Functions  @section Arithmetic Functions
   @cindex Rational arithmetic functions
   @cindex Arithmetic functions
   
 @deftypefun void mpq_add (mpq_t @var{sum}, mpq_t @var{addend1}, mpq_t @var{addend2})  @deftypefun void mpq_add (mpq_t @var{sum}, mpq_t @var{addend1}, mpq_t @var{addend2})
 Set @var{sum} to @var{addend1} + @var{addend2}.  Set @var{sum} to @var{addend1} + @var{addend2}.
Line 1438  Set @var{difference} to @var{minuend} @minus{} @var{su
Line 3742  Set @var{difference} to @var{minuend} @minus{} @var{su
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})  @deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})
 @ifinfo  Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
 Set @var{product} to @var{multiplier} times @var{multiplicand}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{product} to $@var{multiplier} \times @var{multiplicand}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
   Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
   @var{op2}}.
   @end deftypefun
   
 @deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor})  @deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor})
   @cindex Division functions
 Set @var{quotient} to @var{dividend}/@var{divisor}.  Set @var{quotient} to @var{dividend}/@var{divisor}.
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpq_div_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
   Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
   @var{op2}}.
   @end deftypefun
   
 @deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand})  @deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand})
 Set @var{negated_operand} to @minus{}@var{operand}.  Set @var{negated_operand} to @minus{}@var{operand}.
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpq_abs (mpq_t @var{rop}, mpq_t @var{op})
   Set @var{rop} to the absolute value of @var{op}.
   @end deftypefun
   
 @deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number})  @deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number})
 Set @var{inverted_number} to 1/@var{number}.  If the new denominator is  Set @var{inverted_number} to 1/@var{number}.  If the new denominator is
 zero, this routine will divide by zero.  zero, this routine will divide by zero.
 @end deftypefun  @end deftypefun
   
 @node Comparing Rationals, Applying Integer Functions, Assigning Rationals, Rational Number Functions  @node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Comparison Functions  @section Comparison Functions
   @cindex Rational comparison functions
   @cindex Comparison functions
   
 @deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})  @deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})
 @ifinfo  Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
 Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >  @var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
 @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <  @math{@var{op1} < @var{op2}}.
 @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >  
 @var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}  
 < @var{op2}$.  
 @end tex  
 @end iftex  
   
 To determine if two rationals are equal, @code{mpq_equal} is faster than  To determine if two rationals are equal, @code{mpq_equal} is faster than
 @code{mpq_cmp}.  @code{mpq_cmp}.
 @end deftypefun  @end deftypefun
   
 @deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})  @deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
 @ifinfo  @deftypefnx Macro int mpq_cmp_si (mpq_t @var{op1}, long int @var{num2}, unsigned long int @var{den2})
 Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if  Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
 @var{op1} > @var{num2}/@var{den2}, zero if @var{op1} = @var{num2}/@var{den2},  @math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
 and a negative value if @var{op1} < @var{num2}/@var{den2}.  @var{num2}/@var{den2}}, and a negative value if @math{@var{op1} <
 @end ifinfo  @var{num2}/@var{den2}}.
 @iftex  
 @tex  
 Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if  
 $@var{op1} > @var{num2}/@var{den2}$, zero if $@var{op1} =  
 @var{num2}/@var{den2}$, and a negative value if $@var{op1} <  
 @var{num2}/@var{den2}$.  
 @end tex  
 @end iftex  
   
 This routine allows that @var{num2} and @var{den2} have common factors.  @var{num2} and @var{den2} are allowed to have common factors.
   
 This function is actually implemented as a macro.  It evaluates its  These functions are implemented as a macros and evaluate their arguments
 arguments multiple times.  multiple times.
 @end deftypefn  @end deftypefn
   
 @deftypefn Macro int mpq_sgn (mpq_t @var{op})  @deftypefn Macro int mpq_sgn (mpq_t @var{op})
 @ifinfo  @cindex Sign tests
 Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.  @cindex Rational sign tests
 @end ifinfo  Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
 @iftex  @math{-1} if @math{@var{op} < 0}.
 @tex  
 Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.  
 @end tex  
 @end iftex  
   
 This function is actually implemented as a macro.  It evaluates its  This function is actually implemented as a macro.  It evaluates its
 arguments multiple times.  arguments multiple times.
Line 1524  non-equal.  Although @code{mpq_cmp} can be used for th
Line 3817  non-equal.  Although @code{mpq_cmp} can be used for th
 function is much faster.  function is much faster.
 @end deftypefun  @end deftypefun
   
 @node Applying Integer Functions, Miscellaneous Rational Functions, Comparing Rationals, Rational Number Functions  @node Applying Integer Functions, I/O of Rationals, Comparing Rationals, Rational Number Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Applying Integer Functions to Rationals  @section Applying Integer Functions to Rationals
   @cindex Rational numerator and denominator
   @cindex Numerator and denominator
   
 The set of @code{mpq} functions is quite small.  In particular, there are no  The set of @code{mpq} functions is quite small.  In particular, there are few
 functions for either input or output.  But there are two macros that allow us  functions for either input or output.  The following functions give direct
 to apply any @code{mpz} function on the numerator or denominator of a rational  access to the numerator and denominator of an @code{mpq_t}.
 number.  If these macros are used to assign to the rational number,  
 @code{mpq_canonicalize} normally need to be called afterwards.  
   
   Note that if an assignment to the numerator and/or denominator could take an
   @code{mpq_t} out of the canonical form described at the start of this chapter
   (@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be
   called before any other @code{mpq} functions are applied to that @code{mpq_t}.
   
 @deftypefn Macro mpz_t mpq_numref (mpq_t @var{op})  @deftypefn Macro mpz_t mpq_numref (mpq_t @var{op})
 @deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})  @deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})
 Return a reference to the numerator and denominator of @var{op}, respectively.  Return a reference to the numerator and denominator of @var{op}, respectively.
 The @code{mpz} functions can be used on the result of these macros.  The @code{mpz} functions can be used on the result of these macros.
 @end deftypefn  @end deftypefn
   
 @need 2000  @deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
 @node Miscellaneous Rational Functions, , Applying Integer Functions, Rational Number Functions  @deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
 @comment  node-name,  next,  previous,  up  @deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
 @section Miscellaneous Functions  @deftypefunx void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
   Get or set the numerator or denominator of a rational.  These functions are
 @deftypefun double mpq_get_d (mpq_t @var{op})  equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
 Convert @var{op} to a double.  @code{mpq_denref}.  Direct use of @code{mpq_numref} or @code{mpq_denref} is
   recommended instead of these functions.
 @end deftypefun  @end deftypefun
   
 These functions assign between either the numerator or denominator of a  
 rational, and an integer.  Instead of using these functions, it is preferable  
 to use the more general mechanisms @code{mpq_numref} and @code{mpq_denref},  
 together with @code{mpz_set}.  
   
 @deftypefun void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})  @need 2000
 Copy @var{numerator} to the numerator of @var{rational}.  When this risks to  @node I/O of Rationals,  , Applying Integer Functions, Rational Number Functions
 make the numerator and denominator of @var{rational} have common factors, you  @comment  node-name,  next,  previous,  up
 have to pass @var{rational} to @code{mpq_canonicalize} before any operations  @section Input and Output Functions
 are performed on @var{rational}.  @cindex Rational input and output functions
   @cindex Input functions
   @cindex Output functions
   @cindex I/O functions
   
 This function is equivalent to  When using any of these functions, it's a good idea to include @file{stdio.h}
 @code{mpz_set (mpq_numref (@var{rational}), @var{numerator})}.  before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
 @end deftypefun  for these functions.
   
 @deftypefun void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})  Passing a @code{NULL} pointer for a @var{stream} argument to any of these
 Copy @var{denominator} to the denominator of @var{rational}.  When this risks  functions will make them read from @code{stdin} and write to @code{stdout},
 to make the numerator and denominator of @var{rational} have common factors,  respectively.
 or if the denominator might be negative, you have to pass @var{rational} to  
 @code{mpq_canonicalize} before any operations are performed on @var{rational}.  
   
 @strong{In version 1 of the library, negative denominators were handled by  @deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op})
 copying the sign to the numerator.  That is no longer done.}  Output @var{op} on stdio stream @var{stream}, as a string of digits in base
   @var{base}.  The base may vary from 2 to 36.  Output is in the form
   @samp{num/den} or if the denominator is 1 then just @samp{num}.
   
 This function is equivalent to  Return the number of bytes written, or if an error occurred, return 0.
 @code{mpz_set (mpq_denref (@var{rational}), @var{denominators})}.  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})  @deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
 Copy the numerator of @var{rational} to the integer @var{numerator}, to  Read a string of digits from @var{stream} and convert them to a rational in
 prepare for integer operations on the numerator.  @var{rop}.  Any initial white-space characters are read and discarded.  Return
   the number of characters read (including white space), or 0 if a rational
   could not be read.
   
 This function is equivalent to  The input can be a fraction like @samp{17/63} or just an integer like
 @code{mpz_set (@var{numerator}, mpq_numref (@var{rational}))}.  @samp{123}.  Reading stops at the first character not in this form, and white
 @end deftypefun  space is not permitted within the string.  If the input might not be in
   canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational
   Number Functions}).
   
 @deftypefun void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})  The @var{base} can be between 2 and 36, or can be 0 in which case the leading
 Copy the denominator of @var{rational} to the integer @var{denominator}, to  characters of the string determine the base, @samp{0x} or @samp{0X} for
 prepare for integer operations on the denominator.  hexadecimal, @samp{0} for octal, or decimal otherwise.  The leading characters
   are examined separately for the numerator and denominator of a fraction, so
 This function is equivalent to  for instance @samp{0x10/11} is 16/11, whereas @samp{0x10/0x11} is 16/17.
 @code{mpz_set (@var{denominator}, mpq_denref (@var{rational}))}.  
 @end deftypefun  @end deftypefun
   
   
Line 1599  This function is equivalent to
Line 3899  This function is equivalent to
 @chapter Floating-point Functions  @chapter Floating-point Functions
 @cindex Floating-point functions  @cindex Floating-point functions
 @cindex Float functions  @cindex Float functions
   @cindex User-defined precision
   @cindex Precision of floats
   
 This is a description of the @emph{preliminary} interface for floating-point  GMP floating point numbers are stored in objects of type @code{mpf_t} and
 arithmetic in GNU MP 2.  functions operating on them have an @code{mpf_} prefix.
   
 The floating-point functions expect arguments of type @code{mpf_t}.  The mantissa of each float has a user-selectable precision, limited only by
   available memory.  Each variable has its own precision, and that can be
   increased or decreased at any time.
   
 The MP floating-point functions have an interface that is similar to the MP  The exponent of each float is a fixed precision, one machine word on most
 integer functions.  The function prefix for floating-point operations is  systems.  In the current implementation the exponent is a count of limbs, so
 @code{mpf_}.  for example on a 32-bit system this means a range of roughly
   @math{2^@W{-68719476768}} to @math{2^@W{68719476736}}, or on a 64-bit system
   this will be greater.  Note however @code{mpf_get_str} can only return an
   exponent which fits an @code{mp_exp_t} and currently @code{mpf_set_str}
   doesn't accept exponents bigger than a @code{long}.
   
 There is one significant characteristic of floating-point numbers that has  Each variable keeps a size for the mantissa data actually in use.  This means
 motivated a difference between this function class and other MP function  that if a float is exactly represented in only a few bits then only those bits
 classes: the inherent inexactness of floating point arithmetic.  The user has  will be used in a calculation, even if the selected precision is high.
 to specify the precision of each variable.  A computation that assigns a  
 variable will take place with the precision of the assigned variable; the  
 precision of variables used as input is ignored.  
   
 @cindex User-defined precision  All calculations are performed to the precision of the destination variable.
 The precision of a calculation is defined as follows: Compute the requested  Each function is defined to calculate with ``infinite precision'' followed by
 operation exactly (with ``infinite precision''), and truncate the result to  a truncation to the destination precision, but of course the work done is only
 the destination variable precision.  Even if the user has asked for a very  what's needed to determine a result under that definition.
 high precision, MP will not calculate with superfluous digits.  For example,  
 if two low-precision numbers of nearly equal magnitude are added, the  
 precision of the result will be limited to what is required to represent the  
 result accurately.  
   
 The MP floating-point functions are @emph{not} intended as a smooth extension  The precision selected for a variable is a minimum value, GMP may increase it
 to the IEEE P754 arithmetic.  Specifically, the results obtained on one  a little to facilitate efficient calculation.  Currently this means rounding
 computer often differs from the results obtained on a computer with a  up to a whole limb, and then sometimes having a further partial limb,
 different word size.  depending on the high limb of the mantissa.  But applications shouldn't be
   concerned by such details.
   
   The mantissa in stored in binary, as might be imagined from the fact
   precisions are expressed in bits.  One consequence of this is that decimal
   fractions like @math{0.1} cannot be represented exactly.  The same is true of
   plain IEEE @code{double} floats.  This makes both highly unsuitable for
   calculations involving money or other values that should be exact decimal
   fractions.  (Suitably scaled integers, or perhaps rationals, are better
   choices.)
   
   @code{mpf} functions and variables have no special notion of infinity or
   not-a-number, and applications must take care not to overflow the exponent or
   results will be unpredictable.  This might change in a future release.
   
   Note that the @code{mpf} functions are @emph{not} intended as a smooth
   extension to IEEE P754 arithmetic.  In particular results obtained on one
   computer often differ from the results on a computer with a different word
   size.
   
 @menu  @menu
 * Initializing Floats::  * Initializing Floats::
 * Assigning Floats::  * Assigning Floats::
 * Simultaneous Float Init & Assign::  * Simultaneous Float Init & Assign::
 * Converting Floats::  * Converting Floats::
 * Float Arithmetic::  * Float Arithmetic::
 * Float Comparison::  * Float Comparison::
 * I/O of Floats::  * I/O of Floats::
 * Miscellaneous Float Functions::  * Miscellaneous Float Functions::
 @end menu  @end menu
   
 @node Initializing Floats, Assigning Floats, , Floating-point Functions  @node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Initialization and Assignment Functions  @section Initialization Functions
   @cindex Float initialization functions
   @cindex Initialization functions
   
 @deftypefun void mpf_set_default_prec (unsigned long int @var{prec})  @deftypefun void mpf_set_default_prec (unsigned long int @var{prec})
 Set the default precision to be @strong{at least} @var{prec} bits.  All  Set the default precision to be @strong{at least} @var{prec} bits.  All
Line 1651  subsequent calls to @code{mpf_init} will use this prec
Line 3972  subsequent calls to @code{mpf_init} will use this prec
 initialized variables are unaffected.  initialized variables are unaffected.
 @end deftypefun  @end deftypefun
   
   @deftypefun {unsigned long int} mpf_get_default_prec (void)
   Return the default default precision actually used.
   @end deftypefun
   
 An @code{mpf_t} object must be initialized before storing the first value in  An @code{mpf_t} object must be initialized before storing the first value in
 it.  The functions @code{mpf_init} and @code{mpf_init2} are used for that  it.  The functions @code{mpf_init} and @code{mpf_init2} are used for that
 purpose.  purpose.
Line 1678  Here is an example on how to initialize floating-point
Line 4003  Here is an example on how to initialize floating-point
 @example  @example
 @{  @{
   mpf_t x, y;    mpf_t x, y;
   mpf_init (x);                 /* use default precision */    mpf_init (x);           /* use default precision */
   mpf_init2 (y, 256);           /* precision @emph{at least} 256 bits */    mpf_init2 (y, 256);     /* precision @emph{at least} 256 bits */
   @dots{}    @dots{}
   /* Unless the program is about to exit, do ... */    /* Unless the program is about to exit, do ... */
   mpf_clear (x);    mpf_clear (x);
Line 1692  calculation.  A typical use would be for adjusting the
Line 4017  calculation.  A typical use would be for adjusting the
 iterative algorithms like Newton-Raphson, making the computation precision  iterative algorithms like Newton-Raphson, making the computation precision
 closely match the actual accurate part of the numbers.  closely match the actual accurate part of the numbers.
   
 @deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})  @deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})
 Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  Return the current precision of @var{op}, in bits.
 Since changing the precision involves calls to @code{realloc}, this routine  
 should not be called in a tight loop.  
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})  @deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
 Return the precision actually used for assignments of @var{op}.  Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  The
   value in @var{rop} will be truncated to the new precision.
   
   This function requires a call to @code{realloc}, and so should not be used in
   a tight loop.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, unsigned long int @var{prec})  @deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, unsigned long int @var{prec})
 Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  This  Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
 is a low-level function that does not change the allocation.  The @var{prec}  without changing the memory allocated.
 argument must not be larger that the precision previously returned by  
 @code{mpf_get_prec}.  It is crucial that the precision of @var{rop} is  @var{prec} must be no more than the allocated precision for @var{rop}, that
 ultimately reset to exactly the value returned by @code{mpf_get_prec}.  being the precision when @var{rop} was initialized, or in the most recent
   @code{mpf_set_prec}.
   
   The value in @var{rop} is unchanged, and in particular if it had a higher
   precision than @var{prec} it will retain that higher precision.  New values
   written to @var{rop} will use the new @var{prec}.
   
   Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another
   @code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original
   allocated precision.  Failing to do so will have unpredictable results.
   
   @code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the
   original allocated precision.  After @code{mpf_set_prec_raw} it reflects the
   @var{prec} value set.
   
   @code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at
   different precisions during a calculation, perhaps to gradually increase
   precision in an iteration, or just to use various different precisions for
   different purposes during a calculation.
 @end deftypefun  @end deftypefun
   
   
   @need 2000
 @node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions  @node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @subsection Assignment Functions  @section Assignment Functions
 @cindex Float assignment functions  @cindex Float assignment functions
   @cindex Assignment functions
   
 These functions assign new values to already initialized floats  These functions assign new values to already initialized floats
 (@pxref{Initializing Floats}).  (@pxref{Initializing Floats}).
Line 1733  Set the value of @var{rop} from the string in @var{str
Line 4080  Set the value of @var{rop} from the string in @var{str
 form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.  form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.
 @samp{M} is the mantissa and @samp{N} is the exponent.  The mantissa is always  @samp{M} is the mantissa and @samp{N} is the exponent.  The mantissa is always
 in the specified base.  The exponent is either in the specified base or, if  in the specified base.  The exponent is either in the specified base or, if
 @var{base} is negative, in decimal.  @var{base} is negative, in decimal.  The decimal point expected is taken from
   the current locale, on systems providing @code{localeconv}.
   
 The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to  The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
 @minus{}2.  Negative values are used to specify that the exponent is in  @minus{}2.  Negative values are used to specify that the exponent is in
Line 1743  Unlike the corresponding @code{mpz} function, the base
Line 4091  Unlike the corresponding @code{mpz} function, the base
 from the leading characters of the string if @var{base} is 0.  This is so that  from the leading characters of the string if @var{base} is 0.  This is so that
 numbers like @samp{0.23} are not interpreted as octal.  numbers like @samp{0.23} are not interpreted as octal.
   
 White space is allowed in the string, and is simply ignored.  White space is allowed in the string, and is simply ignored.  [This is not
   really true; white-space is ignored in the beginning of the string and within
   the mantissa, but not in other places, such as after a minus sign or in the
   exponent.  We are considering changing the definition of this function, making
   it fail when there is any white-space in the input, since that makes a lot of
   sense.  Please tell us your opinion about this change.  Do you really want it
   to accept @nicode{"3 14"} as meaning 314 as it does now?]
   
 This function returns 0 if the entire string up to the '\0' is a valid number  This function returns 0 if the entire string is a valid number in base
 in base @var{base}.  Otherwise it returns @minus{}1.  @var{base}.  Otherwise it returns @minus{}1.
 @end deftypefun  @end deftypefun
   
   @deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
   Swap @var{rop1} and @var{rop2} efficiently.  Both the values and the
   precisions of the two variables are swapped.
   @end deftypefun
   
   
 @node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions  @node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @subsection Combined Initialization and Assignment Functions  @section Combined Initialization and Assignment Functions
 @cindex Initialization and assignment functions  @cindex Initialization and assignment functions
   @cindex Float init and assign functions
   
 For convenience, MP provides a parallel series of initialize-and-set functions  For convenience, GMP provides a parallel series of initialize-and-set functions
 which initialize the output and then store the value there.  These functions'  which initialize the output and then store the value there.  These functions'
 names have the form @code{mpf_init_set@dots{}}  names have the form @code{mpf_init_set@dots{}}
   
Line 1789  set by @code{mpf_set_default_prec}.
Line 4149  set by @code{mpf_set_default_prec}.
 @node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions  @node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Conversion Functions  @section Conversion Functions
   @cindex Float conversion functions
 @cindex Conversion functions  @cindex Conversion functions
   
 @deftypefun double mpf_get_d (mpf_t @var{op})  @deftypefun double mpf_get_d (mpf_t @var{op})
 Convert @var{op} to a double.  Convert @var{op} to a @code{double}.
 @end deftypefun  @end deftypefun
   
 @deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})  @deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, mpf_t @var{op})
 Convert @var{op} to a string of digits in base @var{base}.  The base may vary  Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
 from 2 to 36.  Generate at most @var{n_digits} significant digits, or if  raised to @var{exp}}, with @math{0.5@le{}@GMPabs{@var{d}}<1}, is a good
 @var{n_digits} is 0, the maximum number of digits accurately representable by  approximation to @var{op}.  This is similar to the standard C function
 @var{op}.  @code{frexp}.
   @end deftypefun
   
 If @var{str} is NULL, space for the mantissa is allocated using the default  @deftypefun long mpf_get_si (mpf_t @var{op})
 allocation function, and a pointer to the string is returned.  @deftypefunx {unsigned long} mpf_get_ui (mpf_t @var{op})
   Convert @var{op} to a @code{long} or @code{unsigned long}, truncating any
   fraction part.  If @var{op} is too big for the return type, the result is
   undefined.
   
 If @var{str} is not NULL, it should point to a block of storage enough large  See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p}
 for the mantissa, i.e., @var{n_digits} + 2.  The two extra bytes are for a  (@pxref{Miscellaneous Float Functions}).
 possible minus sign, and for the terminating null character.  @end deftypefun
   
 The exponent is written through the pointer @var{expptr}.  @deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
   Convert @var{op} to a string of digits in base @var{base}.  @var{base} can be
   2 to 36.  Up to @var{n_digits} digits will be generated.  Trailing zeros are
   not returned.  No more digits than can be accurately represented by @var{op}
   are ever generated.  If @var{n_digits} is 0 then that accurate maximum number
   of digits are generated.
   
 If @var{n_digits} is 0, the maximum number of digits meaningfully achievable  If @var{str} is @code{NULL}, the result string is allocated using the current
 from the precision of @var{op} will be generated.  Note that the space  allocation function (@pxref{Custom Allocation}).  The block will be
 requirements for @var{str} in this case will be impossible for the user to  @code{strlen(str)+1} bytes, that being exactly enough for the string and
 predetermine.  Therefore, you need to pass NULL for the string argument  null-terminator.
 whenever @var{n_digits} is 0.  
   
   If @var{str} is not @code{NULL}, it should point to a block of
   @math{@var{n_digits} + 2} bytes, that being enough for the mantissa, a
   possible minus sign, and a null-terminator.  When @var{n_digits} is 0 to get
   all significant digits, an application won't be able to know the space
   required, and @var{str} should be @code{NULL} in that case.
   
 The generated string is a fraction, with an implicit radix point immediately  The generated string is a fraction, with an implicit radix point immediately
 to the left of the first digit.  For example, the number 3.1416 would be  to the left of the first digit.  The applicable exponent is written through
 returned as "31416" in the string and 1 written at @var{expptr}.  the @var{expptr} pointer.  For example, the number 3.1416 would be returned as
   string @nicode{"31416"} and exponent 1.
   
   When @var{op} is zero, an empty string is produced and the exponent returned
   is 0.
   
   A pointer to the result string is returned, being either the allocated block
   or the given @var{str}.
 @end deftypefun  @end deftypefun
   
   
Line 1830  returned as "31416" in the string and 1 written at @va
Line 4212  returned as "31416" in the string and 1 written at @va
   
 @deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
 @deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @math{@var{op1} + @var{op2}}.
 Set @var{rop} to @var{op1} + @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} + @var{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
Line 1848  Set @var{rop} to @var{op1} @minus{} @var{op2}.
Line 4223  Set @var{rop} to @var{op1} @minus{} @var{op2}.
   
 @deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
 @deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
 Set @var{rop} to @var{op1} times @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} \times @var{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 Division is undefined if the divisor is zero, and passing a zero divisor to  Division is undefined if the divisor is zero, and passing a zero divisor to the
 the divide functions will make these functions intentionally divide by zero.  divide functions will make these functions intentionally divide by zero.  This
 This gives the user the possibility to handle arithmetic exceptions in these  lets the user handle arithmetic exceptions in these functions in the same
 functions in the same manner as other arithmetic exceptions.  manner as other arithmetic exceptions.
   
 @deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
 @deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})  @deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})
 @deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
   @cindex Division functions
 Set @var{rop} to @var{op1}/@var{op2}.  Set @var{rop} to @var{op1}/@var{op2}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op})  @deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op})
 @deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op})  @deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op})
 @ifinfo  @cindex Root extraction functions
 Set @var{rop} to the square root of @var{op}.  Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $\sqrt{@var{op}}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @c @deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
 @c Set @var{rop} to @var{op1} raised to @var{op2}.  @cindex Exponentiation functions
 @c @end deftypefun  @cindex Powering functions
   Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}.
   @end deftypefun
   
 @deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})  @deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})
 Set @var{rop} to @minus{}@var{op}.  Set @var{rop} to @minus{}@var{op}.
Line 1894  Set @var{rop} to the absolute value of @var{op}.
Line 4259  Set @var{rop} to the absolute value of @var{op}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
 Set @var{rop} to @var{op1} times 2 raised to @var{op2}.  @var{op2}}.
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1} \times 2^{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
 @ifinfo  Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
 Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.  @var{op2}}.
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{rop} to $@var{op1}/2^{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions  @node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Comparison Functions  @section Comparison Functions
 @cindex Float comparisons functions  @cindex Float comparison functions
 @cindex Comparison functions  @cindex Comparison functions
   
 @deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2})
   @deftypefunx int mpf_cmp_d (mpf_t @var{op1}, double @var{op2})
 @deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, unsigned long int @var{op2})  @deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, unsigned long int @var{op2})
 @deftypefunx int mpf_cmp_si (mpf_t @var{op1}, signed long int @var{op2})  @deftypefunx int mpf_cmp_si (mpf_t @var{op1}, signed long int @var{op2})
 @ifinfo  Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
 Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >  @var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
 @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <  @math{@var{op1} < @var{op2}}.
 @var{op2}.  
 @end ifinfo  
 @iftex  
 @tex  
 Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >  
 @var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}  
 < @var{op2}$.  
 @end tex  
 @end iftex  
 @end deftypefun  @end deftypefun
   
 @deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, unsigned long int op3)  @deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, unsigned long int op3)
 Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are  Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
 equal, zero otherwise.  I.e., test of @var{op1} and @var{op2} are  equal, zero otherwise.  I.e., test of @var{op1} and @var{op2} are approximately
 approximately equal.  equal.
   
   Caution: Currently only whole limbs are compared, and only in an exact
   fashion.  In the future values like 1000 and 0111 may be considered the same
   to 3 bits (on the basis that their difference is that small).
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})  @deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
 Compute the relative difference between @var{op1} and @var{op2} and store the  Compute the relative difference between @var{op1} and @var{op2} and store the
 result in @var{rop}.  result in @var{rop}.  This is @math{@GMPabs{@var{op1}-@var{op2}}/@var{op1}}.
 @end deftypefun  @end deftypefun
   
 @deftypefn Macro int mpf_sgn (mpf_t @var{op})  @deftypefn Macro int mpf_sgn (mpf_t @var{op})
 @ifinfo  @cindex Sign tests
 Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.  @cindex Float sign tests
 @end ifinfo  Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
 @iftex  @math{-1} if @math{@var{op} < 0}.
 @tex  
 Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.  
 @end tex  
 @end iftex  
   
 This function is actually implemented as a macro.  It evaluates its  This function is actually implemented as a macro.  It evaluates its arguments
 arguments multiple times.  multiple times.
 @end deftypefn  @end deftypefn
   
 @node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions  @node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
Line 1972  arguments multiple times.
Line 4317  arguments multiple times.
 @cindex I/O functions  @cindex I/O functions
   
 Functions that perform input from a stdio stream, and functions that output to  Functions that perform input from a stdio stream, and functions that output to
 a stdio stream.  Passing a NULL pointer for a @var{stream} argument to any of  a stdio stream.  Passing a @code{NULL} pointer for a @var{stream} argument to
 these functions will make them read from @code{stdin} and write to  any of these functions will make them read from @code{stdin} and write to
 @code{stdout}, respectively.  @code{stdout}, respectively.
   
 When using any of these functions, it is a good idea to include @file{stdio.h}  When using any of these functions, it is a good idea to include @file{stdio.h}
Line 1981  before @file{gmp.h}, since that will allow @file{gmp.h
Line 4326  before @file{gmp.h}, since that will allow @file{gmp.h
 for these functions.  for these functions.
   
 @deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})  @deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
 Output @var{op} on stdio stream @var{stream}, as a string of digits in  Print @var{op} to @var{stream}, as a string of digits.  Return the number of
 base @var{base}.  The base may vary from 2 to 36.  Print at most  bytes written, or if an error occurred, return 0.
 @var{n_digits} significant digits, or if @var{n_digits} is 0, the maximum  
 number of digits accurately representable by @var{op}.  
   
 In addition to the significant digits, a leading @samp{0.} and a  The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
 trailing exponent, in the form @samp{eNNN}, are printed.  If @var{base}  which may vary from 2 to 36.  An exponent then printed, separated by an
 is greater than 10, @samp{@@} will be used instead of @samp{e} as  @samp{e}, or if @var{base} is greater than 10 then by an @samp{@@}.  The
 exponent delimiter.  exponent is always in decimal.  The decimal point follows the current locale,
   on systems providing @code{localeconv}.
   
 Return the number of bytes written, or if an error occurred, return 0.  Up to @var{n_digits} will be printed from the mantissa, except that no more
   digits than are accurately representable by @var{op} will be printed.
   @var{n_digits} can be 0 to select that accurate maximum.
 @end deftypefun  @end deftypefun
   
 @deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})  @deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
 Input a string in base @var{base} from stdio stream @var{stream}, and put the  Read a string in base @var{base} from @var{stream}, and put the read float in
 read float in @var{rop}.  The string is of the form @samp{M@@N} or, if the  @var{rop}.  The string is of the form @samp{M@@N} or, if the base is 10 or
 base is 10 or less, alternatively @samp{MeN}.  @samp{M} is the mantissa and  less, alternatively @samp{MeN}.  @samp{M} is the mantissa and @samp{N} is the
 @samp{N} is the exponent.  The mantissa is always in the specified base.  The  exponent.  The mantissa is always in the specified base.  The exponent is
 exponent is either in the specified base or, if @var{base} is negative, in  either in the specified base or, if @var{base} is negative, in decimal.  The
 decimal.  decimal point expected is taken from the current locale, on systems providing
   @code{localeconv}.
   
 The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to  The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
 @minus{}2.  Negative values are used to specify that the exponent is in  @minus{}2.  Negative values are used to specify that the exponent is in
Line 2026  Return the number of bytes read, or if an error occurr
Line 4373  Return the number of bytes read, or if an error occurr
 @c @end deftypefun  @c @end deftypefun
   
   
 @node Miscellaneous Float Functions, , I/O of Floats, Floating-point Functions  @node Miscellaneous Float Functions,  , I/O of Floats, Floating-point Functions
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @section Miscellaneous Functions  @section Miscellaneous Functions
 @cindex Miscellaneous float functions  @cindex Miscellaneous float functions
   @cindex Float miscellaneous functions
   
 @deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{max_exp})  @deftypefun void mpf_ceil (mpf_t @var{rop}, mpf_t @var{op})
   @deftypefunx void mpf_floor (mpf_t @var{rop}, mpf_t @var{op})
   @deftypefunx void mpf_trunc (mpf_t @var{rop}, mpf_t @var{op})
   Set @var{rop} to @var{op} rounded to an integer.  @code{mpf_ceil} rounds to the
   next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc}
   to the integer towards zero.
   @end deftypefun
   
   @deftypefun int mpf_integer_p (mpf_t @var{op})
   Return non-zero if @var{op} is an integer.
   @end deftypefun
   
   @deftypefun int mpf_fits_ulong_p (mpf_t @var{op})
   @deftypefunx int mpf_fits_slong_p (mpf_t @var{op})
   @deftypefunx int mpf_fits_uint_p (mpf_t @var{op})
   @deftypefunx int mpf_fits_sint_p (mpf_t @var{op})
   @deftypefunx int mpf_fits_ushort_p (mpf_t @var{op})
   @deftypefunx int mpf_fits_sshort_p (mpf_t @var{op})
   Return non-zero if @var{op} would fit in the respective C data type, when
   truncated to an integer.
   @end deftypefun
   
   @deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{nbits})
   Generate a uniformly distributed random float in @var{rop}, such that @math{0
   @le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa.
   
   The variable @var{state} must be initialized by calling one of the
   @code{gmp_randinit} functions (@ref{Random State Initialization}) before
   invoking this function.
   @end deftypefun
   
   @deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{exp})
 Generate a random float of at most @var{max_size} limbs, with long strings of  Generate a random float of at most @var{max_size} limbs, with long strings of
 zeros and ones in the binary representation.  The exponent of the number is in  zeros and ones in the binary representation.  The exponent of the number is in
 the interval @minus{}@var{exp} to @var{exp}.  This function is useful for  the interval @minus{}@var{exp} to @var{exp}.  This function is useful for
 testing functions and algorithms, since this kind of random numbers have  testing functions and algorithms, since this kind of random numbers have proven
 proven to be more likely to trigger corner-case bugs.  Negative random numbers  to be more likely to trigger corner-case bugs.  Negative random numbers are
 are generated when @var{max_size} is negative.  generated when @var{max_size} is negative.
 @end deftypefun  @end deftypefun
   
 @c @deftypefun size_t mpf_size (mpf_t @var{op})  @c @deftypefun size_t mpf_size (mpf_t @var{op})
Line 2045  are generated when @var{max_size} is negative.
Line 4424  are generated when @var{max_size} is negative.
 @c zero, the returned value will be zero.  (@xref{Nomenclature}, for an  @c zero, the returned value will be zero.  (@xref{Nomenclature}, for an
 @c explanation of the concept @dfn{limb}.)  @c explanation of the concept @dfn{limb}.)
 @c  @c
 @c @strong{This function is obsolete.  It will disappear from future MP  @c @strong{This function is obsolete.  It will disappear from future GMP
 @c releases.}  @c releases.}
 @c @end deftypefun  @c @end deftypefun
   
 @node Low-level Functions, BSD Compatible Functions, Floating-point Functions, Top  
   @node Low-level Functions, Random Number Functions, Floating-point Functions, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Low-level Functions  @chapter Low-level Functions
 @cindex Low-level functions  @cindex Low-level functions
   
 This chapter describes low-level MP functions, used to implement the high-level  This chapter describes low-level GMP functions, used to implement the
 MP functions, but also intended for time-critical user code.  high-level GMP functions, but also intended for time-critical user code.
   
 These functions start with the prefix @code{mpn_}.  These functions start with the prefix @code{mpn_}.
   
Line 2073  limb count.  A destination operand is specified by jus
Line 4453  limb count.  A destination operand is specified by jus
 responsibility of the caller to ensure that the destination has enough space  responsibility of the caller to ensure that the destination has enough space
 for storing the result.  for storing the result.
   
 With this way of specifying operands, it is possible to perform computations  With this way of specifying operands, it is possible to perform computations on
 on subranges of an argument, and store the result into a subrange of a  subranges of an argument, and store the result into a subrange of a
 destination.  destination.
   
 A common requirement for all functions is that each source area needs at least  A common requirement for all functions is that each source area needs at least
 one limb.  No size argument may be zero.  one limb.  No size argument may be zero.  Unless otherwise stated, in-place
   operations are allowed where source and destination are the same, but not where
   they only partly overlap.
   
 The @code{mpn} functions is the base for the implementation of the @code{mpz_},  The @code{mpn} functions are the base for the implementation of the
 @code{mpf_}, and @code{mpq_} functions.  @code{mpz_}, @code{mpf_}, and @code{mpq_} functions.
   
 This example adds the number beginning at @var{src1_ptr} and the number  This example adds the number beginning at @var{s1p} and the number beginning at
 beginning at @var{src2_ptr} and writes the sum at @var{dest_ptr}.  All areas  @var{s2p} and writes the sum at @var{destp}.  All areas have @var{n} limbs.
 have @var{size} limbs.  
   
 @example  @example
 cy = mpn_add_n (dest_ptr, src1_ptr, src2_ptr, size)  cy = mpn_add_n (destp, s1p, s2p, n)
 @end example  @end example
   
 @noindent  @noindent
 In the notation used here, a source operand is identified by the pointer to  In the notation used here, a source operand is identified by the pointer to
 the least significant limb, and the limb count in braces.  For example,  the least significant limb, and the limb count in braces.  For example,
 @{s1_ptr, s1_size@}.  @{@var{s1p}, @var{s1n}@}.
   
 @deftypefun mp_limb_t mpn_add_n (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{size})  @deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
 Add @{@var{src1_ptr}, @var{size}@} and @{@var{src2_ptr}, @var{size}@}, and  Add @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n}
 write the @var{size} least significant limbs of the result to @var{dest_ptr}.  least significant limbs of the result to @var{rp}.  Return carry, either 0 or
 Return carry, either 0 or 1.  1.
   
 This is the lowest-level function for addition.  It is the preferred function  This is the lowest-level function for addition.  It is the preferred function
 for addition, since it is written in assembly for most targets.  For addition  for addition, since it is written in assembly for most CPUs.  For addition of
 of a variable to itself (i.e., @var{src1_ptr} equals @var{src2_ptr}, use  a variable to itself (i.e., @var{s1p} equals @var{s2p}, use @code{mpn_lshift}
 @code{mpn_lshift} with a count of 1 for optimal speed.  with a count of 1 for optimal speed.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_add_1 (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
 Add @{@var{src1_ptr}, @var{size}@} and @var{src2_limb}, and write the  Add @{@var{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least
 @var{size} least significant limbs of the result to @var{dest_ptr}.  Return  significant limbs of the result to @var{rp}.  Return carry, either 0 or 1.
 carry, either 0 or 1.  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_add (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{src1_size}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{src2_size})  @deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
 Add @{@var{src1_ptr}, @var{src1_size}@} and @{@var{src2_ptr},  Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
 @var{src2_size}@}, and write the @var{src1_size} least significant limbs of  @var{s1n} least significant limbs of the result to @var{rp}.  Return carry,
 the result to @var{dest_ptr}.  Return carry, either 0 or 1.  either 0 or 1.
   
 This function requires that @var{src1_size} is greater than or equal to  This function requires that @var{s1n} is greater than or equal to @var{s2n}.
 @var{src2_size}.  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_sub_n (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{size})  @deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
 Subtract @{@var{src2_ptr}, @var{src2_size}@} from @{@var{src1_ptr},  Subtract @{@var{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the
 @var{size}@}, and write the @var{size} least significant limbs of the result  @var{n} least significant limbs of the result to @var{rp}.  Return borrow,
 to @var{dest_ptr}.  Return borrow, either 0 or 1.  either 0 or 1.
   
 This is the lowest-level function for subtraction.  It is the preferred  This is the lowest-level function for subtraction.  It is the preferred
 function for subtraction, since it is written in assembly for most targets.  function for subtraction, since it is written in assembly for most CPUs.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_sub_1 (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
 Subtract @var{src2_limb} from @{@var{src1_ptr}, @var{size}@}, and write the  Subtract @var{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least
 @var{size} least significant limbs of the result to @var{dest_ptr}.  Return  significant limbs of the result to @var{rp}.  Return borrow, either 0 or 1.
 borrow, either 0 or 1.  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_sub (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{src1_size}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{src2_size})  @deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
 Subtract @{@var{src2_ptr}, @var{src2_size}@} from @{@var{src1_ptr},  Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
 @var{src1_size}@}, and write the @var{src1_size} least significant limbs of  @var{s1n} least significant limbs of the result to @var{rp}.  Return borrow,
 the result to @var{dest_ptr}.  Return borrow, either 0 or 1.  either 0 or 1.
   
 This function requires that @var{src1_size} is greater than or equal to  This function requires that @var{s1n} is greater than or equal to
 @var{src2_size}.  @var{s2n}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpn_mul_n (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{size})  @deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
 Multiply @{@var{src1_ptr}, @var{size}@} and @{@var{src2_ptr}, @var{size}@},  Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
 and write the @strong{entire} result to @var{dest_ptr}.  2*@var{n}-limb result to @var{rp}.
   
 The destination has to have space for 2@var{size} limbs, even if the  The destination has to have space for 2*@var{n} limbs, even if the product's
 significant result might be one limb smaller.  most significant limb is zero.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_mul_1 (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
 Multiply @{@var{src1_ptr}, @var{size}@} and @var{src2_limb}, and write the  Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
 @var{size} least significant limbs of the product to @var{dest_ptr}.  Return  significant limbs of the product to @var{rp}.  Return the most significant
 the most significant limb of the product.  limb of the product.  @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
   allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.
   
 This is a low-level function that is a building block for general  This is a low-level function that is a building block for general
 multiplication as well as other operations in MP.  It is written in assembly  multiplication as well as other operations in GMP.  It is written in assembly
 for most targets.  for most CPUs.
   
 Don't call this function if @var{src2_limb} is a power of 2; use  Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
 @code{mpn_lshift} with a count equal to the logarithm of @var{src2_limb}  with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
 instead, for optimal speed.  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
 Multiply @{@var{src1_ptr}, @var{size}@} and @var{src2_limb}, and add the  Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least
 @var{size} least significant limbs of the product to @{@var{dest_ptr},  significant limbs of the product to @{@var{rp}, @var{n}@} and write the result
 @var{size}@} and write the result to @var{dest_ptr} @var{dest_ptr}.  Return  to @var{rp}.  Return the most significant limb of the product, plus carry-out
 the most significant limb of the product, plus carry-out from the addition.  from the addition.
   
 This is a low-level function that is a building block for general  This is a low-level function that is a building block for general
 multiplication as well as other operations in MP.  It is written in assembly  multiplication as well as other operations in GMP.  It is written in assembly
 for most targets.  for most CPUs.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_submul_1 (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
 Multiply @{@var{src1_ptr}, @var{size}@} and @var{src2_limb}, and subtract the  Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n}
 @var{size} least significant limbs of the product from @{@var{dest_ptr},  least significant limbs of the product from @{@var{rp}, @var{n}@} and write the
 @var{size}@} and write the result to @var{dest_ptr}.  Return the most  result to @var{rp}.  Return the most significant limb of the product, minus
 significant limb of the product, minus borrow-out from the subtraction.  borrow-out from the subtraction.
   
 This is a low-level function that is a building block for general  This is a low-level function that is a building block for general
 multiplication and division as well as other operations in MP.  It is written  multiplication and division as well as other operations in GMP.  It is written
 in assembly for most targets.  in assembly for most CPUs.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_mul (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src1_ptr}, mp_size_t @var{src1_size}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{src2_size})  @deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
 Multiply @{@var{src1_ptr}, @var{src1_size}@} and @{@var{src2_ptr},  Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
 @var{src2_size}@}, and write the result to @var{dest_ptr}.  Return the most  result to @var{rp}.  Return the most significant limb of the result.
 significant limb of the result.  
   
 The destination has to have space for @var{src1_size} + @var{src1_size}  The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
 limbs, even if the result might be one limb smaller.  result might be one limb smaller.
   
 This function requires that @var{src1_size} is greater than or equal to  This function requires that @var{s1n} is greater than or equal to
 @var{src2_size}.  The destination must be distinct from either input operands.  @var{s2n}.  The destination must be distinct from both input operands.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_divrem (mp_limb_t * @var{r1p}, mp_size_t @var{xsize}, mp_limb_t * @var{rs2p}, mp_size_t @var{rs2size}, const mp_limb_t * @var{s3p}, mp_size_t @var{s3size})  @deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn})
 Divide @{@var{rs2p}, @var{rs2size}@} by @{@var{s3p}, @var{s3size}@}, and  Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
 write the quotient at @var{r1p}, with the exception of the most significant  at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
 limb, which is returned.  The remainder replaces the dividend at @var{rs2p}.  @var{dn}@}.  The quotient is rounded towards 0.
   
 In addition to an integer quotient, @var{xsize} fraction limbs are developed,  No overlap is permitted between arguments.  @var{nn} must be greater than or
 and stored after the integral limbs.  For most usages, @var{xsize} will be  equal to @var{dn}.  The most significant limb of @var{dp} must be non-zero.
 zero.  The @var{qxn} operand must be zero.
   @comment FIXME: Relax overlap requirements!
   @end deftypefun
   
 It is required that @var{rs2size} is greater than or equal to @var{s3size}.  @deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
 It is required that the most significant bit of the divisor is set.  [This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
   performance.]
   
 If the quotient is not needed, pass @var{rs2p} + @var{s3size} as @var{r1p}.  Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
 Aside from that special case, no overlap between arguments is permitted.  quotient at @var{r1p}, with the exception of the most significant limb, which
   is returned.  The remainder replaces the dividend at @var{rs2p}; it will be
   @var{s3n} limbs long (i.e., as many limbs as the divisor).
   
   In addition to an integer quotient, @var{qxn} fraction limbs are developed, and
   stored after the integral limbs.  For most usages, @var{qxn} will be zero.
   
   It is required that @var{rs2n} is greater than or equal to @var{s3n}.  It is
   required that the most significant bit of the divisor is set.
   
   If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}.  Aside
   from that special case, no overlap between arguments is permitted.
   
 Return the most significant limb of the quotient, either 0 or 1.  Return the most significant limb of the quotient, either 0 or 1.
   
 The area at @var{r1p} needs to be @var{rs2size} @minus{} @var{s3size} +  The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
 @var{xsize} limbs large.  limbs large.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_divrem_1 (mp_limb_t * @var{r1p}, mp_size_t @var{xsize}, mp_limb_t * @var{s2p}, mp_size_t @var{s2size}, mp_limb_t @var{s3limb})  @deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb})
 Divide @{@var{s2p}, @var{s2size}@} by @var{s3limb}, and write the quotient  @deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}})
 at @var{r1p}.  Return the remainder.  Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
   @var{r1p}.  Return the remainder.
   
 In addition to an integer quotient, @var{xsize} fraction limbs are developed,  The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in
 and stored after the integral limbs.  For most usages, @var{xsize} will be  addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
 zero.  @var{qxn}@}.  Either or both @var{s2n} and @var{qxn} can be zero.  For most
   usages, @var{qxn} will be zero.
   
   @code{mpn_divmod_1} exists for upward source compatibility and is simply a
   macro calling @code{mpn_divrem_1} with a @var{qxn} of 0.
   
 The areas at @var{r1p} and @var{s2p} have to be identical or completely  The areas at @var{r1p} and @var{s2p} have to be identical or completely
 separate, not partially overlapping.  separate, not partially overlapping.
 @end deftypefun  @end deftypefn
   
 @deftypefun mp_size_t mpn_divmod (mp_limb_t * @var{r1p}, mp_limb_t * @var{rs2p}, mp_size_t @var{rs2size}, const mp_limb_t * @var{s3p}, mp_size_t @var{s3size})  @deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
 @strong{This interface is obsolete.  It will disappear from future releases.  [This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
 Use @code{mpn_divrem} in its stead.}  performance.]
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_divmod_1 (mp_limb_t * @var{r1p}, mp_limb_t * @var{s2p}, mp_size_t @var{s2size}, mp_limb_t @var{s3limb})  @deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
 @strong{This interface is obsolete.  It will disappear from future releases.  @deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry})
 Use @code{mpn_divrem_1} in its stead.}  Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
 @end deftypefun  the result to @{@var{rp}, @var{n}@}.  If 3 divides exactly, the return value is
   zero and the result is the quotient.  If not, the return value is non-zero and
   the result won't be anything useful.
   
 @deftypefun mp_limb_t mpn_mod_1 (mp_limb_t * @var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb})  @code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
 Divide @{@var{s1p}, @var{s1size}@} by @var{s2limb}, and return the remainder.  return value from a previous call, so a large calculation can be done piece by
 @end deftypefun  piece from low to high.  @code{mpn_divexact_by3} is simply a macro calling
   @code{mpn_divexact_by3c} with a 0 carry parameter.
   
 @deftypefun mp_limb_t mpn_preinv_mod_1 (mp_limb_t * @var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb}, mp_limb_t @var{s3limb})  These routines use a multiply-by-inverse and will be faster than
 @strong{This interface is obsolete.  It will disappear from future releases.  @code{mpn_divrem_1} on CPUs with fast multiplication but slow division.
 Use @code{mpn_mod_1} in its stead.}  
   The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i},
   and return value @math{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
   @m{b=2\GMPraise{@code{mp\_bits\_per\_limb}}, b=2^mp_bits_per_limb}.  The
   return @math{c} is always 0, 1 or 2, and the initial carry @math{i} must also
   be 0, 1 or 2 (these are both borrows really).  When @math{c=0} clearly
   @math{q=(a-i)/3}.  When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{}
   3} is given by @math{3-c}, because @math{b @equiv{} 1 @bmod{} 3} (when
   @code{mp_bits_per_limb} is even, which is always so currently).
   @end deftypefn
   
   @deftypefun mp_limb_t mpn_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
   Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder.
   @var{s1n} can be zero.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_bdivmod (mp_limb_t * @var{dest_ptr}, mp_limb_t * @var{s1p}, mp_size_t @var{s1size}, const mp_limb_t * @var{s2p}, mp_size_t @var{s2size}, unsigned long int @var{d})  @deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}, unsigned long int @var{d})
 The function puts the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of  This function puts the low
 @var{q} =  @math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of @var{q} =
 @{@var{s1p}, @var{s1size}@}/@{@var{s2p}, @var{s2size}@}  @{@var{s1p}, @var{s1n}@}/@{@var{s2p}, @var{s2n}@} mod @m{2^d,2^@var{d}} at
 mod 2^@var{d}  @var{rp}, and returns the high @var{d} mod @code{mp_bits_per_limb} bits of
 at @var{dest_ptr},  @var{q}.
 and returns the high @var{d} mod @var{BITS_PER_MP_LIMB} bits of @var{q}.  
   
 @{@var{s1p}, @var{s1size}@} - @var{q} * @{@var{s2p}, @var{s2size}@}  @{@var{s1p}, @var{s1n}@} - @var{q} * @{@var{s2p}, @var{s2n}@} mod @m{2
 mod 2^(@var{s1size}*@var{BITS_PER_MP_LIMB})  \GMPraise{@var{s1n}*@code{mp\_bits\_per\_limb}},
 is placed at @var{s1p}.  2^(@var{s1n}*@nicode{mp\_bits\_per\_limb})} is placed at @var{s1p}.  Since the
 Since the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of  low @math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of this
 this difference are zero, it is possible to overwrite the low limbs at  difference are zero, it is possible to overwrite the low limbs at @var{s1p}
 @var{s1p} with this difference,  with this difference, provided @math{@var{rp} @le{} @var{s1p}}.
 provided @var{dest_ptr} <= @var{s1p}.  
   
 This function requires that @var{s1size} * @var{BITS_PER_MP_LIMB} >= @var{D},  This function requires that @math{@var{s1n} * @nicode{mp\_bits\_per\_limb}
 and that @{@var{s2p}, @var{s2size}@} is odd.  @ge{} @var{D}}, and that @{@var{s2p}, @var{s2n}@} is odd.
   
 @strong{This interface is preliminary.  It might change incompatibly in  @strong{This interface is preliminary.  It might change incompatibly in future
 future revisions.}  revisions.}
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_lshift (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})  @deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
 Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the left, and  Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to
 write the @var{src_size} least significant limbs of the result to  @{@var{rp}, @var{n}@}.  The bits shifted out at the left are returned in the
 @var{dest_ptr}.  @var{count} might be in the range 1 to n @minus{} 1, on an  least significant @var{count} bits of the return value (the rest of the return
 n-bit machine. The bits shifted out to the left are returned.  value is zero).
   
 Overlapping of the destination space and the source space is allowed in this  @var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
 function, provided @var{dest_ptr} >= @var{src_ptr}.  regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
   @math{@var{rp} @ge{} @var{sp}}.
   
 This function is written in assembly for most targets.  This function is written in assembly for most CPUs.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limp_t mpn_rshift (mp_limb_t * @var{dest_ptr}, const mp_limb_t * @var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})  @deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
 Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the right, and  Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to
 write the @var{src_size} most significant limbs of the result to  @{@var{rp}, @var{n}@}.  The bits shifted out at the right are returned in the
 @var{dest_ptr}.  @var{count} might be in the range 1 to n @minus{} 1, on an  most significant @var{count} bits of the return value (the rest of the return
 n-bit machine.  The bits shifted out to the right are returned.  value is zero).
   
 Overlapping of the destination space and the source space is allowed in this  @var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
 function, provided @var{dest_ptr} <= @var{src_ptr}.  regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
   @math{@var{rp} @le{} @var{sp}}.
   
 This function is written in assembly for most targets.  This function is written in assembly for most CPUs.
 @end deftypefun  @end deftypefun
   
 @deftypefun int mpn_cmp (const mp_limb_t * @var{src1_ptr}, const mp_limb_t * @var{src2_ptr}, mp_size_t @var{size})  @deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
 Compare @{@var{src1_ptr}, @var{size}@} and @{@var{src2_ptr}, @var{size}@} and  Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
 return a positive value if src1 > src2, 0 of they are equal, and a negative  positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a
 value if src1 < src2.  negative value if @math{@var{s1} < @var{s2}}.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_gcd (mp_limb_t * @var{dest_ptr}, mp_limb_t * @var{src1_ptr}, mp_size_t @var{src1_size}, mp_limb_t * @var{src2_ptr}, mp_size_t @var{src2_size})  @deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
 Puts at @var{dest_ptr} the greatest common divisor of @{@var{src1_ptr},  Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{s1p},
 @var{src1_size}@} and @{@var{src2_ptr}, @var{src2_size}@}; both source  @var{s1n}@} and @{@var{s2p}, @var{s2n}@}.  The result can be up to @var{s2n}
 operands are destroyed by the operation.  The size in limbs of the greatest  limbs, the return value is the actual number produced.  Both source operands
 common divisor is returned.  are destroyed.
   
 @{@var{src1_ptr}, @var{src1_size}@} must be odd, and @{@var{src2_ptr},  @{@var{s1p}, @var{s1n}@} must have at least as many bits as @{@var{s2p},
 @var{src2_size}@} must have at least as many bits as @{@var{src1_ptr},  @var{s2n}@}.  @{@var{s2p}, @var{s2n}@} must be odd.  Both operands must have
 @var{src1_size}@}.  non-zero most significant limbs.  No overlap is permitted between @{@var{s1p},
   @var{s1n}@} and @{@var{s2p}, @var{s2n}@}.
 @strong{This interface is preliminary.  It might change incompatibly in  
 future revisions.}  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t * @var{src1_ptr}, mp_size_t @var{src1_size}, mp_limb_t @var{src2_limb})  @deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
 Return the greatest common divisor of @{@var{src1_ptr}, @var{src1_size}@}  Return the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
 and @var{src2_limb}, where @var{src2_limb} (as well as @var{src1_size})  @var{s2limb}.  Both operands must be non-zero.
 must be different from 0.  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_gcdext (mp_limb_t * @var{r1p}, mp_limb_t * @var{r2p}, mp_limb_t * @var{s1p}, mp_size_t @var{s1size}, mp_limb_t * @var{s2p}, mp_size_t @var{s2size})  @deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, mp_size_t *@var{r2n}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
 Puts at @var{r1p} the greatest common divisor of @{@var{s1p}, @var{s1size}@}  Calculate the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
 and @{@var{s2p}, @var{s2size}@}.  The first cofactor is written at  @{@var{s2p}, @var{s2n}@}.  Store the gcd at @{@var{r1p}, @var{retval}@} and
 @var{r2p}.  Both source operands are destroyed by the operation.  The size  the first cofactor at @{@var{r2p}, *@var{r2n}@}, with *@var{r2n} negative if
 in limbs of the greatest common divisor is returned.  the cofactor is negative.  @var{r1p} and @var{r2p} should each have room for
   @math{@var{s1n}+1} limbs, but the return value and value stored through
   @var{r2n} indicate the actual number produced.
   
 @strong{This interface is preliminary.  It might change incompatibly in  @math{@{@var{s1p}, @var{s1n}@} @ge{} @{@var{s2p}, @var{s2n}@}} is required,
 future revisions.}  and both must be non-zero.  The regions @{@var{s1p}, @math{@var{s1n}+1}@} and
   @{@var{s2p}, @math{@var{s2n}+1}@} are destroyed (i.e. the operands plus an
   extra limb past the end of each).
   
   The cofactor @var{r1} will satisfy @m{r_2 s_1 + k s_2 = r_1, @var{r2}*@var{s1}
   + @var{k}*@var{s2} = @var{r1}}.  The second cofactor @var{k} is not calculated
   but can easily be obtained from @m{(r_1 - r_2 s_1) / s_2, (@var{r1} -
   @var{r2}*@var{s1}) / @var{s2}}.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_sqrtrem (mp_limb_t * @var{r1p}, mp_limb_t * @var{r2p}, const mp_limb_t * @var{sp}, mp_size_t @var{size})  @deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
 Compute the square root of @{@var{sp}, @var{size}@} and put the result at  Compute the square root of @{@var{sp}, @var{n}@} and put the result at
 @var{r1p}.  Write the remainder at @var{r2p}, unless @var{r2p} is NULL.  @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p},
   @var{retval}@}.  @var{r2p} needs space for @var{n} limbs, but the return value
   indicates how many are produced.
   
 Return the size of the remainder, whether @var{r2p} was NULL or non-NULL.  The most significant limb of @{@var{sp}, @var{n}@} must be non-zero.  The
 Iff the operand was a perfect square, the return value will be 0.  areas @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must
   be completely separate.  The areas @{@var{r2p}, @var{n}@} and @{@var{sp},
   @var{n}@} must be either identical or completely separate.
   
 The areas at @var{r1p} and @var{sp} have to be distinct.  The areas at  If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
 @var{r2p} and @var{sp} have to be identical or completely separate, not  case the return value is zero or non-zero according to whether the remainder
 partially overlapping.  would have been zero or non-zero.
   
 @ifinfo  A return value of zero indicates a perfect square.  See also
 The area at @var{r1p} needs to have space for ceil(@var{size}/2) limbs.  @code{mpz_perfect_square_p}.
 @end ifinfo  
 @iftex  
 @tex  
 The area at @var{r1p} needs to have space for $\lceil@var{size}/2\rceil$ limbs.  
 @end tex  
 @end iftex  
 The area at @var{r2p} needs to be @var{size} limbs large.  
   
 @strong{This interface is preliminary.  It might change incompatibly in  
 future revisions.}  
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t * @var{s1p}, mp_size_t @var{s1size})  @deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n})
 Convert @{@var{s1p}, @var{s1size}@} to a raw unsigned char array in base  Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in
 @var{base}.  The string is not in ASCII; to convert it to printable format,  base @var{base}, and return the number of characters produced.  There may be
 add the ASCII codes for @samp{0} or @samp{A}, depending on the base and  leading zeros in the string.  The string is not in ASCII; to convert it to
 range.  There may be leading zeros in the string.  printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on
   the base and range.  @var{base} can vary from 2 to 256.
   
 The area at @var{s1p} is clobbered.  The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be
   non-zero.  The input @{@var{s1p}, @var{s1n}@} is clobbered, except when
   @var{base} is a power of 2, in which case it's unchanged.
   
 Return the number of characters in @var{str}.  
   
 The area at @var{str} has to have space for the largest possible number  The area at @var{str} has to have space for the largest possible number
 represented by a @var{s1size} long limb array, plus one extra character.  represented by a @var{s1n} long limb array, plus one extra character.
 @end deftypefun  @end deftypefun
   
 @deftypefun mp_size_t mpn_set_str (mp_limb_t * @var{r1p}, const char *@var{str}, size_t {strsize}, int @var{base})  @deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{rp}, const unsigned char *@var{str}, size_t @var{strsize}, int @var{base})
 Convert the raw unsigned char array at @var{str} of length @var{strsize} to  Convert bytes @{@var{str},@var{strsize}@} in the given @var{base} to limbs at
 a limb array @{@var{s1p}, @var{s1size}@}.  The base of @var{str} is  @var{rp}.
 @var{base}.  
   
 Return the number of limbs stored in @var{r1p}.  @math{@var{str}[0]} is the most significant byte and
   @math{@var{str}[@var{strsize}-1]} is the least significant.  Each byte should
   be a value in the range 0 to @math{@var{base}-1}, not an ASCII character.
   @var{base} can vary from 2 to 256.
   
   The return value is the number of limbs written to @var{rp}.  If the most
   significant input byte is non-zero then the high limb at @var{rp} will be
   non-zero, and only that exact number of limbs will be required there.
   
   If the most significant input byte is zero then there may be high zero limbs
   written to @var{rp} and included in the return value.
   
   @var{strsize} must be at least 1, and no overlap is permitted between
   @{@var{str},@var{strsize}@} and the result at @var{rp}.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpn_scan0 (const mp_limb_t * @var{s1p}, unsigned long int @var{bit})  @deftypefun {unsigned long int} mpn_scan0 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
 Scan @var{s1p} from bit position @var{bit} for the next clear bit.  Scan @var{s1p} from bit position @var{bit} for the next clear bit.
   
 It is required that there be a clear bit within the area at @var{s1p} at or  It is required that there be a clear bit within the area at @var{s1p} at or
 beyond bit position @var{bit}, so that the function has something to return.  beyond bit position @var{bit}, so that the function has something to return.
   
 @strong{This interface is preliminary.  It might change incompatibly in  
 future revisions.}  
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpn_scan1 (const mp_limb_t * @var{s1p}, unsigned long int @var{bit})  @deftypefun {unsigned long int} mpn_scan1 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
 Scan @var{s1p} from bit position @var{bit} for the next set bit.  Scan @var{s1p} from bit position @var{bit} for the next set bit.
   
 It is required that there be a set bit within the area at @var{s1p} at or  It is required that there be a set bit within the area at @var{s1p} at or
 beyond bit position @var{bit}, so that the function has something to return.  beyond bit position @var{bit}, so that the function has something to return.
   @end deftypefun
   
 @strong{This interface is preliminary.  It might change incompatibly in  @deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
 future revisions.}  @deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
   Generate a random number of length @var{r1n} and store it at @var{r1p}.  The
   most significant limb is always non-zero.  @code{mpn_random} generates
   uniformly distributed limb data, @code{mpn_random2} generates long strings of
   zeros and ones in the binary representation.
   
   @code{mpn_random2} is intended for testing the correctness of the @code{mpn}
   routines.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mpn_random2 (mp_limb_t * @var{r1p}, mp_size_t @var{r1size})  @deftypefun {unsigned long int} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
 Generate a random number of length @var{r1size} with long strings of zeros  Count the number of set bits in @{@var{s1p}, @var{n}@}.
 and ones in the binary representation, and store it at @var{r1p}.  @end deftypefun
   
 The generated random numbers are intended for testing the correctness of the  @deftypefun {unsigned long int} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
 implementation of the @code{mpn} routines.  Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p},
   @var{n}@}.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpn_popcount (const mp_limb_t * @var{s1p}, unsigned long int @var{size})  @deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
 Count the number of set bits in @{@var{s1p}, @var{size}@}.  Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square.
 @end deftypefun  @end deftypefun
   
 @deftypefun {unsigned long int} mpn_hamdist (const mp_limb_t * @var{s1p}, const mp_limb_t * @var{s2p}, unsigned long int @var{size})  
 Compute the hamming distance between @{@var{s1p}, @var{size}@} and  @sp 1
 @{@var{s2p}, @var{size}@}.  @section Nails
   @cindex Nails
   
   @strong{Everything in this section is highly experimental and may disappear or
   be subject to incompatible changes in a future version of GMP.}
   
   Nails are an experimental feature whereby a few bits are left unused at the
   top of each @code{mp_limb_t}.  This can significantly improve carry handling
   on some processors.
   
   All the @code{mpn} functions accepting limb data will expect the nail bits to
   be zero on entry, and will return data with the nails similarly all zero.
   This applies both to limb vectors and to single limb arguments.
   
   Nails can be enabled by configuring with @samp{--enable-nails}.  By default
   the number of bits will be chosen according to what suits the host processor,
   but a particular number can be selected with @samp{--enable-nails=N}.
   
   At the mpn level, a nail build is neither source nor binary compatible with a
   non-nail build, strictly speaking.  But programs acting on limbs only through
   the mpn functions are likely to work equally well with either build, and
   judicious use of the definitions below should make any program compatible with
   either build, at the source level.
   
   For the higher level routines, meaning @code{mpz} etc, a nail build should be
   fully source and binary compatible with a non-nail build.
   
   @defmac GMP_NAIL_BITS
   @defmacx GMP_NUMB_BITS
   @defmacx GMP_LIMB_BITS
   @code{GMP_NAIL_BITS} is the number of nail bits, or 0 when nails are not in
   use.  @code{GMP_NUMB_BITS} is the number of data bits in a limb.
   @code{GMP_LIMB_BITS} is the total number of bits in an @code{mp_limb_t}.  In
   all cases
   
   @example
   GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
   @end example
   @end defmac
   
   @defmac GMP_NAIL_MASK
   @defmacx GMP_NUMB_MASK
   Bit masks for the nail and number parts of a limb.  @code{GMP_NAIL_MASK} is 0
   when nails are not in use.
   
   @code{GMP_NAIL_MASK} is not often needed, since the nail part can be obtained
   with @code{x >> GMP_NUMB_BITS}, and that means one less large constant, which
   can help various RISC chips.
   @end defmac
   
   @defmac GMP_NUMB_MAX
   The maximum value that can be stored in the number part of a limb.  This is
   the same as @code{GMP_NUMB_MASK}, but can be used for clarity when doing
   comparisons rather than bit-wise operations.
   @end defmac
   
   The term ``nails'' comes from finger or toe nails, which are at the ends of a
   limb (arm or leg).  ``numb'' is short for number, but is also how the
   developers felt after trying for a long time to come up with sensible names
   for these things.
   
   In the future (the distant future most likely) a non-zero nail might be
   permitted, giving non-unique representations for numbers in a limb vector.
   This would help vector processors since carries would only ever need to
   propagate one or two limbs.
   
   
   @node Random Number Functions, Formatted Output, Low-level Functions, Top
   @chapter Random Number Functions
   @cindex Random number functions
   
   Sequences of pseudo-random numbers in GMP are generated using a variable of
   type @code{gmp_randstate_t}, which holds an algorithm selection and a current
   state.  Such a variable must be initialized by a call to one of the
   @code{gmp_randinit} functions, and can be seeded with one of the
   @code{gmp_randseed} functions.
   
   The functions actually generating random numbers are described in @ref{Integer
   Random Numbers}, and @ref{Miscellaneous Float Functions}.
   
   The older style random number functions don't accept a @code{gmp_randstate_t}
   parameter but instead share a global variable of that type.  They use a
   default algorithm and are currently not seeded (though perhaps that will
   change in the future).  The new functions accepting a @code{gmp_randstate_t}
   are recommended for applications that care about randomness.
   
   @menu
   * Random State Initialization::
   * Random State Seeding::
   @end menu
   
   @node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
   @section Random State Initialization
   @cindex Random number state
   
   @deftypefun void gmp_randinit_default (gmp_randstate_t @var{state})
   Initialize @var{state} with a default algorithm.  This will be a compromise
   between speed and randomness, and is recommended for applications with no
   special requirements.
 @end deftypefun  @end deftypefun
   
 @deftypefun int mpn_perfect_square_p (const mp_limb_t * @var{s1p}, mp_size_t @var{size})  @deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a}, @w{unsigned long @var{c}}, @w{unsigned long @var{m2exp}})
 Return non-zero iff @{@var{s1p}, @var{size}@} is a perfect square.  Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X +
   @var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}.
   
   The low bits of @math{X} in this algorithm are not very random.  The least
   significant bit will have a period no more than 2, and the second bit no more
   than 4, etc.  For this reason only the high half of each @math{X} is actually
   used.
   
   When a random number of more than @math{@var{m2exp}/2} bits is to be
   generated, multiple iterations of the recurrence are used and the results
   concatenated.
 @end deftypefun  @end deftypefun
   
   @deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, unsigned long @var{size})
   Initialize @var{state} for a linear congruential algorithm as per
   @code{gmp_randinit_lc_2exp}.  @var{a}, @var{c} and @var{m2exp} are selected
   from a table, chosen so that @var{size} bits (or more) of each @math{X} will
   be used, ie. @math{@var{m2exp}/2 @ge{} @var{size}}.
   
 @node BSD Compatible Functions, Custom Allocation, Low-level Functions, Top  If successful the return value is non-zero.  If @var{size} is bigger than the
   table data provides then the return value is zero.  The maximum @var{size}
   currently supported is 128.
   @end deftypefun
   
   @deftypefun void gmp_randinit (gmp_randstate_t @var{state}, @w{gmp_randalg_t @var{alg}}, ...)
   @strong{This function is obsolete.}
   
   Initialize @var{state} with an algorithm selected by @var{alg}.  The only
   choice is @code{GMP_RAND_ALG_LC}, which is @code{gmp_randinit_lc_2exp_size}.
   A third parameter of type @code{unsigned long} is required, this is the
   @var{size} for that function.  @code{GMP_RAND_ALG_DEFAULT} or 0 are the same
   as @code{GMP_RAND_ALG_LC}.
   
   @code{gmp_randinit} sets bits in @code{gmp_errno} to indicate an error.
   @code{GMP_ERROR_UNSUPPORTED_ARGUMENT} if @var{alg} is unsupported, or
   @code{GMP_ERROR_INVALID_ARGUMENT} if the @var{size} parameter is too big.
   @end deftypefun
   
   @c  Not yet in the library.
   @ignore
   @deftypefun void gmp_randinit_lc (gmp_randstate_t @var{state}, mpz_t @var{a}, unsigned long int @var{c}, mpz_t @var{m})
   Initialize @var{state} for a linear congruential scheme @m{X = (@var{a}X +
   @var{c}) @bmod @var{m}, X = (@var{a}*X + @var{c}) mod 2^@var{m}}.
   @end deftypefun
   @end ignore
   
   @deftypefun void gmp_randclear (gmp_randstate_t @var{state})
   Free all memory occupied by @var{state}.
   @end deftypefun
   
   
   @node Random State Seeding,  , Random State Initialization, Random Number Functions
   @section Random State Seeding
   @cindex Random number seeding
   
   @deftypefun void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed})
   @deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, @w{unsigned long int @var{seed}})
   Set an initial seed value into @var{state}.
   
   The size of a seed determines how many different sequences of random numbers
   that it's possible to generate.  The ``quality'' of the seed is the randomness
   of a given seed compared to the previous seed used, and this affects the
   randomness of separate number sequences.  The method for choosing a seed is
   critical if the generated numbers are to be used for important applications,
   such as generating cryptographic keys.
   
   Traditionally the system time has been used to seed, but care needs to be
   taken with this.  If an application seeds often and the resolution of the
   system clock is low, then the same sequence of numbers might be repeated.
   Also, the system time is quite easy to guess, so if unpredictability is
   required then it should definitely not be the only source for the seed value.
   On some systems there's a special device @file{/dev/random} which provides
   random data better suited for use as a seed.
   @end deftypefun
   
   
   @node Formatted Output, Formatted Input, Random Number Functions, Top
   @chapter Formatted Output
   @cindex Formatted output
   @cindex @code{printf} formatted output
   
   @menu
   * Formatted Output Strings::
   * Formatted Output Functions::
   * C++ Formatted Output::
   @end menu
   
   @node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
   @section Format Strings
   
   @code{gmp_printf} and friends accept format strings similar to the standard C
   @code{printf} (@pxref{Formatted Output,,,libc,The GNU C Library Reference
   Manual}).  A format specification is of the form
   
   @example
   % [flags] [width] [.[precision]] [type] conv
   @end example
   
   GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
   and @code{mpf_t} respectively, and @samp{N} for an @code{mp_limb_t} array.
   @samp{Z}, @samp{Q} and @samp{N} behave like integers.  @samp{Q} will print a
   @samp{/} and a denominator, if needed.  @samp{F} behaves like a float.  For
   example,
   
   @example
   mpz_t z;
   gmp_printf ("%s is an mpz %Zd\n", "here", z);
   
   mpq_t q;
   gmp_printf ("a hex rational: %#40Qx\n", q);
   
   mpf_t f;
   int   n;
   gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
   
   const mp_limb_t *ptr;
   mp_size_t       size;
   gmp_printf ("limb array %Nx\n", ptr, size);
   @end example
   
   For @samp{N} the limbs are expected least significant first, as per the
   @code{mpn} functions (@pxref{Low-level Functions}).  A negative size can be
   given to print the value as a negative.
   
   All the standard C @code{printf} types behave the same as the C library
   @code{printf}, and can be freely intermixed with the GMP extensions.  In the
   current implementation the standard parts of the format string are simply
   handed to @code{printf} and only the GMP extensions handled directly.
   
   The flags accepted are as follows.  GLIBC style @nisamp{'} is only for the
   standard C types (not the GMP types), and only if the C library supports it.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{0} @tab pad with zeros (rather than spaces)
   @item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0}
   @item @nicode{+} @tab always show a sign
   @item (space)    @tab show a space or a @samp{-} sign
   @item @nicode{'} @tab group digits, GLIBC style (not GMP types)
   @end multitable
   @end quotation
   
   The optional width and precision can be given as a number within the format
   string, or as a @samp{*} to take an extra parameter of type @code{int}, the
   same as the standard @code{printf}.
   
   The standard types accepted are as follows.  @samp{h} and @samp{l} are
   portable, the rest will depend on the compiler (or include files) for the type
   and the C library for the output.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{h}  @tab @nicode{short}
   @item @nicode{hh} @tab @nicode{char}
   @item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
   @item @nicode{l}  @tab @nicode{long} or @nicode{wchar_t}
   @item @nicode{ll} @tab @nicode{long long}
   @item @nicode{L}  @tab @nicode{long double}
   @item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
   @item @nicode{t}  @tab @nicode{ptrdiff_t}
   @item @nicode{z}  @tab @nicode{size_t}
   @end multitable
   @end quotation
   
   @noindent
   The GMP types are
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{F}  @tab @nicode{mpf_t}, float conversions
   @item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
   @item @nicode{N}  @tab @nicode{mp_limb_t} array, integer conversions
   @item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
   @end multitable
   @end quotation
   
   The conversions accepted are as follows.  @samp{a} and @samp{A} are always
   supported for @code{mpf_t} but depend on the C library for standard C float
   types.  @samp{m} and @samp{p} depend on the C library.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{a} @nicode{A} @tab hex floats, C99 style
   @item @nicode{c}            @tab character
   @item @nicode{d}            @tab decimal integer
   @item @nicode{e} @nicode{E} @tab scientific format float
   @item @nicode{f}            @tab fixed point float
   @item @nicode{i}            @tab same as @nicode{d}
   @item @nicode{g} @nicode{G} @tab fixed or scientific float
   @item @nicode{m}            @tab @code{strerror} string, GLIBC style
   @item @nicode{n}            @tab store characters written so far
   @item @nicode{o}            @tab octal integer
   @item @nicode{p}            @tab pointer
   @item @nicode{s}            @tab string
   @item @nicode{u}            @tab unsigned integer
   @item @nicode{x} @nicode{X} @tab hex integer
   @end multitable
   @end quotation
   
   @samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for
   types @samp{Z}, @samp{Q} and @samp{N} they are signed.  @samp{u} is not
   meaningful for @samp{Z}, @samp{Q} and @samp{N}.
   
   @samp{n} can be used with any type, even the GMP types.
   
   Other types or conversions that might be accepted by the C library
   @code{printf} cannot be used through @code{gmp_printf}, this includes for
   instance extensions registered with GLIBC @code{register_printf_function}.
   Also currently there's no support for POSIX @samp{$} style numbered arguments
   (perhaps this will be added in the future).
   
   The precision field has it's usual meaning for integer @samp{Z} and float
   @samp{F} types, but is currently undefined for @samp{Q} and should not be used
   with that.
   
   @code{mpf_t} conversions only ever generate as many digits as can be
   accurately represented by the operand, the same as @code{mpf_get_str} does.
   Zeros will be used if necessary to pad to the requested precision.  This
   happens even for an @samp{f} conversion of an @code{mpf_t} which is an
   integer, for instance @math{2^@W{1024}} in an @code{mpf_t} of 128 bits
   precision will only produce about 40 digits, then pad with zeros to the
   decimal point.  An empty precision field like @samp{%.Fe} or @samp{%.Ff} can
   be used to specifically request just the significant digits.
   
   The decimal point character (or string) is taken from the current locale
   settings on systems which provide @code{localeconv} (@pxref{Locales,,Locales
   and Internationalization,libc,The GNU C Library Reference Manual}).  The C
   library will normally do the same for standard float output.
   
   The format string is only interpreted as plain @code{char}s, multibyte
   characters are not recognised.  Perhaps this will change in the future.
   
   
   @node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output
   @section Functions
   
   Each of the following functions is similar to the corresponding C library
   function.  The basic @code{printf} forms take a variable argument list.  The
   @code{vprintf} forms take an argument pointer, see @ref{Variadic
   Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3
   va_start}.
   
   It should be emphasised that if a format string is invalid, or the arguments
   don't match what the format specifies, then the behaviour of any of these
   functions will be unpredictable.  GCC format string checking is not available,
   since it doesn't recognise the GMP extensions.
   
   The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return
   @math{-1} to indicate a write error.  All the functions can return @math{-1}
   if the C library @code{printf} variant in use returns @math{-1}, but this
   shouldn't normally occur.
   
   @deftypefun int gmp_printf (const char *@var{fmt}, ...)
   @deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap})
   Print to the standard output @code{stdout}.  Return the number of characters
   written, or @math{-1} if an error occurred.
   @end deftypefun
   
   @deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
   Print to the stream @var{fp}.  Return the number of characters written, or
   @math{-1} if an error occurred.
   @end deftypefun
   
   @deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap})
   Form a null-terminated string in @var{buf}.  Return the number of characters
   written, excluding the terminating null.
   
   No overlap is permitted between the space at @var{buf} and the string
   @var{fmt}.
   
   These functions are not recommended, since there's no protection against
   exceeding the space available at @var{buf}.
   @end deftypefun
   
   @deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap})
   Form a null-terminated string in @var{buf}.  No more than @var{size} bytes
   will be written.  To get the full output, @var{size} must be enough for the
   string and null-terminator.
   
   The return value is the total number of characters which ought to have been
   produced, excluding the terminating null.  If @math{@var{retval} @ge{}
   @var{size}} then the actual output has been truncated to the first
   @math{@var{size}-1} characters, and a null appended.
   
   No overlap is permitted between the region @{@var{buf},@var{size}@} and the
   @var{fmt} string.
   
   Notice the return value is in ISO C99 @code{snprintf} style.  This is so even
   if the C library @code{vsnprintf} is the older GLIBC 2.0.x style.
   @end deftypefun
   
   @deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vasprintf (char *@var{pp}, const char *@var{fmt}, va_list @var{ap})
   Form a null-terminated string in a block of memory obtained from the current
   memory allocation function (@pxref{Custom Allocation}).  The block will be the
   size of the string and null-terminator.  Put the address of the block in
   *@var{pp}.  Return the number of characters produced, excluding the
   null-terminator.
   
   Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return
   @math{-1} if there's no more memory available, it lets the current allocation
   function handle that.
   @end deftypefun
   
   @deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap})
   Append to the current obstack object, in the same style as
   @code{obstack_printf}.  Return the number of characters written.  A
   null-terminator is not written.
   
   @var{fmt} cannot be within the current obstack object, since the object might
   move as it grows.
   
   These functions are available only when the C library provides the obstack
   feature, which probably means only on GNU systems, see
   @ref{Obstacks,,,libc,The GNU C Library Reference Manual}.
   @end deftypefun
   
   
   @node C++ Formatted Output,  , Formatted Output Functions, Formatted Output
   @section C++ Formatted Output
   @cindex C++ @code{ostream} output
   @cindex @code{ostream} output
   
   The following functions are provided in @file{libgmpxx}, which is built if C++
   support is enabled (@pxref{Build Options}).  Prototypes are available from
   @code{<gmp.h>}.
   
   @deftypefun ostream& operator<< (ostream& @var{stream}, mpz_t @var{op})
   Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
   @code{ios::width} is reset to 0 after output, the same as the standard
   @code{ostream operator<<} routines do.
   
   In hex or octal, @var{op} is printed as a signed number, the same as for
   decimal.  This is unlike the standard @code{operator<<} routines on @code{int}
   etc, which instead give twos complement.
   @end deftypefun
   
   @deftypefun ostream& operator<< (ostream& @var{stream}, mpq_t @var{op})
   Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
   @code{ios::width} is reset to 0 after output, the same as the standard
   @code{ostream operator<<} routines do.
   
   Output will be a fraction like @samp{5/9}, or if the denominator is 1 then
   just a plain integer like @samp{123}.
   
   In hex or octal, @var{op} is printed as a signed value, the same as for
   decimal.  If @code{ios::showbase} is set then a base indicator is shown on
   both the numerator and denominator (if the denominator is required).
   @end deftypefun
   
   @deftypefun ostream& operator<< (ostream& @var{stream}, mpf_t @var{op})
   Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
   @code{ios::width} is reset to 0 after output, the same as the standard
   @code{ostream operator<<} routines do.  The decimal point follows the current
   locale, on systems providing @code{localeconv}.
   
   Hex and octal are supported, unlike the standard @code{operator<<} on
   @code{double}.  The mantissa will be in hex or octal, the exponent will be in
   decimal.  For hex the exponent delimiter is an @samp{@@}.  This is as per
   @code{mpf_out_str}.
   
   @code{ios::showbase} is supported, and will put a base on the mantissa, for
   example hex @samp{0x1.8} or @samp{0x0.8}, or octal @samp{01.4} or @samp{00.4}.
   This last form is slightly strange, but at least differentiates itself from
   decimal.
   @end deftypefun
   
   These operators mean that GMP types can be printed in the usual C++ way, for
   example,
   
   @example
   mpz_t  z;
   int    n;
   ...
   cout << "iteration " << n << " value " << z << "\n";
   @end example
   
   But note that @code{ostream} output (and @code{istream} input, @pxref{C++
   Formatted Input}) is the only overloading available and using for instance
   @code{+} with an @code{mpz_t} will have unpredictable results.
   
   
   @node Formatted Input, C++ Class Interface, Formatted Output, Top
   @chapter Formatted Input
   @cindex Formatted input
   @cindex @code{scanf} formatted input
   
   @menu
   * Formatted Input Strings::
   * Formatted Input Functions::
   * C++ Formatted Input::
   @end menu
   
   
   @node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input
   @section Formatted Input Strings
   
   @code{gmp_scanf} and friends accept format strings similar to the standard C
   @code{scanf} (@pxref{Formatted Input,,,libc,The GNU C Library Reference
   Manual}).  A format specification is of the form
   
   @example
   % [flags] [width] [type] conv
   @end example
   
   GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
   and @code{mpf_t} respectively.  @samp{Z} and @samp{Q} behave like integers.
   @samp{Q} will read a @samp{/} and a denominator, if present.  @samp{F} behaves
   like a float.
   
   GMP variables don't require an @code{&} when passed to @code{gmp_scanf}, since
   they're already ``call-by-reference''.  For example,
   
   @example
   /* to read say "a(5) = 1234" */
   int   n;
   mpz_t z;
   gmp_scanf ("a(%d) = %Zd\n", &n, z);
   
   mpq_t q1, q2;
   gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
   
   /* to read say "topleft (1.55,-2.66)" */
   mpf_t x, y;
   char  buf[32];
   gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
   @end example
   
   All the standard C @code{scanf} types behave the same as in the C library
   @code{scanf}, and can be freely intermixed with the GMP extensions.  In the
   current implementation the standard parts of the format string are simply
   handed to @code{scanf} and only the GMP extensions handled directly.
   
   The flags accepted are as follows.  @samp{a} and @samp{'} will depend on
   support from the C library, and @samp{'} cannot be used with GMP types.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{*} @tab read but don't store
   @item @nicode{a} @tab allocate a buffer (string conversions)
   @item @nicode{'} @tab group digits, GLIBC style (not GMP types)
   @end multitable
   @end quotation
   
   The standard types accepted are as follows.  @samp{h} and @samp{l} are
   portable, the rest will depend on the compiler (or include files) for the type
   and the C library for the input.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{h}  @tab @nicode{short}
   @item @nicode{hh} @tab @nicode{char}
   @item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
   @item @nicode{l}  @tab @nicode{long int}, @nicode{double} or @nicode{wchar_t}
   @item @nicode{ll} @tab @nicode{long long}
   @item @nicode{L}  @tab @nicode{long double}
   @item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
   @item @nicode{t}  @tab @nicode{ptrdiff_t}
   @item @nicode{z}  @tab @nicode{size_t}
   @end multitable
   @end quotation
   
   @noindent
   The GMP types are
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{F}  @tab @nicode{mpf_t}, float conversions
   @item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
   @item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
   @end multitable
   @end quotation
   
   The conversions accepted are as follows.  @samp{p} and @samp{[} will depend on
   support from the C library, the rest are standard.
   
   @quotation
   @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item @nicode{c}            @tab character or characters
   @item @nicode{d}            @tab decimal integer
   @item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G}
                               @tab float
   @item @nicode{i}            @tab integer with base indicator
   @item @nicode{n}            @tab characters read so far
   @item @nicode{o}            @tab octal integer
   @item @nicode{p}            @tab pointer
   @item @nicode{s}            @tab string of non-whitespace characters
   @item @nicode{u}            @tab decimal integer
   @item @nicode{x} @nicode{X} @tab hex integer
   @item @nicode{[}            @tab string of characters in a set
   @end multitable
   @end quotation
   
   @samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all
   read either fixed point or scientific format, and either @samp{e} or @samp{E}
   for the exponent in scientific format.
   
   @samp{x} and @samp{X} are identical, both accept both upper and lower case
   hexadecimal.
   
   @samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative
   values.  For the standard C types these are described as ``unsigned''
   conversions, but that merely affects certain overflow handling, negatives are
   still allowed (see @code{strtoul}, @ref{Parsing of Integers,,,libc,The GNU C
   Library Reference Manual}).  For GMP types there are no overflows, and
   @samp{d} and @samp{u} are identical.
   
   @samp{Q} type reads the numerator and (optional) denominator as given.  If the
   value might not be in canonical form then @code{mpq_canonicalize} must be
   called before using it in any calculations (@pxref{Rational Number
   Functions}).
   
   @samp{Qi} will read a base specification separately for the numerator and
   denominator.  For example @samp{0x10/11} would be 16/11, whereas
   @samp{0x10/0x11} would be 16/17.
   
   @samp{n} can be used with any of the types above, even the GMP types.
   @samp{*} to suppress assignment is allowed, though the field would then do
   nothing at all.
   
   Other conversions or types that might be accepted by the C library
   @code{scanf} cannot be used through @code{gmp_scanf}.
   
   Whitespace is read and discarded before a field, except for @samp{c} and
   @samp{[} conversions.
   
   For float conversions, the decimal point character (or string) expected is
   taken from the current locale settings on systems which provide
   @code{localeconv} (@pxref{Locales,,Locales and Internationalization,libc,The
   GNU C Library Reference Manual}).  The C library will normally do the same for
   standard float input.
   
   The format string is only interpreted as plain @code{char}s, multibyte
   characters are not recognised.  Perhaps this will change in the future.
   
   
   @node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input
   @section Formatted Input Functions
   
   Each of the following functions is similar to the corresponding C library
   function.  The plain @code{scanf} forms take a variable argument list.  The
   @code{vscanf} forms take an argument pointer, see @ref{Variadic
   Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3
   va_start}.
   
   It should be emphasised that if a format string is invalid, or the arguments
   don't match what the format specifies, then the behaviour of any of these
   functions will be unpredictable.  GCC format string checking is not available,
   since it doesn't recognise the GMP extensions.
   
   No overlap is permitted between the @var{fmt} string and any of the results
   produced.
   
   @deftypefun int gmp_scanf (const char *@var{fmt}, ...)
   @deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap})
   Read from the standard input @code{stdin}.
   @end deftypefun
   
   @deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
   Read from the stream @var{fp}.
   @end deftypefun
   
   @deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, ...)
   @deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap})
   Read from a null-terminated string @var{s}.
   @end deftypefun
   
   The return value from each of these functions is the same as the standard C99
   @code{scanf}, namely the number of fields successfully parsed and stored.
   @samp{%n} fields and fields read but suppressed by @samp{*} don't count
   towards the return value.
   
   If end of file or file error, or end of string, is reached when a match is
   required, and when no previous non-suppressed fields have matched, then the
   return value is EOF instead of 0.  A match is required for a literal character
   in the format string or a field other than @samp{%n}.  Whitespace in the
   format string is only an optional match and won't induce an EOF in this
   fashion.  Leading whitespace read and discarded for a field doesn't count as a
   match.
   
   
   @node C++ Formatted Input,  , Formatted Input Functions, Formatted Input
   @section C++ Formatted Input
   @cindex C++ @code{istream} input
   @cindex @code{istream} input
   
   The following functions are provided in @file{libgmpxx}, which is built only
   if C++ support is enabled (@pxref{Build Options}).  Prototypes are available
   from @code{<gmp.h>}.
   
   @deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop})
   Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
   @end deftypefun
   
   @deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop})
   Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
   
   An integer like @samp{123} will be read, or a fraction like @samp{5/9}.  If
   the fraction is not in canonical form then @code{mpq_canonicalize} must be
   called (@pxref{Rational Number Functions}).
   @end deftypefun
   
   @deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop})
   Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
   
   Hex or octal floats are not supported, but might be in the future.
   @end deftypefun
   
   These operators mean that GMP types can be read in the usual C++ way, for
   example,
   
   @example
   mpz_t  z;
   ...
   cin >> z;
   @end example
   
   But note that @code{istream} input (and @code{ostream} output, @pxref{C++
   Formatted Output}) is the only overloading available and using for instance
   @code{+} with an @code{mpz_t} will have unpredictable results.
   
   
   @node C++ Class Interface, BSD Compatible Functions, Formatted Input, Top
   @chapter C++ Class Interface
   @cindex C++ Interface
   
   This chapter describes the C++ class based interface to GMP.
   
   All GMP C language types and functions can be used in C++ programs, since
   @file{gmp.h} has @code{extern "C"} qualifiers, but the class interface offers
   overloaded functions and operators which may be more convenient.
   
   Due to the implementation of this interface, a reasonably recent C++ compiler
   is required, one supporting namespaces, partial specialization of templates
   and member templates.  For GCC this means version 2.91 or later.
   
   @strong{Everything described in this chapter is to be considered preliminary
   and might be subject to incompatible changes if some unforeseen difficulty
   reveals itself.}
   
   @menu
   * C++ Interface General::
   * C++ Interface Integers::
   * C++ Interface Rationals::
   * C++ Interface Floats::
   * C++ Interface MPFR::
   * C++ Interface Random Numbers::
   * C++ Interface Limitations::
   @end menu
   
   
   @node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface
   @section C++ Interface General
   
   @noindent
   All the C++ classes and functions are available with
   
   @cindex gmpxx.h
   @example
   #include <gmpxx.h>
   @end example
   
   Programs should be linked with the @file{libgmpxx} and @file{libgmp}
   libraries.  For example,
   
   @example
   g++ mycxxprog.cc -lgmpxx -lgmp
   @end example
   
   @noindent
   The classes defined are
   
   @deftp Class mpz_class
   @deftpx Class mpq_class
   @deftpx Class mpf_class
   @end deftp
   
   The standard operators and various standard functions are overloaded to allow
   arithmetic with these classes.  For example,
   
   @example
   int
   main (void)
   @{
     mpz_class a, b, c;
   
     a = 1234;
     b = "-5678";
     c = a+b;
     cout << "sum is " << c << "\n";
     cout << "absolute value is " << abs(c) << "\n";
   
     return 0;
   @}
   @end example
   
   An important feature of the implementation is that an expression like
   @code{a=b+c} results in a single call to the corresponding @code{mpz_add},
   without using a temporary for the @code{b+c} part.  Expressions which by their
   nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries
   though.
   
   The classes can be freely intermixed in expressions, as can the classes and
   the standard types @code{long}, @code{unsigned long} and @code{double}.
   Smaller types like @code{int} or @code{float} can also be intermixed, since
   C++ will promote them.
   
   Note that @code{bool} is not accepted directly, but must be explicitly cast to
   an @code{int} first.  This is because C++ will automatically convert any
   pointer to a @code{bool}, so if GMP accepted @code{bool} it would make all
   sorts of invalid class and pointer combinations compile but almost certainly
   not do anything sensible.
   
   Conversions back from the classes to standard C++ types aren't done
   automatically, instead member functions like @code{get_si} are provided (see
   the following sections for details).
   
   Also there are no automatic conversions from the classes to the corresponding
   GMP C types, instead a reference to the underlying C object can be obtained
   with the following functions,
   
   @deftypefun mpz_t mpz_class::get_mpz_t ()
   @deftypefunx mpq_t mpq_class::get_mpq_t ()
   @deftypefunx mpf_t mpf_class::get_mpf_t ()
   @end deftypefun
   
   These can be used to call a C function which doesn't have a C++ class
   interface.  For example to set @code{a} to the GCD of @code{b} and @code{c},
   
   @example
   mpz_class a, b, c;
   ...
   mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
   @end example
   
   In the other direction, a class can be initialized from the corresponding GMP
   C type, or assigned to if an explicit constructor is used.  In both cases this
   makes a copy of the value, it doesn't create any sort of association.  For
   example,
   
   @example
   mpz_t z;
   // ... init and calculate z ...
   mpz_class x(z);
   mpz_class y;
   y = mpz_class (z);
   @end example
   
   There are no namespace setups in @file{gmpxx.h}, all types and functions are
   simply put into the global namespace.  This is what @file{gmp.h} has done in
   the past, and continues to do for compatibility.  The extras provided by
   @file{gmpxx.h} follow GMP naming conventions and are unlikely to clash with
   anything.
   
   
   @node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface
   @section C++ Interface Integers
   
   @deftypefun void mpz_class::mpz_class (type @var{n})
   Construct an @code{mpz_class}.  All the standard C++ types may be used, except
   @code{long long} and @code{long double}, and all the GMP C++ classes can be
   used.  Any necessary conversion follows the corresponding C function, for
   example @code{double} follows @code{mpz_set_d} (@pxref{Assigning Integers}).
   @end deftypefun
   
   @deftypefun void mpz_class::mpz_class (mpz_t @var{z})
   Construct an @code{mpz_class} from an @code{mpz_t}.  The value in @var{z} is
   copied into the new @code{mpz_class}, there won't be any permanent association
   between it and @var{z}.
   @end deftypefun
   
   @deftypefun void mpz_class::mpz_class (const char *@var{s})
   @deftypefunx void mpz_class::mpz_class (const char *@var{s}, int base)
   @deftypefunx void mpz_class::mpz_class (const string& @var{s})
   @deftypefunx void mpz_class::mpz_class (const string& @var{s}, int base)
   Construct an @code{mpz_class} converted from a string using
   @code{mpz_set_str}, (@pxref{Assigning Integers}).  If the @var{base} is not
   given then 0 is used.
   @end deftypefun
   
   @deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d})
   @deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d})
   Divisions involving @code{mpz_class} round towards zero, as per the
   @code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}).
   This corresponds to the rounding used for plain @code{int} calculations on
   most machines.
   
   The @code{mpz_fdiv...} or @code{mpz_cdiv...} functions can always be called
   directly if desired.  For example,
   
   @example
   mpz_class q, a, d;
   ...
   mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
   @end example
   @end deftypefun
   
   @deftypefun mpz_class abs (mpz_class @var{op1})
   @deftypefunx int cmp (mpz_class @var{op1}, type @var{op2})
   @deftypefunx int cmp (type @var{op1}, mpz_class @var{op2})
   @deftypefunx double mpz_class::get_d (void)
   @deftypefunx long mpz_class::get_si (void)
   @deftypefunx {unsigned long} mpz_class::get_ui (void)
   @maybepagebreak
   @deftypefunx bool mpz_class::fits_sint_p (void)
   @deftypefunx bool mpz_class::fits_slong_p (void)
   @deftypefunx bool mpz_class::fits_sshort_p (void)
   @maybepagebreak
   @deftypefunx bool mpz_class::fits_uint_p (void)
   @deftypefunx bool mpz_class::fits_ulong_p (void)
   @deftypefunx bool mpz_class::fits_ushort_p (void)
   @maybepagebreak
   @deftypefunx int sgn (mpz_class @var{op})
   @deftypefunx mpz_class sqrt (mpz_class @var{op})
   These functions provide a C++ class interface to the corresponding GMP C
   routines.
   
   @code{cmp} can be used with any of the classes or the standard C++ types,
   except @code{long long} and @code{long double}.
   @end deftypefun
   
   @sp 1
   Overloaded operators for combinations of @code{mpz_class} and @code{double}
   are provided for completeness, but it should be noted that if the given
   @code{double} is not an integer then the way any rounding is done is currently
   unspecified.  The rounding might take place at the start, in the middle, or at
   the end of the operation, and it might change in the future.
   
   Conversions between @code{mpz_class} and @code{double}, however, are defined
   to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}.
   And comparisons are always made exactly, as per @code{mpz_cmp_d}.
   
   
   @node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface
   @section C++ Interface Rationals
   
   In all the following constructors, if a fraction is given then it should be in
   canonical form, or if not then @code{mpq_class::canonicalize} called.
   
   @deftypefun void mpq_class::mpq_class (type @var{op})
   @deftypefunx void mpq_class::mpq_class (integer @var{num}, integer @var{den})
   Construct an @code{mpq_class}.  The initial value can be a single value of any
   type, or a pair of integers (@code{mpz_class} or standard C++ integer types)
   representing a fraction, except that @code{long long} and @code{long double}
   are not supported.  For example,
   
   @example
   mpq_class q (99);
   mpq_class q (1.75);
   mpq_class q (1, 3);
   @end example
   @end deftypefun
   
   @deftypefun void mpq_class::mpq_class (mpq_t @var{q})
   Construct an @code{mpq_class} from an @code{mpq_t}.  The value in @var{q} is
   copied into the new @code{mpq_class}, there won't be any permanent association
   between it and @var{q}.
   @end deftypefun
   
   @deftypefun void mpq_class::mpq_class (const char *@var{s})
   @deftypefunx void mpq_class::mpq_class (const char *@var{s}, int base)
   @deftypefunx void mpq_class::mpq_class (const string& @var{s})
   @deftypefunx void mpq_class::mpq_class (const string& @var{s}, int base)
   Construct an @code{mpq_class} converted from a string using
   @code{mpq_set_str}, (@pxref{Initializing Rationals}).  If the @var{base} is
   not given then 0 is used.
   @end deftypefun
   
   @deftypefun void mpq_class::canonicalize ()
   Put an @code{mpq_class} into canonical form, as per @ref{Rational Number
   Functions}.  All arithmetic operators require their operands in canonical
   form, and will return results in canonical form.
   @end deftypefun
   
   @deftypefun mpq_class abs (mpq_class @var{op})
   @deftypefunx int cmp (mpq_class @var{op1}, type @var{op2})
   @deftypefunx int cmp (type @var{op1}, mpq_class @var{op2})
   @maybepagebreak
   @deftypefunx double mpq_class::get_d (void)
   @deftypefunx int sgn (mpq_class @var{op})
   These functions provide a C++ class interface to the corresponding GMP C
   routines.
   
   @code{cmp} can be used with any of the classes or the standard C++ types,
   except @code{long long} and @code{long double}.
   @end deftypefun
   
   @deftypefun {mpz_class&} mpq_class::get_num ()
   @deftypefunx {mpz_class&} mpq_class::get_den ()
   Get a reference to an @code{mpz_class} which is the numerator or denominator
   of an @code{mpq_class}.  This can be used both for read and write access.  If
   the object returned is modified, it modifies the original @code{mpq_class}.
   
   If direct manipulation might produce a non-canonical value, then
   @code{mpq_class::canonicalize} must be called before further operations.
   @end deftypefun
   
   @deftypefun mpz_t mpq_class::get_num_mpz_t ()
   @deftypefunx mpz_t mpq_class::get_den_mpz_t ()
   Get a reference to the underlying @code{mpz_t} numerator or denominator of an
   @code{mpq_class}.  This can be passed to C functions expecting an
   @code{mpz_t}.  Any modifications made to the @code{mpz_t} will modify the
   original @code{mpq_class}.
   
   If direct manipulation might produce a non-canonical value, then
   @code{mpq_class::canonicalize} must be called before further operations.
   @end deftypefun
   
   @deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop});
   Read @var{rop} from @var{stream}, using its @code{ios} formatting settings,
   the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}).
   
   If the @var{rop} read might not be in canonical form then
   @code{mpq_class::canonicalize} must be called.
   @end deftypefun
   
   
   @node C++ Interface Floats, C++ Interface MPFR, C++ Interface Rationals, C++ Class Interface
   @section C++ Interface Floats
   
   When an expression requires the use of temporary intermediate @code{mpf_class}
   values, like @code{f=g*h+x*y}, those temporaries will have the same precision
   as the destination @code{f}.  Explicit constructors can be used if this
   doesn't suit.
   
   @deftypefun {} mpf_class::mpf_class (type @var{op})
   @deftypefunx {} mpf_class::mpf_class (type @var{op}, unsigned long @var{prec})
   Construct an @code{mpf_class}.  Any standard C++ type can be used, except
   @code{long long} and @code{long double}, and any of the GMP C++ classes can be
   used.
   
   If @var{prec} is given, the initial precision is that value, in bits.  If
   @var{prec} is not given, then the initial precision is determined by the type
   of @var{op} given.  An @code{mpz_class}, @code{mpq_class}, string, or C++
   builtin type will give the default @code{mpf} precision (@pxref{Initializing
   Floats}).  An @code{mpf_class} or expression will give the precision of that
   value.  The precision of a binary expression is the higher of the two
   operands.
   
   @example
   mpf_class f(1.5);        // default precision
   mpf_class f(1.5, 500);   // 500 bits (at least)
   mpf_class f(x);          // precision of x
   mpf_class f(abs(x));     // precision of x
   mpf_class f(-g, 1000);   // 1000 bits (at least)
   mpf_class f(x+y);        // greater of precisions of x and y
   @end example
   @end deftypefun
   
   @deftypefun mpf_class abs (mpf_class @var{op})
   @deftypefunx mpf_class ceil (mpf_class @var{op})
   @deftypefunx int cmp (mpf_class @var{op1}, type @var{op2})
   @deftypefunx int cmp (type @var{op1}, mpf_class @var{op2})
   @maybepagebreak
   @deftypefunx mpf_class floor (mpf_class @var{op})
   @deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2})
   @deftypefunx double mpf_class::get_d (void)
   @deftypefunx long mpf_class::get_si (void)
   @deftypefunx {unsigned long} mpf_class::get_ui (void)
   @maybepagebreak
   @deftypefunx bool mpf_class::fits_sint_p (void)
   @deftypefunx bool mpf_class::fits_slong_p (void)
   @deftypefunx bool mpf_class::fits_sshort_p (void)
   @maybepagebreak
   @deftypefunx bool mpf_class::fits_uint_p (void)
   @deftypefunx bool mpf_class::fits_ulong_p (void)
   @deftypefunx bool mpf_class::fits_ushort_p (void)
   @maybepagebreak
   @deftypefunx int sgn (mpf_class @var{op})
   @deftypefunx mpf_class sqrt (mpf_class @var{op})
   @deftypefunx mpf_class trunc (mpf_class @var{op})
   These functions provide a C++ class interface to the corresponding GMP C
   routines.
   
   @code{cmp} can be used with any of the classes or the standard C++ types,
   except @code{long long} and @code{long double}.
   
   The accuracy provided by @code{hypot} is not currently guaranteed.
   @end deftypefun
   
   @deftypefun {unsigned long int} mpf_class::get_prec ()
   @deftypefunx void mpf_class::set_prec (unsigned long @var{prec})
   @deftypefunx void mpf_class::set_prec_raw (unsigned long @var{prec})
   Get or set the current precision of an @code{mpf_class}.
   
   The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing
   Floats}) apply to @code{mpf_class::set_prec_raw}.  Note in particular that the
   @code{mpf_class} must be restored to it's allocated precision before being
   destroyed.  This must be done by application code, there's no automatic
   mechanism for it.
   @end deftypefun
   
   
   @node C++ Interface MPFR, C++ Interface Random Numbers, C++ Interface Floats, C++ Class Interface
   @section C++ Interface MPFR
   
   The C++ class interface to MPFR is provided if MPFR is enabled (@pxref{Build
   Options}).  This interface must be regarded as preliminary and possibly
   subject to incompatible changes in the future, since MPFR itself is
   preliminary.  All definitions can be obtained with
   
   @cindex mpfrxx.h
   @example
   #include <mpfrxx.h>
   @end example
   
   @noindent
   This defines
   
   @deftp Class mpfr_class
   @end deftp
   
   @noindent
   which behaves similarly to @code{mpf_class} (@pxref{C++ Interface Floats}).
   
   
   @node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface MPFR, C++ Class Interface
   @section C++ Interface Random Numbers
   
   @deftp Class gmp_randclass
   The C++ class interface to the GMP random number functions uses
   @code{gmp_randclass} to hold an algorithm selection and current state, as per
   @code{gmp_randstate_t}.
   @end deftp
   
   @deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, ...), ...)
   Construct a @code{gmp_randclass}, using a call to the given @var{randinit}
   function (@pxref{Random State Initialization}).  The arguments expected are
   the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}.
   For example,
   
   @example
   gmp_randclass r1 (gmp_randinit_default);
   gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
   gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
   @end example
   
   @code{gmp_randinit_lc_2exp_size} can fail if the size requested is too big,
   the behaviour of @code{gmp_randclass::gmp_randclass} is undefined in this case
   (perhaps this will change in the future).
   @end deftypefun
   
   @deftypefun {} gmp_randclass::gmp_randclass (gmp_randalg_t @var{alg}, ...)
   Construct a @code{gmp_randclass} using the same parameters as
   @code{gmp_randinit} (@pxref{Random State Initialization}).  This function is
   obsolete and the above @var{randinit} style should be preferred.
   @end deftypefun
   
   @deftypefun void gmp_randclass::seed (unsigned long int @var{s})
   @deftypefunx void gmp_randclass::seed (mpz_class @var{s})
   Seed a random number generator.  See @pxref{Random Number Functions}, for how
   to choose a good seed.
   @end deftypefun
   
   @deftypefun mpz_class gmp_randclass::get_z_bits (unsigned long @var{bits})
   @deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits})
   Generate a random integer with a specified number of bits.
   @end deftypefun
   
   @deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n})
   Generate a random integer in the range 0 to @math{@var{n}-1} inclusive.
   @end deftypefun
   
   @deftypefun mpf_class gmp_randclass::get_f ()
   @deftypefunx mpf_class gmp_randclass::get_f (unsigned long @var{prec})
   Generate a random float @var{f} in the range @math{0 <= @var{f} < 1}.  @var{f}
   will be to @var{prec} bits precision, or if @var{prec} is not given then to
   the precision of the destination.  For example,
   
   @example
   gmp_randclass  r;
   ...
   mpf_class  f (0, 512);   // 512 bits precision
   f = r.get_f();           // random number, 512 bits
   @end example
   @end deftypefun
   
   
   
   @node C++ Interface Limitations,  , C++ Interface Random Numbers, C++ Class Interface
   @section C++ Interface Limitations
   
   @table @asis
   @item @code{mpq_class} and Templated Reading
   A generic piece of template code probably won't know that @code{mpq_class}
   requires a @code{canonicalize} call if inputs read with @code{operator>>}
   might be non-canonical.  This can lead to incorrect results.
   
   @code{operator>>} behaves as it does for reasons of efficiency.  A
   canonicalize can be quite time consuming on large operands, and is best
   avoided if it's not necessary.
   
   But this potential difficulty reduces the usefulness of @code{mpq_class}.
   Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in
   the future, maybe a preprocessor define, a global flag, or an @code{ios} flag
   pressed into service.  Or maybe, at the risk of inconsistency, the
   @code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t}
   @code{operator>>} not doing so, for use on those occasions when that's
   acceptable.  Send feedback or alternate ideas to @email{bug-gmp@@gnu.org}.
   
   @item Subclassing
   Subclassing the GMP C++ classes works, but is not currently recommended.
   
   Expressions involving subclasses resolve correctly (or seem to), but in normal
   C++ fashion the subclass doesn't inherit constructors and assignments.
   There's many of those in the GMP classes, and a good way to reestablish them
   in a subclass is not yet provided.
   
   @item Templated Expressions
   
   A subtle difficulty exists when using expressions together with
   application-defined template functions.  Consider the following, with @code{T}
   intended to be some numeric type,
   
   @example
   template <class T>
   T fun (const T &, const T &);
   @end example
   
   @noindent
   When used with, say, plain @code{mpz_class} variables, it works fine: @code{T}
   is resolved as @code{mpz_class}.
   
   @example
   mpz_class f(1), g(2);
   fun (f, g);    // Good
   @end example
   
   @noindent
   But when one of the arguments is an expression, it doesn't work.
   
   @example
   mpz_class f(1), g(2), h(3);
   fun (f, g+h);  // Bad
   @end example
   
   This is because @code{g+h} ends up being a certain expression template type
   internal to @code{gmpxx.h}, which the C++ template resolution rules are unable
   to automatically convert to @code{mpz_class}.  The workaround is simply to add
   an explicit cast.
   
   @example
   mpz_class f(1), g(2), h(3);
   fun (f, mpz_class(g+h));  // Good
   @end example
   
   Similarly, within @code{fun} it may be necessary to cast an expression to type
   @code{T} when calling a templated @code{fun2}.
   
   @example
   template <class T>
   void fun (T f, T g)
   @{
     fun2 (f, f+g);     // Bad
   @}
   
   template <class T>
   void fun (T f, T g)
   @{
     fun2 (f, T(f+g));  // Good
   @}
   @end example
   @end table
   
   
   @node BSD Compatible Functions, Custom Allocation, C++ Class Interface, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Berkeley MP Compatible Functions  @chapter Berkeley MP Compatible Functions
   @cindex Berkeley MP compatible functions
 @cindex BSD MP compatible functions  @cindex BSD MP compatible functions
   
 These functions are intended to be fully compatible with the Berkeley MP  These functions are intended to be fully compatible with the Berkeley MP
 library which is available on many BSD derived U*ix systems.  library which is available on many BSD derived U*ix systems.  The
   @samp{--enable-mpbsd} option must be used when building GNU MP to make these
   available (@pxref{Installing GMP}).
   
 The original Berkeley MP library has a usage restriction: you cannot use the  The original Berkeley MP library has a usage restriction: you cannot use the
 same variable as both source and destination in a single function call.  The  same variable as both source and destination in a single function call.  The
Line 2445  Apart from the incomplete set of functions, the interf
Line 6143  Apart from the incomplete set of functions, the interf
 with @code{pow} in @file{libm.a}.  with @code{pow} in @file{libm.a}.
   
 @cindex @file{mp.h}  @cindex @file{mp.h}
 Include the header @file{mp.h} to get the definition of the necessary types  Include the header @file{mp.h} to get the definition of the necessary types and
 and functions.  If you are on a BSD derived system, make sure to include GNU  functions.  If you are on a BSD derived system, make sure to include GNU
 @file{mp.h} if you are going to link the GNU @file{libmp.a} to you program.  @file{mp.h} if you are going to link the GNU @file{libmp.a} to your program.
 This means that you probably need to give the -I<dir> option to the compiler,  This means that you probably need to give the @samp{-I<dir>} option to the
 where <dir> is the directory where you have GNU @file{mp.h}.  compiler, where @samp{<dir>} is the directory where you have GNU @file{mp.h}.
   
 @deftypefun {MINT *} itom (signed short int @var{initial_value})  @deftypefun {MINT *} itom (signed short int @var{initial_value})
 Allocate an integer consisting of a @code{MINT} object and dynamic limb space.  Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
Line 2459  Initialize the integer to @var{initial_value}.  Return
Line 6157  Initialize the integer to @var{initial_value}.  Return
   
 @deftypefun {MINT *} xtom (char *@var{initial_value})  @deftypefun {MINT *} xtom (char *@var{initial_value})
 Allocate an integer consisting of a @code{MINT} object and dynamic limb space.  Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
 Initialize the integer from @var{initial_value}, a hexadecimal, '\0'-terminate  Initialize the integer from @var{initial_value}, a hexadecimal,
 C string.  Return a pointer to the @code{MINT} object.  null-terminated C string.  Return a pointer to the @code{MINT} object.
 @end deftypefun  @end deftypefun
   
 @deftypefun void move (MINT *@var{src}, MINT *@var{dest})  @deftypefun void move (MINT *@var{src}, MINT *@var{dest})
Line 2478  Subtract @var{src_2} from @var{src_1} and put the diff
Line 6176  Subtract @var{src_2} from @var{src_1} and put the diff
 @end deftypefun  @end deftypefun
   
 @deftypefun void mult (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination})  @deftypefun void mult (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination})
 Multiply @var{src_1} and @var{src_2} and put the product in  Multiply @var{src_1} and @var{src_2} and put the product in @var{destination}.
 @var{destination}.  
 @end deftypefun  @end deftypefun
   
 @deftypefun void mdiv (MINT *@var{dividend}, MINT *@var{divisor}, MINT *@var{quotient}, MINT *@var{remainder})  @deftypefun void mdiv (MINT *@var{dividend}, MINT *@var{divisor}, MINT *@var{quotient}, MINT *@var{remainder})
Line 2492  Some implementations of these functions work different
Line 6189  Some implementations of these functions work different
 negative arguments.  negative arguments.
 @end deftypefun  @end deftypefun
   
 @deftypefun void msqrt (MINT *@var{operand}, MINT *@var{root}, MINT *@var{remainder})  @deftypefun void msqrt (MINT *@var{op}, MINT *@var{root}, MINT *@var{remainder})
 @ifinfo  Set @var{root} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
 Set @var{root} to the truncated integer part of the square root of  of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{remainder} to
 @var{operand}.  Set @var{remainder} to  @m{(@var{op} - @var{root}^2), @var{op}@minus{}@var{root}*@var{root}}, i.e.
 @var{operand}@minus{}@var{root}*@var{root},  zero if @var{op} is a perfect square.
 @end ifinfo  
 @iftex  
 @tex  
 Set @var{root} to $\lfloor\sqrt{@var{operand}}\rfloor$, like  
 @code{mpz_sqrt}.  Set @var{remainder} to $(operand - root^2)$,  
 @end tex  
 @end iftex  
 (i.e., zero if @var{operand} is a perfect square).  
   
 If @var{root} and @var{remainder} are the same variable, the results are  If @var{root} and @var{remainder} are the same variable, the results are
 undefined.  undefined.
Line 2518  Set @var{dest} to (@var{base} raised to @var{exp}) mod
Line 6207  Set @var{dest} to (@var{base} raised to @var{exp}) mod
 Set @var{dest} to @var{base} raised to @var{exp}.  Set @var{dest} to @var{base} raised to @var{exp}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void gcd (MINT *@var{operand1}, MINT *@var{operand2}, MINT *@var{res})  @deftypefun void gcd (MINT *@var{op1}, MINT *@var{op2}, MINT *@var{res})
 Set @var{res} to the greatest common divisor of @var{operand1} and  Set @var{res} to the greatest common divisor of @var{op1} and @var{op2}.
 @var{operand2}.  
 @end deftypefun  @end deftypefun
   
 @deftypefun int mcmp (MINT *@var{operand1}, MINT *@var{operand2})  @deftypefun int mcmp (MINT *@var{op1}, MINT *@var{op2})
 Compare @var{operand1} and @var{operand2}.  Return a positive value if  Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
 @var{operand1} > @var{operand2}, zero if @var{operand1} =  @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
 @var{operand2}, and a negative value if @var{operand1} < @var{operand2}.  @var{op2}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void min (MINT *@var{dest})  @deftypefun void min (MINT *@var{dest})
Line 2538  Input a decimal string from @code{stdin}, and put the 
Line 6226  Input a decimal string from @code{stdin}, and put the 
 Output @var{src} to @code{stdout}, as a decimal string.  Also output a newline.  Output @var{src} to @code{stdout}, as a decimal string.  Also output a newline.
 @end deftypefun  @end deftypefun
   
 @deftypefun {char *} mtox (MINT *@var{operand})  @deftypefun {char *} mtox (MINT *@var{op})
 Convert @var{operand} to a hexadecimal string, and return a pointer to the  Convert @var{op} to a hexadecimal string, and return a pointer to the string.
 string.  The returned string is allocated using the default memory allocation  The returned string is allocated using the default memory allocation function,
 function, @code{malloc} by default.  @code{malloc} by default.  It will be @code{strlen(str)+1} bytes, that being
   exactly enough for the string and null-terminator.
 @end deftypefun  @end deftypefun
   
 @deftypefun void mfree (MINT *@var{operand})  @deftypefun void mfree (MINT *@var{op})
 De-allocate, the space used by @var{operand}.  @strong{This function should  De-allocate, the space used by @var{op}.  @strong{This function should only be
 only be passed a value returned by @code{itom} or @code{xtom}.}  passed a value returned by @code{itom} or @code{xtom}.}
 @end deftypefun  @end deftypefun
   
 @node Custom Allocation, Contributors, BSD Compatible Functions, Top  
   @node Custom Allocation, Language Bindings, BSD Compatible Functions, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @chapter Custom Allocation  @chapter Custom Allocation
   @cindex Custom allocation
   @cindex Memory allocation
   @cindex Allocation of memory
   
 By default, the MP functions use @code{malloc}, @code{realloc}, and  By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory
 @code{free} for memory allocation.  If @code{malloc} or @code{realloc} fails,  allocation, and if they fail GMP prints a message to the standard error output
 the MP library terminates execution after printing a fatal error message to  and terminates the program.
 standard error.  
   
 For some applications, you may wish to allocate memory in other ways, or you  Alternate functions can be specified to allocate memory in a different way or
 may not want to have a fatal error when there is no more memory available.  To  to have a different error action on running out of memory.
 accomplish this, you can specify alternative memory allocation functions.  
   
   This feature is available in the Berkeley compatibility library (@pxref{BSD
   Compatible Functions}) as well as the main GMP library.
   
 @deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t))  @deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t))
 Replace the current allocation functions from the arguments.  If an argument  Replace the current allocation functions from the arguments.  If an argument
 is NULL, the corresponding default function is retained.  is @code{NULL}, the corresponding default function is used.
   
 @strong{Make sure to call this function in such a way that there are no active  These functions will be used for all memory allocation done by GMP, apart from
 MP objects that were allocated using the previously active allocation  temporary space from @code{alloca} if that function is available and GMP is
 function!  Usually, that means that you have to call this function before any  configured to use it (@pxref{Build Options}).
 other MP function.}  
   @strong{Be sure to call @code{mp_set_memory_functions} only when there are no
   active GMP objects allocated using the previous memory functions!  Usually
   that means calling it before any other GMP function.}
 @end deftypefun  @end deftypefun
   
 The functions you supply should fit the following declarations:  The functions supplied should fit the following declarations:
   
 @deftypefun {void *} allocate_function (size_t @var{alloc_size})  @deftypefun {void *} allocate_function (size_t @var{alloc_size})
 This function should return a pointer to newly allocated space with at least  Return a pointer to newly allocated space with at least @var{alloc_size}
 @var{alloc_size} storage units.  bytes.
 @end deftypefun  @end deftypefun
   
 @deftypefun {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})  @deftypefun {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})
 This function should return a pointer to newly allocated space of at least  Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be
 @var{new_size} storage units, after copying at least the first @var{old_size}  @var{new_size} bytes.
 storage units from @var{ptr}.  It should also de-allocate the space at  
 @var{ptr}.  
   
 You can assume that the space at @var{ptr} was formerly returned from  The block may be moved if necessary or if desired, and in that case the
 @code{allocate_function} or @code{reallocate_function}, for a request for  smaller of @var{old_size} and @var{new_size} bytes must be copied to the new
 @var{old_size} storage units.  location.  The return value is a pointer to the resized block, that being the
   new location if moved or just @var{ptr} if not.
   
   @var{ptr} is never @code{NULL}, it's always a previously allocated block.
   @var{new_size} may be bigger or smaller than @var{old_size}.
 @end deftypefun  @end deftypefun
   
 @deftypefun void deallocate_function (void *@var{ptr}, size_t @var{size})  @deftypefun void deallocate_function (void *@var{ptr}, size_t @var{size})
 De-allocate the space pointed to by @var{ptr}.  De-allocate the space pointed to by @var{ptr}.
   
 You can assume that the space at @var{ptr} was formerly returned from  @var{ptr} is never @code{NULL}, it's always a previously allocated block of
 @code{allocate_function} or @code{reallocate_function}, for a request for  @var{size} bytes.
 @var{size} storage units.  
 @end deftypefun  @end deftypefun
   
 (A @dfn{storage unit} is the unit in which the @code{sizeof} operator returns  A @dfn{byte} here means the unit used by the @code{sizeof} operator.
 the size of an object, normally an 8 bit byte.)  
   
   The @var{old_size} parameters to @var{reallocate_function} and
   @var{deallocate_function} are passed for convenience, but of course can be
   ignored if not needed.  The default functions using @code{malloc} and friends
   for instance don't use them.
   
 @node Contributors, References, Custom Allocation, Top  No error return is allowed from any of these functions, if they return then
   they must have performed the specified operation.  In particular note that
   @var{allocate_function} or @var{reallocate_function} mustn't return
   @code{NULL}.
   
   Getting a different fatal error action is a good use for custom allocation
   functions, for example giving a graphical dialog rather than the default print
   to @code{stderr}.  How much is possible when genuinely out of memory is
   another question though.
   
   There's currently no defined way for the allocation functions to recover from
   an error such as out of memory, they must terminate program execution.  A
   @code{longjmp} or throwing a C++ exception will have undefined results.  This
   may change in the future.
   
   GMP may use allocated blocks to hold pointers to other allocated blocks.  This
   will limit the assumptions a conservative garbage collection scheme can make.
   
   Since the default GMP allocation uses @code{malloc} and friends, those
   functions will be linked in even if the first thing a program does is an
   @code{mp_set_memory_functions}.  It's necessary to change the GMP sources if
   this is a problem.
   
   
   @node Language Bindings, Algorithms, Custom Allocation, Top
   @chapter Language Bindings
   
   The following packages and projects offer access to GMP from languages other
   than C, though perhaps with varying levels of functionality and efficiency.
   
   @c  GNUstep Base Library @uref{http://www.gnustep.org} (version 0.9.1) is
   @c  intending to use GMP for its NSDecimal class, which would be an Objective
   @c  C binding for GMP.  Has some configure stuff ready, but no code.
   
   @c  @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces
   @c  in tex, just to separate the URL from the preceding text a bit.
   @iftex
   @macro spaceuref {U}
   @ @ @uref{\U\}
   @end macro
   @end iftex
   @ifnottex
   @macro spaceuref {U}
   @uref{\U\}
   @end macro
   @end ifnottex
   
   @sp 1
   @table @asis
   @item C++
   @itemize @bullet
   @item
   GMP C++ class interface, @pxref{C++ Class Interface} @* Straightforward
   interface, expression templates to eliminate temporaries.
   @item
   ALP @spaceuref{http://www.inria.fr/saga/logiciels/ALP} @* Linear algebra and
   polynomials using templates.
   @item
   Arithmos @spaceuref{http://win-www.uia.ac.be/u/cant/arithmos} @* Rationals
   with infinities and square roots.
   @item
   CLN @spaceuref{http://clisp.cons.org/~haible/packages-cln.html} @* High level
   classes for arithmetic.
   @item
   LiDIA @spaceuref{http://www.informatik.tu-darmstadt.de/TI/LiDIA} @* A C++
   library for computational number theory.
   @item
   Linbox @spaceuref{http://www.linalg.org} @* Sparse vectors and matrices.
   @item
   NTL @spaceuref{http://www.shoup.net/ntl} @* A C++ number theory library.
   @end itemize
   
   @item Fortran
   @itemize @bullet
   @item
   Omni F77 @spaceuref{http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html} @*
   Arbitrary precision floats.
   @end itemize
   
   @item Haskell
   @itemize @bullet
   @item
   Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc}
   @end itemize
   
   @item Java
   @itemize @bullet
   @item
   Kaffe @spaceuref{http://www.kaffe.org}
   @item
   Kissme @spaceuref{http://kissme.sourceforge.net}
   @end itemize
   
   @item Lisp
   @itemize @bullet
   @item
   GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html} @* In the
   process of switching to GMP for bignums.
   @item
   Librep @spaceuref{http://librep.sourceforge.net}
   @end itemize
   
   @item M4
   @itemize @bullet
   @item
   GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu} @* Optionally provides
   an arbitrary precision @code{mpeval}.
   @end itemize
   
   @item ML
   @itemize @bullet
   @item
   MLton compiler @spaceuref{http://www.mlton.org}
   @end itemize
   
   @item Oz
   @itemize @bullet
   @item
   Mozart @spaceuref{http://www.mozart-oz.org}
   @end itemize
   
   @item Pascal
   @itemize @bullet
   @item
   GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de} @* GMP unit.
   @end itemize
   
   @item Perl
   @itemize @bullet
   @item
   GMP module, see @file{demos/perl} in the GMP sources.
   @item
   Math::GMP @spaceuref{http://www.cpan.org} @* Compatible with Math::BigInt, but
   not as many functions as the GMP module above.
   @item
   Math::BigInt::GMP @spaceuref{http://www.cpan.org} @* Plug Math::GMP into
   normal Math::BigInt operations.
   @end itemize
   
   @need 1000
   @item Pike
   @itemize @bullet
   @item
   mpz module in the standard distribution, @uref{http://pike.idonex.com}
   @end itemize
   
   @need 500
   @item Prolog
   @itemize @bullet
   @item
   SWI Prolog @spaceuref{http://www.swi.psy.uva.nl/projects/SWI-Prolog} @*
   Arbitrary precision floats.
   @end itemize
   
   @item Python
   @itemize @bullet
   @item
   mpz module in the standard distribution, @uref{http://www.python.org}
   @item
   GMPY @uref{http://gmpy.sourceforge.net}
   @end itemize
   
   @item Scheme
   @itemize @bullet
   @item
   RScheme @spaceuref{http://www.rscheme.org}
   @item
   STklos @spaceuref{http://kaolin.unice.fr/STklos}
   @end itemize
   
   @item Smalltalk
   @itemize @bullet
   @item
   GNU Smalltalk @spaceuref{http://www.smalltalk.org/versions/GNUSmalltalk.html}
   @end itemize
   
   @item Other
   @itemize @bullet
   @item
   DrGenius @spaceuref{http://drgenius.seul.org} @* Geometry system and
   mathematical programming language.
   @item
   GiNaC @spaceuref{http://www.ginac.de} @* C++ computer algebra using CLN.
   @item
   Maxima @uref{http://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma
   computer algebra using GCL.
   @item
   Q @spaceuref{http://www.musikwissenschaft.uni-mainz.de/~ag/q} @* Equational
   programming system.
   @item
   Regina @spaceuref{http://regina.sourceforge.net} @* Topological calculator.
   @item
   Yacas @spaceuref{http://www.xs4all.nl/~apinkus/yacas.html} @* Yet another
   computer algebra system.
   @end itemize
   
   @end table
   
   
   @node Algorithms, Internals, Language Bindings, Top
   @chapter Algorithms
   @cindex Algorithms
   
   This chapter is an introduction to some of the algorithms used for various GMP
   operations.  The code is likely to be hard to understand without knowing
   something about the algorithms.
   
   Some GMP internals are mentioned, but applications that expect to be
   compatible with future GMP releases should take care to use only the
   documented functions.
   
   @menu
   * Multiplication Algorithms::
   * Division Algorithms::
   * Greatest Common Divisor Algorithms::
   * Powering Algorithms::
   * Root Extraction Algorithms::
   * Radix Conversion Algorithms::
   * Other Algorithms::
   * Assembler Coding::
   @end menu
   
   
   @node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms
   @section Multiplication
   @cindex Multiplication algorithms
   
   N@cross{}N limb multiplications and squares are done using one of four
   algorithms, as the size N increases.
   
   @quotation
   @multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item Algorithm @tab Threshold
   @item Basecase  @tab (none)
   @item Karatsuba @tab @code{MUL_KARATSUBA_THRESHOLD}
   @item Toom-3    @tab @code{MUL_TOOM3_THRESHOLD}
   @item FFT       @tab @code{MUL_FFT_THRESHOLD}
   @end multitable
   @end quotation
   
   Similarly for squaring, with the @code{SQR} thresholds.  Note though that the
   FFT is only used if GMP is configured with @samp{--enable-fft}, @pxref{Build
   Options}.
   
   N@cross{}M multiplications of operands with different sizes above
   @code{MUL_KARATSUBA_THRESHOLD} are currently done by splitting into M@cross{}M
   pieces.  The Karatsuba and Toom-3 routines then operate only on equal size
   operands.  This is not very efficient, and is slated for improvement in the
   future.
   
   @menu
   * Basecase Multiplication::
   * Karatsuba Multiplication::
   * Toom-Cook 3-Way Multiplication::
   * FFT Multiplication::
   * Other Multiplication::
   @end menu
   
   
   @node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms
   @subsection Basecase Multiplication
   
   Basecase N@cross{}M multiplication is a straightforward rectangular set of
   cross-products, the same as long multiplication done by hand and for that
   reason sometimes known as the schoolbook or grammar school method.  This is an
   @m{O(NM),O(N*M)} algorithm.  See Knuth section 4.3.1 algorithm M
   (@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code.
   
   Assembler implementations of @code{mpn_mul_basecase} are essentially the same
   as the generic C code, but have all the usual assembler tricks and
   obscurities introduced for speed.
   
   A square can be done in roughly half the time of a multiply, by using the fact
   that the cross products above and below the diagonal are the same.  A triangle
   of products below the diagonal is formed, doubled (left shift by one bit), and
   then the products on the diagonal added.  This can be seen in
   @file{mpn/generic/sqr_basecase.c}.  Again the assembler implementations take
   essentially the same approach.
   
   @tex
   \def\GMPline#1#2#3#4#5#6{%
     \hbox {%
       \vrule height 2.5ex depth 1ex
              \hbox to 2em {\hfil{#2}\hfil}%
       \vrule \hbox to 2em {\hfil{#3}\hfil}%
       \vrule \hbox to 2em {\hfil{#4}\hfil}%
       \vrule \hbox to 2em {\hfil{#5}\hfil}%
       \vrule \hbox to 2em {\hfil{#6}\hfil}%
       \vrule}}
   \GMPdisplay{
     \hbox{%
       \vbox{%
         \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}%
         \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}%
         \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}%
         \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}%
         \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}%
         \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}%
         \vfill}%
       \vbox{%
         \hbox{%
           \hbox to 2em {\hfil u0\hfil}%
           \hbox to 2em {\hfil u1\hfil}%
           \hbox to 2em {\hfil u2\hfil}%
           \hbox to 2em {\hfil u3\hfil}%
           \hbox to 2em {\hfil u4\hfil}}%
         \vskip 0.7ex
         \hrule
         \GMPline{u0}{d}{}{}{}{}%
         \hrule
         \GMPline{u1}{}{d}{}{}{}%
         \hrule
         \GMPline{u2}{}{}{d}{}{}%
         \hrule
         \GMPline{u3}{}{}{}{d}{}%
         \hrule
         \GMPline{u4}{}{}{}{}{d}%
         \hrule}}}
   @end tex
   @ifnottex
   @example
   @group
        u0  u1  u2  u3  u4
      +---+---+---+---+---+
   u0 | d |   |   |   |   |
      +---+---+---+---+---+
   u1 |   | d |   |   |   |
      +---+---+---+---+---+
   u2 |   |   | d |   |   |
      +---+---+---+---+---+
   u3 |   |   |   | d |   |
      +---+---+---+---+---+
   u4 |   |   |   |   | d |
      +---+---+---+---+---+
   @end group
   @end example
   @end ifnottex
   
   In practice squaring isn't a full 2@cross{} faster than multiplying, it's
   usually around 1.5@cross{}.  Less than 1.5@cross{} probably indicates
   @code{mpn_sqr_basecase} wants improving on that CPU.
   
   On some CPUs @code{mpn_mul_basecase} can be faster than the generic C
   @code{mpn_sqr_basecase}.  @code{SQR_BASECASE_THRESHOLD} is the size at which
   to use @code{mpn_sqr_basecase}, this will be zero if that routine should be
   used always.
   
   
   @node Karatsuba Multiplication, Toom-Cook 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms
   @subsection Karatsuba Multiplication
   
   The Karatsuba multiplication algorithm is described in Knuth section 4.3.3
   part A, and various other textbooks.  A brief description is given here.
   
   The inputs @math{x} and @math{y} are treated as each split into two parts of
   equal length (or the most significant part one limb shorter if N is odd).
   
   @tex
   % GMPboxwidth used for all the multiplication pictures
   \global\newdimen\GMPboxwidth \global\GMPboxwidth=5em
   % GMPboxdepth and GMPboxheight are also used for the float pictures
   \global\newdimen\GMPboxdepth  \global\GMPboxdepth=1ex
   \global\newdimen\GMPboxheight \global\GMPboxheight=2ex
   \gdef\GMPvrule{\vrule height \GMPboxheight depth \GMPboxdepth}
   \def\GMPbox#1#2{%
     \vbox {%
       \hrule
       \hbox to 2\GMPboxwidth{%
         \GMPvrule \hfil $#1$\hfil \vrule \hfil $#2$\hfil \vrule}%
       \hrule}}
   \GMPdisplay{%
   \vbox{%
     \hbox to 2\GMPboxwidth {high \hfil low}
     \vskip 0.7ex
     \GMPbox{x_1}{x_0}
     \vskip 0.5ex
     \GMPbox{y_1}{y_0}
   }}
   @end tex
   @ifnottex
   @example
   @group
    high              low
   +----------+----------+
   |    x1    |    x0    |
   +----------+----------+
   
   +----------+----------+
   |    y1    |    y0    |
   +----------+----------+
   @end group
   @end example
   @end ifnottex
   
   Let @math{b} be the power of 2 where the split occurs, ie.@: if @ms{x,0} is
   @math{k} limbs (@ms{y,0} the same) then
   @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
   With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the
   following holds,
   
   @display
   @m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0,
     x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0}
   @end display
   
   This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs,
   whereas a basecase multiply of N@cross{}N limbs is equivalent to four
   multiplies of (N/2)@cross{}(N/2).  The factors @math{(b^2+b)} etc represent
   the positions where the three products must be added.
   
   @tex
   \def\GMPboxA#1#2{%
     \vbox{%
       \hrule
       \hbox{%
         \GMPvrule
         \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
         \vrule
         \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
         \vrule}
       \hrule}}
   \def\GMPboxB#1#2{%
     \hbox{%
       \raise \GMPboxdepth \hbox to \GMPboxwidth {\hfil #1\hskip 0.5em}%
       \vbox{%
         \hrule
         \hbox{%
           \GMPvrule
           \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
           \vrule}%
         \hrule}}}
   \GMPdisplay{%
   \vbox{%
     \hbox to 4\GMPboxwidth {high \hfil low}
     \vskip 0.7ex
     \GMPboxA{x_1y_1}{x_0y_0}
     \vskip 0.5ex
     \GMPboxB{$+$}{x_1y_1}
     \vskip 0.5ex
     \GMPboxB{$+$}{x_0y_0}
     \vskip 0.5ex
     \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)}
   }}
   @end tex
   @ifnottex
   @example
   @group
    high                              low
   +--------+--------+ +--------+--------+
   |      x1*y1      | |      x0*y0      |
   +--------+--------+ +--------+--------+
             +--------+--------+
         add |      x1*y1      |
             +--------+--------+
             +--------+--------+
         add |      x0*y0      |
             +--------+--------+
             +--------+--------+
         sub | (x1-x0)*(y1-y0) |
             +--------+--------+
   @end group
   @end example
   @end ifnottex
   
   The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an
   absolute value, and the sign used to choose to add or subtract.  Notice the
   sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1),
   high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb
   additions, rather than @m{6k,6*k}, but in GMP extra function call overheads
   outweigh the saving.
   
   Squaring is similar to multiplying, but with @math{x=y} the formula reduces to
   an equivalent with three squares,
   
   @display
   @m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2,
      x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2}
   @end display
   
   The final result is accumulated from those three squares the same way as for
   the three multiplies above.  The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now
   always positive.
   
   A similar formula for both multiplying and squaring can be constructed with a
   middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}.  But those sums can exceed
   @math{k} limbs, leading to more carry handling and additions than the form
   above.
   
   Karatsuba multiplication is asymptotically an @math{O(N^@W{1.585})} algorithm,
   the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies
   each 1/2 the size of the inputs.  This is a big improvement over the basecase
   multiply at @math{O(N^2)} and the advantage soon overcomes the extra additions
   Karatsuba performs.
   
   @code{MUL_KARATSUBA_THRESHOLD} can be as little as 10 limbs.  The @code{SQR}
   threshold is usually about twice the @code{MUL}.  The basecase algorithm will
   take a time of the form @m{M(N) = aN^2 + bN + c, M(N) = a*N^2 + b*N + c} and
   the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N +
   e}.  Clearly per-crossproduct speedups in the basecase code reduce @math{a}
   and decrease the threshold, but linear style speedups reducing @math{b} will
   actually increase the threshold.  The latter can be seen for instance when
   adding an optimized @code{mpn_sqr_diagonal} to @code{mpn_sqr_basecase}.  Of
   course all speedups reduce total time, and in that sense the algorithm
   thresholds are merely of academic interest.
   
   
   @node Toom-Cook 3-Way Multiplication, FFT Multiplication, Karatsuba Multiplication, Multiplication Algorithms
   @subsection Toom-Cook 3-Way Multiplication
   
   The Karatsuba formula is the simplest case of a general approach to splitting
   inputs that leads to both Toom-Cook and FFT algorithms.  A description of
   Toom-Cook can be found in Knuth section 4.3.3, with an example 3-way
   calculation after Theorem A.  The 3-way form used in GMP is described here.
   
   The operands are each considered split into 3 pieces of equal length (or the
   most significant part 1 or 2 limbs shorter than the others).
   
   @tex
   \def\GMPbox#1#2#3{%
     \vbox{%
       \hrule \vfil
       \hbox to 3\GMPboxwidth {%
         \GMPvrule
         \hfil$#1$\hfil
         \vrule
         \hfil$#2$\hfil
         \vrule
         \hfil$#3$\hfil
         \vrule}%
       \vfil \hrule
   }}
   \GMPdisplay{%
   \vbox{%
     \hbox to 3\GMPboxwidth {high \hfil low}
     \vskip 0.7ex
     \GMPbox{x_2}{x_1}{x_0}
     \vskip 0.5ex
     \GMPbox{y_2}{y_1}{y_0}
     \vskip 0.5ex
   }}
   @end tex
   @ifnottex
   @example
   @group
    high                         low
   +----------+----------+----------+
   |    x2    |    x1    |    x0    |
   +----------+----------+----------+
   
   +----------+----------+----------+
   |    y2    |    y1    |    y0    |
   +----------+----------+----------+
   @end group
   @end example
   @end ifnottex
   
   @noindent
   These parts are treated as the coefficients of two polynomials
   
   @display
   @group
   @m{X(t) = x_2t^2 + x_1t + x_0,
      X(t) = x2*t^2 + x1*t + x0}
   @m{Y(t) = y_2t^2 + y_1t + y_0,
      Y(t) = y2*t^2 + y1*t + y0}
   @end group
   @end display
   
   Again let @math{b} equal the power of 2 which is the size of the @ms{x,0},
   @ms{x,1}, @ms{y,0} and @ms{y,1} pieces, ie.@: if they're @math{k} limbs each
   then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}},
   b=2^(k*mp_bits_per_limb)}.  With this @math{x=X(b)} and @math{y=Y(b)}.
   
   Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients
   are
   
   @display
   @m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0,
      W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}
   @end display
   
   @noindent
   The @m{w_i,w[i]} are going to be determined, and when they are they'll give
   the final result using @math{w=W(b)}, since
   @m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}.  The coefficients will be roughly
   @math{b^2} each, and the final @math{W(b)} will be an addition like,
   
   @tex
   \def\GMPbox#1#2{%
     \moveright #1\GMPboxwidth
     \vbox{%
       \hrule
       \hbox{%
         \GMPvrule
         \hbox to 2\GMPboxwidth {\hfil$#2$\hfil}%
         \vrule}%
       \hrule
   }}
   \GMPdisplay{%
   \vbox{%
     \hbox to 6\GMPboxwidth {high \hfil low}%
     \vskip 0.7ex
     \GMPbox{0}{w_4}
     \vskip 0.5ex
     \GMPbox{1}{w_3}
     \vskip 0.5ex
     \GMPbox{2}{w_2}
     \vskip 0.5ex
     \GMPbox{3}{w_1}
     \vskip 0.5ex
     \GMPbox{4}{w_1}
   }}
   @end tex
   @ifnottex
   @example
   @group
    high                                        low
   +-------+-------+
   |       w4      |
   +-------+-------+
          +--------+-------+
          |        w3      |
          +--------+-------+
                  +--------+-------+
                  |        w2      |
                  +--------+-------+
                          +--------+-------+
                          |        w1      |
                          +--------+-------+
                                   +-------+-------+
                                   |       w0      |
                                   +-------+-------+
   @end group
   @end example
   @end ifnottex
   
   The @m{w_i,w[i]} coefficients could be formed by a simple set of cross
   products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2},
   @m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all
   nine @m{x_iy_j,x[i]*y[j]} for @math{i,j=0,1,2}, and would be equivalent merely
   to a basecase multiply.  Instead the following approach is used.
   
   @math{X(t)} and @math{Y(t)} are evaluated and multiplied at 5 points, giving
   values of @math{W(t)} at those points.  The points used can be chosen in
   various ways, but in GMP the following are used
   
   @quotation
   @multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
   @item Point                 @tab Value
   @item @math{t=0}            @tab @m{x_0y_0,x0*y0}, which gives @ms{w,0} immediately
   @item @math{t=2}            @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0)*(4*y2+2*y1+y0)}
   @item @math{t=1}            @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0)*(y2+y1+y0)}
   @item @m{t={1\over2},t=1/2} @tab @m{(x_2+2x_1+4x_0)(y_2+2y_1+4y_0),(x2+2*x1+4*x0)*(y2+2*y1+4*y0)}
   @item @m{t=\infty,t=inf}    @tab @m{x_2y_2,x2*y2}, which gives @ms{w,4} immediately
   @end multitable
   @end quotation
   
   At @m{t={1\over2},t=1/2} the value calculated is actually
   @m{16X({1\over2})Y({1\over2}), 16*X(1/2)*Y(1/2)}, giving a value for
   @m{16W({1\over2}),16*W(1/2)}, and this is always an integer.  At
   @m{t=\infty,t=inf} the value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over
   t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much
   easier to think of as simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately
   (much like @m{x_0y_0,x0*y0} at @math{t=0} gives @ms{w,0} immediately).
   
   Now each of the points substituted into
   @m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+@dots{}+w0} gives a linear combination
   of the @m{w_i,w[i]} coefficients, and the value of those combinations has just
   been calculated.
   
   @tex
   \GMPdisplay{%
   $\matrix{%
   W(0)           & = &       &   &      &   &      &   &      &   &   w_0 \cr
   16W({1\over2}) & = &   w_4 & + & 2w_3 & + & 4w_2 & + & 8w_1 & + & 16w_0 \cr
   W(1)           & = &   w_4 & + &  w_3 & + &  w_2 & + &  w_1 & + &   w_0 \cr
   W(2)           & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + &   w_0 \cr
   W(\infty)      & = &   w_4 \cr
   }$}
   @end tex
   @ifnottex
   @example
   @group
      W(0)   =                                 w0
   16*W(1/2) =    w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0
      W(1)   =    w4 +   w3 +   w2 +   w1 +    w0
      W(2)   = 16*w4 + 8*w3 + 4*w2 + 2*w1 +    w0
      W(inf) =    w4
   @end group
   @end example
   @end ifnottex
   
   This is a set of five equations in five unknowns, and some elementary linear
   algebra quickly isolates each @m{w_i,w[i]}, by subtracting multiples of one
   equation from another.
   
   In the code the set of five values @math{W(0)},@dots{},@m{W(\infty),W(inf)}
   will represent those certain linear combinations.  By adding or subtracting
   one from another as necessary, values which are each @m{w_i,w[i]} alone are
   arrived at.  This involves only a few subtractions of small multiples (some of
   which are powers of 2), and so is fast.  A couple of divisions remain by
   powers of 2 and one division by 3 (or by 6 rather), and that last uses the
   special @code{mpn_divexact_by3} (@pxref{Exact Division}).
   
   In the code the values @ms{w,4}, @ms{w,2} and @ms{w,0} are formed in the
   destination with pointers @code{E}, @code{C} and @code{A}, and @ms{w,3} and
   @ms{w,1} in temporary space @code{D} and @code{B} are added to them.  There
   are extra limbs @code{tD}, @code{tC} and @code{tB} at the high end of
   @ms{w,3}, @ms{w,2} and @ms{w,1} which are handled separately.  The final
   addition then is as follows.
   
   @tex
   \def\GMPboxT#1{%
     \vbox{%
       \hrule
       \hbox {\GMPvrule\hskip 0.4em #1\hskip 0.4em \vrule}%
       \hrule
   }}
   \GMPdisplay{%
   \vbox{%
     \hbox to 6\GMPboxwidth {high \hfil low}%
     \vskip 0.7ex
     \vbox{%
       \hrule
       \hbox{%
         \GMPvrule
         \hbox to 2\GMPboxwidth {\hfil@code{E}\hfil}
         \vrule
         \hbox to 2\GMPboxwidth {\hfil@code{C}\hfil}
         \vrule
         \hbox to 2\GMPboxwidth {\hfil@code{A}\hfil}
         \vrule}%
       \hrule}%
     \vskip 0.5ex
     \moveright \GMPboxwidth \vbox{%
       \hrule
       \hbox to 4\GMPboxwidth {%
         \GMPvrule \hfil @code{D}\hfil
         \vrule \hfil @code{B}\hfil
         \vrule}
       \hrule}%
     \vskip 0.5ex
     \hbox{%
       \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tD}}}%
       \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tC}}}%
       \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tB}}}}
   }}
   @end tex
   @ifnottex
   @example
   @group
    high                                        low
   +-------+-------+-------+-------+-------+-------+
   |       E       |       C       |       A       |
   +-------+-------+-------+-------+-------+-------+
            +------+-------++------+-------+
            |      D       ||      B       |
            +------+-------++------+-------+
         --      --      --
        |tD|    |tC|    |tB|
         --      --      --
   @end group
   @end example
   @end ifnottex
   
   The conversion of @math{W(t)} values to the coefficients is interpolation.  A
   polynomial of degree 4 like @math{W(t)} is uniquely determined by values known
   at 5 different points.  The points can be chosen to make the linear equations
   come out with a convenient set of steps for isolating the @m{w_i,w[i]}.
   
   In @file{mpn/generic/mul_n.c} the @code{interpolate3} routine performs the
   interpolation.  The open-coded one-pass version may be a bit hard to
   understand, the steps performed can be better seen in the @code{USE_MORE_MPN}
   version.
   
   Squaring follows the same procedure as multiplication, but there's only one
   @math{X(t)} and it's evaluated at 5 points, and those values squared to give
   values of @math{W(t)}.  The interpolation is then identical, and in fact the
   same @code{interpolate3} subroutine is used for both squaring and multiplying.
   
   Toom-3 is asymptotically @math{O(N^@W{1.465})}, the exponent being
   @m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the
   original size.  This is an improvement over Karatsuba at @math{O(N^@W{1.585})},
   though Toom-Cook does more work in the evaluation and interpolation and so it
   only realizes its advantage above a certain size.
   
   Near the crossover between Toom-3 and Karatsuba there's generally a range of
   sizes where the difference between the two is small.
   @code{MUL_TOOM3_THRESHOLD} is a somewhat arbitrary point in that range and
   successive runs of the tune program can give different values due to small
   variations in measuring.  A graph of time versus size for the two shows the
   effect, see @file{tune/README}.
   
   At the fairly small sizes where the Toom-3 thresholds occur it's worth
   remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be
   expected to make accurate predictions, due of course to the big influence of
   all sorts of overheads, and the fact that only a few recursions of each are
   being performed.  Even at large sizes there's a good chance machine dependent
   effects like cache architecture will mean actual performance deviates from
   what might be predicted.
   
   The formula given above for the Karatsuba algorithm has an equivalent for
   Toom-3 involving only five multiplies, but this would be complicated and
   unenlightening.
   
   An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using
   a vector to represent the @math{x} and @math{y} splits and a matrix
   multiplication for the evaluation and interpolation stages.  The matrix
   inverses are not meant to be actually used, and they have elements with values
   much greater than in fact arise in the interpolation steps.  The diagram shown
   for the 3-way is attractive, but again doesn't have to be implemented that way
   and for example with a bit of rearrangement just one division by 6 can be
   done.
   
   
   @node FFT Multiplication, Other Multiplication, Toom-Cook 3-Way Multiplication, Multiplication Algorithms
   @subsection FFT Multiplication
   
   At large to very large sizes a Fermat style FFT multiplication is used,
   following Sch@"onhage and Strassen (@pxref{References}).  Descriptions of FFTs
   in various forms can be found in many textbooks, for instance Knuth section
   4.3.3 part C or Lipson chapter IX.  A brief description of the form used in
   GMP is given here.
   
   The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given
   @math{N}.  A full product @m{xy,x*y} is obtained by choosing @m{N \ge
   \mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding
   @math{x} and @math{y} with high zero limbs.  The modular product is the native
   form for the algorithm, so padding to get a full product is unavoidable.
   
   The algorithm follows a split, evaluate, pointwise multiply, interpolate and
   combine similar to that described above for Karatsuba and Toom-3.  A @math{k}
   parameter controls the split, with an FFT-@math{k} splitting into @math{2^k}
   pieces of @math{M=N/2^k} bits each.  @math{N} must be a multiple of
   @m{2^k\times@code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so
   the split falls on limb boundaries, avoiding bit shifts in the split and
   combine stages.
   
   The evaluations, pointwise multiplications, and interpolation, are all done
   modulo @m{2^{N'}+1, 2^N'+1} where @math{N'} is @math{2M+k+3} rounded up to a
   multiple of @math{2^k} and of @code{mp_bits_per_limb}.  The results of
   interpolation will be the following negacyclic convolution of the input
   pieces, and the choice of @math{N'} ensures these sums aren't truncated.
   @tex
   $$ w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j $$
   @end tex
   @ifnottex
   
   @example
              ---
              \         b
   w[n] =     /     (-1) * x[i] * y[j]
              ---
          i+j==b*2^k+n
             b=0,1
   @end example
   
   @end ifnottex
   The points used for the evaluation are @math{g^i} for @math{i=0} to
   @math{2^k-1} where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}.  @math{g} is a
   @m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary
   cancellations at the interpolation stage, and it's also a power of 2 so the
   fast fourier transforms used for the evaluation and interpolation do only
   shifts, adds and negations.
   
   The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either
   recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or
   basecase), whichever is optimal at the size @math{N'}.  The interpolation is
   an inverse fast fourier transform.  The resulting set of sums of @m{x_iy_j,
   x[i]*y[j]} are added at appropriate offsets to give the final result.
   
   Squaring is the same, but @math{x} is the only input so it's one transform at
   the evaluate stage and the pointwise multiplies are squares.  The
   interpolation is the same.
   
   For a mod @math{2^N+1} product, an FFT-@math{k} is an @m{O(N^{k/(k-1)}),
   O(N^(k/(k-1)))} algorithm, the exponent representing @math{2^k} recursed
   modular multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original.
   Each successive @math{k} is an asymptotic improvement, but overheads mean each
   is only faster at bigger and bigger sizes.  In the code, @code{MUL_FFT_TABLE}
   and @code{SQR_FFT_TABLE} are the thresholds where each @math{k} is used.  Each
   new @math{k} effectively swaps some multiplying for some shifts, adds and
   overheads.
   
   A mod @math{2^N+1} product can be formed with a normal
   @math{N@cross{}N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT
   and Toom-3 etc can be compared directly.  A @math{k=4} FFT at
   @math{O(N^@W{1.333})} can be expected to be the first faster than Toom-3 at
   @math{O(N^@W{1.465})}.  In practice this is what's found, with
   @code{MUL_FFT_MODF_THRESHOLD} and @code{SQR_FFT_MODF_THRESHOLD} being between
   300 and 1000 limbs, depending on the CPU.  So far it's been found that only
   very large FFTs recurse into pointwise multiplies above these sizes.
   
   When an FFT is to give a full product, the change of @math{N} to @math{2N}
   doesn't alter the theoretical complexity for a given @math{k}, but for the
   purposes of considering where an FFT might be first used it can be assumed
   that the FFT is recursing into a normal multiply and that on that basis it's
   doing @math{2^k} recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of
   the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}.  This would mean
   @math{k=7} at @math{O(N^@W{1.4})} would be the first FFT faster than Toom-3.
   In practice @code{MUL_FFT_THRESHOLD} and @code{SQR_FFT_THRESHOLD} have been
   found to be in the @math{k=8} range, somewhere between 3000 and 10000 limbs.
   
   The way @math{N} is split into @math{2^k} pieces and then @math{2M+k+3} is
   rounded up to a multiple of @math{2^k} and @code{mp_bits_per_limb} means that
   when @math{2^k@ge{}@nicode{mp\_bits\_per\_limb}} the effective @math{N} is a
   multiple of @m{2^{2k-1},2^(2k-1)} bits.  The @math{+k+3} means some values of
   @math{N} just under such a multiple will be rounded to the next.  The
   complexity calculations above assume that a favourable size is used, meaning
   one which isn't padded through rounding, and it's also assumed that the extra
   @math{+k+3} bits are negligible at typical FFT sizes.
   
   The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a
   step-effect into measured speeds.  For example @math{k=8} will round @math{N}
   up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
   groups of sizes for which @code{mpn_mul_n} runs at the same speed.  Or for
   @math{k=9} groups of 2048 limbs, @math{k=10} groups of 8192 limbs, etc.  In
   practice it's been found each @math{k} is used at quite small multiples of its
   size constraint and so the step effect is quite noticeable in a time versus
   size graph.
   
   The threshold determinations currently measure at the mid-points of size
   steps, but this is sub-optimal since at the start of a new step it can happen
   that it's better to go back to the previous @math{k} for a while.  Something
   more sophisticated for @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} will be
   needed.
   
   
   @node Other Multiplication,  , FFT Multiplication, Multiplication Algorithms
   @subsection Other Multiplication
   
   The 3-way Toom-Cook algorithm described above (@pxref{Toom-Cook 3-Way
   Multiplication}) generalizes to split into an arbitrary number of pieces, as
   per Knuth section 4.3.3 algorithm C.  This is not currently used, though it's
   possible a Toom-4 might fit in between Toom-3 and the FFTs.  The notes here
   are merely for interest.
   
   In general a split into @math{r+1} pieces is made, and evaluations and
   pointwise multiplications done at @m{2r+1,2*r+1} points.  A 4-way split does 7
   pointwise multiplies, 5-way does 9, etc.  Asymptotically an @math{(r+1)}-way
   algorithm is @m{O(N^{log(2r+1)/log(r+1)}, O(N^(log(2*r+1)/log(r+1)))}.  Only
   the pointwise multiplications count towards big-@math{O} complexity, but the
   time spent in the evaluate and interpolate stages grows with @math{r} and has
   a significant practical impact, with the asymptotic advantage of each @math{r}
   realized only at bigger and bigger sizes.  The overheads grow as
   @m{O(Nr),O(N*r)}, whereas in an @math{r=2^k} FFT they grow only as @m{O(N \log
   r), O(N*log(r))}.
   
   Knuth algorithm C evaluates at points 0,1,2,@dots{},@m{2r,2*r}, but exercise 4
   uses @math{-r},@dots{},0,@dots{},@math{r} and the latter saves some small
   multiplies in the evaluate stage (or rather trades them for additions), and
   has a further saving of nearly half the interpolate steps.  The idea is to
   separate odd and even final coefficients and then perform algorithm C steps C7
   and C8 on them separately.  The divisors at step C7 become @math{j^2} and the
   multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}.
   
   Splitting odd and even parts through positive and negative points can be
   thought of as using @math{-1} as a square root of unity.  If a 4th root of
   unity was available then a further split and speedup would be possible, but no
   such root exists for plain integers.  Going to complex integers with
   @m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in cartesian
   form it takes three real multiplies to do a complex multiply.  The existence
   of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast
   fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}.
   
   Floating point FFTs use complex numbers approximating Nth roots of unity.
   Some processors have special support for such FFTs.  But these are not used in
   GMP since it's very difficult to guarantee an exact result (to some number of
   bits).  An occasional difference of 1 in the last bit might not matter to a
   typical signal processing algorithm, but is of course of vital importance to
   GMP.
   
   
   @node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms
   @section Division Algorithms
   @cindex Division algorithms
   
   @menu
   * Single Limb Division::
   * Basecase Division::
   * Divide and Conquer Division::
   * Exact Division::
   * Exact Remainder::
   * Small Quotient Division::
   @end menu
   
   
   @node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms
   @subsection Single Limb Division
   
   N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from
   high to low, either with a hardware divide instruction or a multiplication by
   inverse, whichever is best on a given CPU.
   
   The multiply by inverse follows section 8 of ``Division by Invariant Integers
   using Multiplication'' by Granlund and Montgomery (@pxref{References}) and is
   implemented as @code{udiv_qrnnd_preinv} in @file{gmp-impl.h}.  The idea is to
   have a fixed-point approximation to @math{1/d} (see @code{invert_limb}) and
   then multiply by the high limb (plus one bit) of the dividend to get a
   quotient @math{q}.  With @math{d} normalized (high bit set), @math{q} is no
   more than 1 too small.  Subtracting @m{qd,q*d} from the dividend gives a
   remainder, and reveals whether @math{q} or @math{q-1} is correct.
   
   The result is a division done with two multiplications and four or five
   arithmetic operations.  On CPUs with low latency multipliers this can be much
   faster than a hardware divide, though the cost of calculating the inverse at
   the start may mean it's only better on inputs bigger than say 4 or 5 limbs.
   
   When a divisor must be normalized, either for the generic C
   @code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is
   actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @math{a} is the dividend and
   @math{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set.
   The bit shifts for the dividend are usually accomplished ``on the fly''
   meaning by extracting the appropriate bits at each step.  Done this way the
   quotient limbs come out aligned ready to store.  When only the remainder is
   wanted, an alternative is to take the dividend limbs unshifted and calculate
   @m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k
   \bmod d2^k, r*2^k mod d*2^k}.  This can help on CPUs with poor bit shifts or
   few registers.
   
   The multiply by inverse can be done two limbs at a time.  The calculation is
   basically the same, but the inverse is two limbs and the divisor treated as if
   padded with a low zero limb.  This means more work, since the inverse will
   need a 2@cross{}2 multiply, but the four 1@cross{}1s to do that are
   independent and can therefore be done partly or wholly in parallel.  Likewise
   for a 2@cross{}1 calculating @m{qd,q*d}.  The net effect is to process two
   limbs with roughly the same two multiplies worth of latency that one limb at a
   time gives.  This extends to 3 or 4 limbs at a time, though the extra work to
   apply the inverse will almost certainly soon reach the limits of multiplier
   throughput.
   
   A similar approach in reverse can be taken to process just half a limb at a
   time if the divisor is only a half limb.  In this case the 1@cross{}1 multiply
   for the inverse effectively becomes two @m{1\over2@cross{}1, (1/2)x1} for each
   limb, which can be a saving on CPUs with a fast half limb multiply, or in fact
   if the only multiply is a half limb, and especially if it's not pipelined.
   
   
   @node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms
   @subsection Basecase Division
   
   Basecase N@cross{}M division is like long division done by hand, but in base
   @m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}.  See Knuth
   section 4.3.1 algorithm D, and @file{mpn/generic/sb_divrem_mn.c}.
   
   Briefly stated, while the dividend remains larger than the divisor, a high
   quotient limb is formed and the N@cross{}1 product @m{qd,q*d} subtracted at
   the top end of the dividend.  With a normalized divisor (most significant bit
   set), each quotient limb can be formed with a 2@cross{}1 division and a
   1@cross{}1 multiplication plus some subtractions.  The 2@cross{}1 division is
   by the high limb of the divisor and is done either with a hardware divide or a
   multiply by inverse (the same as in @ref{Single Limb Division}) whichever is
   faster.  Such a quotient is sometimes one too big, requiring an addback of the
   divisor, but that happens rarely.
   
   With Q=N@minus{}M being the number of quotient limbs, this is an
   @m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase
   Q@cross{}M multiplication, differing in fact only in the extra multiply and
   divide for each of the Q quotient limbs.
   
   
   @node Divide and Conquer Division, Exact Division, Basecase Division, Division Algorithms
   @subsection Divide and Conquer Division
   
   For divisors larger than @code{DIV_DC_THRESHOLD}, division is done by dividing.
   Or to be precise by a recursive divide and conquer algorithm based on work by
   Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}).
   
   The algorithm consists essentially of recognising that a 2N@cross{}N division
   can be done with the basecase division algorithm (@pxref{Basecase Division}),
   but using N/2 limbs as a base, not just a single limb.  This way the
   multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of
   Karatsuba and higher multiplication algorithms (@pxref{Multiplication
   Algorithms}).  The ``digits'' of the quotient are formed by recursive
   N@cross{}(N/2) divisions.
   
   If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication
   then the work is about the same as a basecase division, but with more function
   call overheads and with some subtractions separated from the multiplies.
   These overheads mean that it's only when N/2 is above
   @code{MUL_KARATSUBA_THRESHOLD} that divide and conquer is of use.
   
   @code{DIV_DC_THRESHOLD} is based on the divisor size N, so it will be somewhere
   above twice @code{MUL_KARATSUBA_THRESHOLD}, but how much above depends on the
   CPU.  An optimized @code{mpn_mul_basecase} can lower @code{DIV_DC_THRESHOLD} a
   little by offering a ready-made advantage over repeated @code{mpn_submul_1}
   calls.
   
   Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where
   @math{M(N)} is the time for an N@cross{}N multiplication done with FFTs.  The
   actual time is a sum over multiplications of the recursed sizes, as can be
   seen near the end of section 2.2 of Burnikel and Ziegler.  For example, within
   the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}.  With higher
   algorithms the @math{M(N)} term improves and the multiplier tends to @m{\log
   N, log(N)}.  In practice, at moderate to large sizes, a 2N@cross{}N division
   is about 2 to 4 times slower than an N@cross{}N multiplication.
   
   Newton's method used for division is asymptotically @math{O(M(N))} and should
   therefore be superior to divide and conquer, but it's believed this would only
   be for large to very large N.
   
   
   @node Exact Division, Exact Remainder, Divide and Conquer Division, Division Algorithms
   @subsection Exact Division
   
   A so-called exact division is when the dividend is known to be an exact
   multiple of the divisor.  Jebelean's exact division algorithm uses this
   knowledge to make some significant optimizations (@pxref{References}).
   
   The idea can be illustrated in decimal for example with 368154 divided by
   543.  Because the low digit of the dividend is 4, the low digit of the
   quotient must be 8.  This is arrived at from @m{4 \mathord{\times} 7 \bmod 10,
   4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of
   the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7
   @equiv{} 1 mod 10}.  So @m{8\mathord{\times}543 = 4344,8*543=4344} can be
   subtracted from the dividend leaving 363810.  Notice the low digit has become
   zero.
   
   The procedure is repeated at the second digit, with the next quotient digit 7
   (@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting
   @m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800.  And finally at
   the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7
   mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0.
   So the quotient is 678.
   
   Notice however that the multiplies and subtractions don't need to extend past
   the low three digits of the dividend, since that's enough to determine the
   three quotient digits.  For the last quotient digit no subtraction is needed
   at all.  On a 2N@cross{}N division like this one, only about half the work of
   a normal basecase division is necessary.
   
   For an N@cross{}M exact division producing Q=N@minus{}M quotient limbs, the
   saving over a normal basecase division is in two parts.  Firstly, each of the
   Q quotient limbs needs only one multiply, not a 2@cross{}1 divide and
   multiply.  Secondly, the crossproducts are reduced when @math{Q>M} to
   @m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q@le{}M} to @m{Q(Q-1)/2,
   Q*(Q-1)/2}.  Notice the savings are complementary.  If Q is big then many
   divisions are saved, or if Q is small then the crossproducts reduce to a small
   number.
   
   The modular inverse used is calculated efficiently by @code{modlimb_invert} in
   @file{gmp-impl.h}.  This does four multiplies for a 32-bit limb, or six for a
   64-bit limb.  @file{tune/modlinv.c} has some alternate implementations that
   might suit processors better at bit twiddling than multiplying.
   
   The sub-quadratic exact division described by Jebelean in ``Exact Division
   with Karatsuba Complexity'' is not currently implemented.  It uses a
   rearrangement similar to the divide and conquer for normal division
   (@pxref{Divide and Conquer Division}), but operating from low to high.  A
   further possibility not currently implemented is ``Bidirectional Exact Integer
   Division'' by Krandick and Jebelean which forms quotient limbs from both the
   high and low ends of the dividend, and can halve once more the number of
   crossproducts needed in a 2N@cross{}N division.
   
   A special case exact division by 3 exists in @code{mpn_divexact_by3},
   supporting Toom-3 multiplication and @code{mpq} canonicalizations.  It forms
   quotient digits with a multiply by the modular inverse of 3 (which is
   @code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next
   limb.  The multiplications don't need to be on the dependent chain, as long as
   the effect of the borrows is applied.  Only a few optimized assembler
   implementations currently exist.
   
   
   @node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms
   @subsection Exact Remainder
   
   If the exact division algorithm is done with a full subtraction at each stage
   and the dividend isn't a multiple of the divisor, then low zero limbs are
   produced but with a remainder in the high limbs.  For dividend @math{a},
   divisor @math{d}, quotient @math{q}, and @m{b = 2
   \GMPraise{@code{mp\_bits\_per\_limb}}, b = 2^mp_bits_per_limb}, then this
   remainder @math{r} is of the form
   @tex
   $$ a = qd + r b^n $$
   @end tex
   @ifnottex
   
   @example
   a = q*d + r*b^n
   @end example
   
   @end ifnottex
   @math{n} represents the number of zero limbs produced by the subtractions,
   that being the number of limbs produced for @math{q}.  @math{r} will be in the
   range @math{0@le{}r<d} and can be viewed as a remainder, but one shifted up by
   a factor of @math{b^n}.
   
   Carrying out full subtractions at each stage means the same number of cross
   products must be done as a normal division, but there's still some single limb
   divisions saved.  When @math{d} is a single limb some simplifications arise,
   providing good speedups on a number of processors.
   
   @code{mpn_bdivmod}, @code{mpn_divexact_by3}, @code{mpn_modexact_1_odd} and the
   @code{redc} function in @code{mpz_powm} differ subtly in how they return
   @math{r}, leading to some negations in the above formula, but all are
   essentially the same.
   
   Clearly @math{r} is zero when @math{a} is a multiple of @math{d}, and this
   leads to divisibility or congruence tests which are potentially more efficient
   than a normal division.
   
   The factor of @math{b^n} on @math{r} can be ignored in a GCD when @math{d} is
   odd, hence the use of @code{mpn_bdivmod} in @code{mpn_gcd}, and the use of
   @code{mpn_modexact_1_odd} by @code{mpn_gcd_1} and @code{mpz_kronecker_ui} etc
   (@pxref{Greatest Common Divisor Algorithms}).
   
   Montgomery's REDC method for modular multiplications uses operands of the form
   of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n})
   (yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of @math{b^n} in the exact
   remainder to reach a product in the same form @m{(xy)b^{-n}, (x*y)*b^-n}
   (@pxref{Modular Powering Algorithm}).
   
   Notice that @math{r} generally gives no useful information about the ordinary
   remainder @math{a @bmod d} since @math{b^n @bmod d} could be anything.  If
   however @math{b^n @equiv{} 1 @bmod d}, then @math{r} is the negative of the
   ordinary remainder.  This occurs whenever @math{d} is a factor of
   @math{b^n-1}, as for example with 3 in @code{mpn_divexact_by3}.  Other such
   factors include 5, 17 and 257, but no particular use has been found for this.
   
   
   @node Small Quotient Division,  , Exact Remainder, Division Algorithms
   @subsection Small Quotient Division
   
   An N@cross{}M division where the number of quotient limbs Q=N@minus{}M is
   small can be optimized somewhat.
   
   An ordinary basecase division normalizes the divisor by shifting it to make
   the high bit set, shifting the dividend accordingly, and shifting the
   remainder back down at the end of the calculation.  This is wasteful if only a
   few quotient limbs are to be formed.  Instead a division of just the top
   @m{\rm2Q,2*Q} limbs of the dividend by the top Q limbs of the divisor can be
   used to form a trial quotient.  This requires only those limbs normalized, not
   the whole of the divisor and dividend.
   
   A multiply and subtract then applies the trial quotient to the M@minus{}Q
   unused limbs of the divisor and N@minus{}Q dividend limbs (which includes Q
   limbs remaining from the trial quotient division).  The starting trial
   quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1
   too big are detected by first comparing the most significant limbs that will
   arise from the subtraction.  An addback is done if the quotient still turns
   out to be 1 too big.
   
   This whole procedure is essentially the same as one step of the basecase
   algorithm done in a Q limb base, though with the trial quotient test done only
   with the high limbs, not an entire Q limb ``digit'' product.  The correctness
   of this weaker test can be established by following the argument of Knuth
   section 4.3.1 exercise 20 but with the @m{v_2 \GMPhat q > b \GMPhat r
   + u_2, v2*q>b*r+u2} condition appropriately relaxed.
   
   
   @need 1000
   @node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms
   @section Greatest Common Divisor
   @cindex Greatest common divisor algorithms
   
   @menu
   * Binary GCD::
   * Accelerated GCD::
   * Extended GCD::
   * Jacobi Symbol::
   @end menu
   
   
   @node Binary GCD, Accelerated GCD, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms
   @subsection Binary GCD
   
   At small sizes GMP uses an @math{O(N^2)} binary style GCD.  This is described
   in many textbooks, for example Knuth section 4.5.2 algorithm B.  It simply
   consists of successively reducing operands @math{a} and @math{b} using
   @math{@gcd{}(a,b) = @gcd{}(@min{}(a,b),@abs{}(a-b))}, and also that if
   @math{a} and @math{b} are first made odd then @math{@abs{}(a-b)} is even and
   factors of two can be discarded.
   
   Variants like letting @math{a-b} become negative and doing a different next
   step are of interest only as far as they suit particular CPUs, since on small
   operands it's machine dependent factors that determine performance.
   
   The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using
   @math{a @bmod b} but this has so far been found to be slower everywhere.  One
   reason the binary method does well is that the implied quotient at each step
   is usually small, so often only one or two subtractions are needed to get the
   same effect as a division.  Quotients 1, 2 and 3 for example occur 67.7% of
   the time, see Knuth section 4.5.3 Theorem E.
   
   When the implied quotient is large, meaning @math{b} is much smaller than
   @math{a}, then a division is worthwhile.  This is the basis for the initial
   @math{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter
   for both N@cross{}1 and 1@cross{}1 cases).  But after that initial reduction,
   big quotients occur too rarely to make it worth checking for them.
   
   
   @node Accelerated GCD, Extended GCD, Binary GCD, Greatest Common Divisor Algorithms
   @subsection Accelerated GCD
   
   For sizes above @code{GCD_ACCEL_THRESHOLD}, GMP uses the Accelerated GCD
   algorithm described independently by Weber and Jebelean (the latter as the
   ``Generalized Binary'' algorithm), @pxref{References}.  This algorithm is
   still @math{O(N^2)}, but is much faster than the binary algorithm since it
   does fewer multi-precision operations.  It consists of alternating the
   @math{k}-ary reduction by Sorenson, and a ``dmod'' exact remainder reduction.
   
   For operands @math{u} and @math{v} the @math{k}-ary reduction replaces
   @math{u} with @m{nv-du,n*v-d*u} where @math{n} and @math{d} are single limb
   values chosen to give two trailing zero limbs on that value, which can be
   stripped.  @math{n} and @math{d} are calculated using an algorithm similar to
   half of a two limb GCD (see @code{find_a} in @file{mpn/generic/gcd.c}).
   
   When @math{u} and @math{v} differ in size by more than a certain number of
   bits, a dmod is performed to zero out bits at the low end of the larger.  It
   consists of an exact remainder style division applied to an appropriate number
   of bits (@pxref{Exact Division}, and @pxref{Exact Remainder}).  This is faster
   than a @math{k}-ary reduction but useful only when the operands differ in
   size.  There's a dmod after each @math{k}-ary reduction, and if the dmod
   leaves the operands still differing in size then it's repeated.
   
   The @math{k}-ary reduction step can introduce spurious factors into the GCD
   calculated, and these are eliminated at the end by taking GCDs with the
   original inputs @math{@gcd{}(u,@gcd{}(v,g))} using the binary algorithm.
   Since @math{g} is almost always small this takes very little time.
   
   At small sizes the algorithm needs a good implementation of @code{find_a}.  At
   larger sizes it's dominated by @code{mpn_addmul_1} applying @math{n} and
   @math{d}.
   
   
   @node Extended GCD, Jacobi Symbol, Accelerated GCD, Greatest Common Divisor Algorithms
   @subsection Extended GCD
   
   The extended GCD calculates @math{@gcd{}(a,b)} and also cofactors @math{x} and
   @math{y} satisfying @m{ax+by=\gcd(a@C{}b), a*x+b*y=gcd(a@C{}b)}.  Lehmer's
   multi-step improvement of the extended Euclidean algorithm is used.  See Knuth
   section 4.5.2 algorithm L, and @file{mpn/generic/gcdext.c}.  This is an
   @math{O(N^2)} algorithm.
   
   The multipliers at each step are found using single limb calculations for
   sizes up to @code{GCDEXT_THRESHOLD}, or double limb calculations above that.
   The single limb code is faster but doesn't produce full-limb multipliers,
   hence not making full use of the @code{mpn_addmul_1} calls.
   
   When a CPU has a data-dependent multiplier, meaning one which is faster on
   operands with fewer bits, the extra work in the double-limb calculation might
   only save some looping overheads, leading to a large @code{GCDEXT_THRESHOLD}.
   
   Currently the single limb calculation doesn't optimize for the small quotients
   that often occur, and this can lead to unusually low values of
   @code{GCDEXT_THRESHOLD}, depending on the CPU.
   
   An analysis of double-limb calculations can be found in ``A Double-Digit
   Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}).  The code in GMP
   was developed independently.
   
   It should be noted that when a double limb calculation is used, it's used for
   the whole of that GCD, it doesn't fall back to single limb part way through.
   This is because as the algorithm proceeds, the inputs @math{a} and @math{b}
   are reduced, but the cofactors @math{x} and @math{y} grow, so the multipliers
   at each step are applied to a roughly constant total number of limbs.
   
   
   @node Jacobi Symbol,  , Extended GCD, Greatest Common Divisor Algorithms
   @subsection Jacobi Symbol
   
   @code{mpz_jacobi} and @code{mpz_kronecker} are currently implemented with a
   simple binary algorithm similar to that described for the GCDs (@pxref{Binary
   GCD}).  They're not very fast when both inputs are large.  Lehmer's multi-step
   improvement or a binary based multi-step algorithm is likely to be better.
   
   When one operand fits a single limb, and that includes @code{mpz_kronecker_ui}
   and friends, an initial reduction is done with either @code{mpn_mod_1} or
   @code{mpn_modexact_1_odd}, followed by the binary algorithm on a single limb.
   The binary algorithm is well suited to a single limb, and the whole
   calculation in this case is quite efficient.
   
   In all the routines sign changes for the result are accumulated using some bit
   twiddling, avoiding table lookups or conditional jumps.
   
   
   @need 1000
   @node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms
   @section Powering Algorithms
   @cindex Powering algorithms
   
   @menu
   * Normal Powering Algorithm::
   * Modular Powering Algorithm::
   @end menu
   
   
   @node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms
   @subsection Normal Powering
   
   Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm,
   successively squaring and then multiplying by the base when a 1 bit is seen in
   the exponent, as per Knuth section 4.6.3.  The ``left to right''
   variant described there is used rather than algorithm A, since it's just as
   easy and can be done with somewhat less temporary memory.
   
   
   @node Modular Powering Algorithm,  , Normal Powering Algorithm, Powering Algorithms
   @subsection Modular Powering
   
   Modular powering is implemented using a @math{2^k}-ary sliding window
   algorithm, as per ``Handbook of Applied Cryptography'' algorithm 14.85
   (@pxref{References}).  @math{k} is chosen according to the size of the
   exponent.  Larger exponents use larger values of @math{k}, the choice being
   made to minimize the average number of multiplications that must supplement
   the squaring.
   
   The modular multiplies and squares use either a simple division or the REDC
   method by Montgomery (@pxref{References}).  REDC is a little faster,
   essentially saving N single limb divisions in a fashion similar to an exact
   remainder (@pxref{Exact Remainder}).  The current REDC has some limitations.
   It's only @math{O(N^2)} so above @code{POWM_THRESHOLD} division becomes faster
   and is used.  It doesn't attempt to detect small bases, but rather always uses
   a REDC form, which is usually a full size operand.  And lastly it's only
   applied to odd moduli.
   
   
   @node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms
   @section Root Extraction Algorithms
   @cindex Root extraction algorithms
   
   @menu
   * Square Root Algorithm::
   * Nth Root Algorithm::
   * Perfect Square Algorithm::
   * Perfect Power Algorithm::
   @end menu
   
   
   @node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms
   @subsection Square Root
   
   Square roots are taken using the ``Karatsuba Square Root'' algorithm by Paul
   Zimmermann (@pxref{References}).  This is expressed in a divide and conquer
   form, but as noted in the paper it can also be viewed as a discrete variant of
   Newton's method.
   
   In the Karatsuba multiplication range this is an @m{O({3\over2}
   M(N/2)),O(1.5*M(N/2))} algorithm, where @math{M(n)} is the time to multiply
   two numbers of @math{n} limbs.  In the FFT multiplication range this grows to
   a bound of @m{O(6 M(N/2)),O(6*M(N/2))}.  In practice a factor of about 1.5 to
   1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT
   range.
   
   The algorithm does all its calculations in integers and the resulting
   @code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}.
   The extended precision given by @code{mpf_sqrt_ui} is obtained by
   padding with zero limbs.
   
   
   @node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms
   @subsection Nth Root
   
   Integer Nth roots are taken using Newton's method with the following
   iteration, where @math{A} is the input and @math{n} is the root to be taken.
   @tex
   $$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$
   @end tex
   @ifnottex
   
   @example
            1         A
   a[i+1] = - * ( --------- + (n-1)*a[i] )
            n     a[i]^(n-1)
   @end example
   
   @end ifnottex
   The initial approximation @m{a_1,a[1]} is generated bitwise by successively
   powering a trial root with or without new 1 bits, aiming to be just above the
   true root.  The iteration converges quadratically when started from a good
   approximation.  When @math{n} is large more initial bits are needed to get
   good convergence.  The current implementation is not particularly well
   optimized.
   
   
   @node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms
   @subsection Perfect Square
   
   @code{mpz_perfect_square_p} is able to quickly exclude most non-squares by
   checking whether the input is a quadratic residue modulo some small integers.
   
   The first test is modulo 256 which means simply examining the least
   significant byte.  Only 44 different values occur as the low byte of a square,
   so 82.8% of non-squares can be immediately excluded.  Similar tests modulo
   primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits
   then primes up to 53 are used, excluding 99.99%.  A single N@cross{}1
   remainder using @code{PP} from @file{gmp-impl.h} quickly gives all these
   remainders.
   
   A square root must still be taken for any value that passes the residue tests,
   to verify it's really a square and not one of the 0.086% (or 0.000156% for 64
   bits) non-squares that get through.  @xref{Square Root Algorithm}.
   
   
   @node Perfect Power Algorithm,  , Perfect Square Algorithm, Root Extraction Algorithms
   @subsection Perfect Power
   
   Detecting perfect powers is required by some factorization algorithms.
   Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root
   extractions, though naturally only prime roots need to be considered.
   (@xref{Nth Root Algorithm}.)
   
   If a prime divisor @math{p} with multiplicity @math{e} can be found, then only
   roots which are divisors of @math{e} need to be considered, much reducing the
   work necessary.  To this end divisibility by a set of small primes is checked.
   
   
   @node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms
   @section Radix Conversion
   @cindex Radix conversion algorithms
   
   Radix conversions are less important than other algorithms.  A program
   dominated by conversions should probably use a different data representation.
   
   @menu
   * Binary to Radix::
   * Radix to Binary::
   @end menu
   
   
   @node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms
   @subsection Binary to Radix
   
   Conversions from binary to a power-of-2 radix use a simple and fast
   @math{O(N)} bit extraction algorithm.
   
   Conversions from binary to other radices use one of two algorithms.  Sizes
   below @code{GET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method.
   Repeated divisions by @math{b^n} are made, where @math{b} is the radix and
   @math{n} is the biggest power that fits in a limb.  But instead of simply
   using the remainder @math{r} from such divisions, an extra divide step is done
   to give a fractional limb representing @math{r/b^n}.  The digits of @math{r}
   can then be extracted using multiplications by @math{b} rather than divisions.
   Special case code is provided for decimal, allowing multiplications by 10 to
   optimize to shifts and adds.
   
   Above @code{GET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used.
   For an input @math{t}, powers @m{b^{n2^i},b^(n*2^i)} of the radix are
   calculated, until a power between @math{t} and @m{\sqrt{t},sqrt(t)} is
   reached.  @math{t} is then divided by that largest power, giving a quotient
   which is the digits above that power, and a remainder which is those below.
   These two parts are in turn divided by the second highest power, and so on
   recursively.  When a piece has been divided down to less than
   @code{GET_STR_DC_THRESHOLD} limbs, the basecase algorithm described above is
   used.
   
   The advantage of this algorithm is that big divisions can make use of the
   sub-quadratic divide and conquer division (@pxref{Divide and Conquer
   Division}), and big divisions tend to have less overheads than lots of
   separate single limb divisions anyway.  But in any case the cost of
   calculating the powers @m{b^{n2^i},b^(n*2^i)} must first be overcome.
   
   @code{GET_STR_PRECOMPUTE_THRESHOLD} and @code{GET_STR_DC_THRESHOLD} represent
   the same basic thing, the point where it becomes worth doing a big division to
   cut the input in half.  @code{GET_STR_PRECOMPUTE_THRESHOLD} includes the cost
   of calculating the radix power required, whereas @code{GET_STR_DC_THRESHOLD}
   assumes that's already available, which is the case when recursing.
   
   Since the base case produces digits from least to most significant but they
   want to be stored from most to least, it's necessary to calculate in advance
   how many digits there will be, or at least be sure not to underestimate that.
   For GMP the number of input bits is multiplied by @code{chars_per_bit_exactly}
   from @code{mp_bases}, rounding up.  The result is either correct or one too
   big.
   
   Examining some of the high bits of the input could increase the chance of
   getting the exact number of digits, but an exact result every time would not
   be practical, since in general the difference between numbers 100@dots{} and
   99@dots{} is only in the last few bits and the work to identify 99@dots{}
   might well be almost as much as a full conversion.
   
   @code{mpf_get_str} doesn't currently use the algorithm described here, it
   multiplies or divides by a power of @math{b} to move the radix point to the
   just above the highest non-zero digit (or at worst one above that location),
   then multiplies by @math{b^n} to bring out digits.  This is @math{O(N^2)} and
   is certainly not optimal.
   
   The @math{r/b^n} scheme described above for using multiplications to bring out
   digits might be useful for more than a single limb.  Some brief experiments
   with it on the base case when recursing didn't give a noticable improvement,
   but perhaps that was only due to the implementation.  Something similar would
   work for the sub-quadratic divisions too, though there would be the cost of
   calculating a bigger radix power.
   
   Another possible improvement for the sub-quadratic part would be to arrange
   for radix powers that balanced the sizes of quotient and remainder produced,
   ie. the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to
   @m{\sqrt{t},sqrt(t)}, not restricted to a @math{2^i} factor.  That ought to
   smooth out a graph of times against sizes, but may or may not be a net
   speedup.
   
   
   @node Radix to Binary,  , Binary to Radix, Radix Conversion Algorithms
   @subsection Radix to Binary
   
   Conversions from a power-of-2 radix into binary use a simple and fast
   @math{O(N)} bitwise concatenation algorithm.
   
   Conversions from other radices use one of two algorithms.  Sizes below
   @code{SET_STR_THRESHOLD} use a basic @math{O(N^2)} method.  Groups of @math{n}
   digits are converted to limbs, where @math{n} is the biggest power of the base
   @math{b} which will fit in a limb, then those groups are accumulated into the
   result by multiplying by @math{b^n} and adding.  This saves multi-precision
   operations, as per Knuth section 4.4 part E (@pxref{References}).  Some
   special case code is provided for decimal, giving the compiler a chance to
   optimize multiplications by 10.
   
   Above @code{SET_STR_THRESHOLD} a sub-quadratic algorithm is used.  First
   groups of @math{n} digits are converted into limbs.  Then adjacent limbs are
   combined into limb pairs with @m{xb^n+y,x*b^n+y}, where @math{x} and @math{y}
   are the limbs.  Adjacent limb pairs are combined into quads similarly with
   @m{xb^{2n}+y,x*b^(2n)+y}.  This continues until a single block remains, that
   being the result.
   
   The advantage of this method is that the multiplications for each @math{x} are
   big blocks, allowing Karatsuba and higher algorithms to be used.  But the cost
   of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome.
   @code{SET_STR_THRESHOLD} usually ends up quite big, around 5000 digits, and on
   some processors much bigger still.
   
   @code{SET_STR_THRESHOLD} is based on the input digits (and tuned for decimal),
   though it might be better based on a limb count, so as to be independent of
   the base.  But that sort of count isn't used by the base case and so would
   need some sort of initial calculation or estimate.
   
   The main reason @code{SET_STR_THRESHOLD} is so much bigger than the
   corresponding @code{GET_STR_PRECOMPUTE_THRESHOLD} is that @code{mpn_mul_1} is
   much faster than @code{mpn_divrem_1} (often by a factor of 10, or more).
   
   
   @need 1000
   @node Other Algorithms, Assembler Coding, Radix Conversion Algorithms, Algorithms
   @section Other Algorithms
   
   @menu
   * Factorial Algorithm::
   * Binomial Coefficients Algorithm::
   * Fibonacci Numbers Algorithm::
   * Lucas Numbers Algorithm::
   @end menu
   
   
   @node Factorial Algorithm, Binomial Coefficients Algorithm, Other Algorithms, Other Algorithms
   @subsection Factorial
   
   Factorials @math{n!} are calculated by a simple product from @math{1} to
   @math{n}, but arranged into certain sub-products.
   
   First as many factors as fit in a limb are accumulated, then two of those
   multiplied to give a 2-limb product.  When two 2-limb products are ready
   they're multiplied to a 4-limb product, and when two 4-limbs are ready they're
   multiplied to an 8-limb product, etc.  A stack of outstanding products is
   built up, with two of the same size multiplied together when ready.
   
   Arranging for multiplications to have operands the same (or nearly the same)
   size means the Karatsuba and higher multiplication algorithms can be used.
   And even on sizes below the Karatsuba threshold an N@cross{}N multiply will
   give a basecase multiply more to work on.
   
   An obvious improvement not currently implemented would be to strip factors of
   2 from the products and apply them at the end with a bit shift.  Another
   possibility would be to determine the prime factorization of the result (which
   can be done easily), and use a powering method, at each stage squaring then
   multiplying in those primes with a 1 in their exponent at that point.  The
   advantage would be some multiplies turned into squares.
   
   
   @node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms
   @subsection Binomial Coefficients
   
   Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated
   by first arranging @math{k @le{} n/2} using @m{\left({n}\atop{k}\right) =
   \left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then
   evaluating the following product simply from @math{i=2} to @math{i=k}.
   @tex
   $$ \left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i} $$
   @end tex
   @ifnottex
   
   @example
                         k  (n-k+i)
   C(n,k) =  (n-k+1) * prod -------
                        i=2    i
   @end example
   
   @end ifnottex
   It's easy to show that each denominator @math{i} will divide the product so
   far, so the exact division algorithm is used (@pxref{Exact Division}).
   
   The numerators @math{n-k+i} and denominators @math{i} are first accumulated
   into as many fit a limb, to save multi-precision operations, though for
   @code{mpz_bin_ui} this applies only to the divisors, since @math{n} is an
   @code{mpz_t} and @math{n-k+i} in general won't fit in a limb at all.
   
   An obvious improvement would be to strip factors of 2 from each multiplier and
   divisor and count them separately, to be applied with a bit shift at the end.
   Factors of 3 and perhaps 5 could even be handled similarly.  Another
   possibility, if @math{n} is not too big, would be to determine the prime
   factorization of the result based on the factorials involved, and power up
   those primes appropriately.  This would help most when @math{k} is near
   @math{n/2}.
   
   
   @node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms
   @subsection Fibonacci Numbers
   
   The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed
   for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]}
   values efficiently.
   
   For small @math{n}, a table of single limb values in @code{__gmp_fib_table} is
   used.  On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb
   up to @m{F_{93},F[93]}.  For convenience the table starts at @m{F_{-1},F[-1]}.
   
   Beyond the table, values are generated with a binary powering algorithm,
   calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to
   low across the bits of @math{n}.  The formulas used are
   @tex
   $$\eqalign{
     F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr
     F_{2k-1} &=  F_k^2 + F_{k-1}^2           \cr
     F_{2k}   &= F_{2k+1} - F_{2k-1}
   }$$
   @end tex
   @ifnottex
   
   @example
   F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
   F[2k-1] =   F[k]^2 + F[k-1]^2
   
   F[2k] = F[2k+1] - F[2k-1]
   @end example
   
   @end ifnottex
   At each step, @math{k} is the high @math{b} bits of @math{n}.  If the next bit
   of @math{n} is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if
   it's a 1 then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process
   repeated until all bits of @math{n} are incorporated.  Notice these formulas
   require just two squares per bit of @math{n}.
   
   It'd be possible to handle the first few @math{n} above the single limb table
   with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} =
   F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually
   turns out to be faster for only about 10 or 20 values of @math{n}, and
   including a block of code for just those doesn't seem worthwhile.  If they
   really mattered it'd be better to extend the data table.
   
   Using a table avoids lots of calculations on small numbers, and makes small
   @math{n} go fast.  A bigger table would make more small @math{n} go fast, it's
   just a question of balancing size against desired speed.  For GMP the code is
   kept compact, with the emphasis primarily on a good powering algorithm.
   
   @code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but
   @code{mpz_fib_ui} is only interested in @m{F_n,F[n]}.  In this case the last
   step of the algorithm can become one multiply instead of two squares.  One of
   the following two formulas is used, according as @math{n} is odd or even.
   @tex
   $$\eqalign{
     F_{2k}   &= F_k (F_k + 2F_{k-1}) \cr
     F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k
   }$$
   @end tex
   @ifnottex
   
   @example
   F[2k]   = F[k]*(F[k]+2F[k-1])
   
   F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
   @end example
   
   @end ifnottex
   @m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a
   multiply.  For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above
   can be applied just to the low limb of the calculation, without a carry or
   borrow into further limbs, which saves some code size.  See comments with
   @code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done.
   
   
   @node Lucas Numbers Algorithm,  , Fibonacci Numbers Algorithm, Other Algorithms
   @subsection Lucas Numbers
   
   @code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci
   numbers with the following simple formulas.
   @tex
   $$\eqalign{
     L_k     &=  F_k + 2F_{k-1} \cr
     L_{k-1} &= 2F_k -  F_{k-1}
   }$$
   @end tex
   @ifnottex
   
   @example
   L[k]   =   F[k] + 2*F[k-1]
   L[k-1] = 2*F[k] -   F[k-1]
   @end example
   
   @end ifnottex
   @code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be
   saved.  Trailing zero bits on @math{n} can be handled with a single square
   each.
   @tex
   $$ L_{2k} = L_k^2 - 2(-1)^k $$
   @end tex
   @ifnottex
   
   @example
   L[2k] = L[k]^2 - 2*(-1)^k
   @end example
   
   @end ifnottex
   And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
   numbers, similar to what @code{mpz_fib_ui} does.
   @tex
   $$ L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k $$
   @end tex
   @ifnottex
   
   @example
   L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
   @end example
   
   @end ifnottex
   
   
   @node Assembler Coding,  , Other Algorithms, Algorithms
   @section Assembler Coding
   
   The assembler subroutines in GMP are the most significant source of speed at
   small to moderate sizes.  At larger sizes algorithm selection becomes more
   important, but of course speedups in low level routines will still speed up
   everything proportionally.
   
   Carry handling and widening multiplies that are important for GMP can't be
   easily expressed in C.  GCC @code{asm} blocks help a lot and are provided in
   @file{longlong.h}, but hand coding low level routines invariably offers a
   speedup over generic C by a factor of anything from 2 to 10.
   
   @menu
   * Assembler Code Organisation::
   * Assembler Basics::
   * Assembler Carry Propagation::
   * Assembler Cache Handling::
   * Assembler Floating Point::
   * Assembler SIMD Instructions::
   * Assembler Software Pipelining::
   * Assembler Loop Unrolling::
   @end menu
   
   
   @node Assembler Code Organisation, Assembler Basics, Assembler Coding, Assembler Coding
   @subsection Code Organisation
   
   The various @file{mpn} subdirectories contain machine-dependent code, written
   in C or assembler.  The @file{mpn/generic} subdirectory contains default code,
   used when there's no machine-specific version of a particular file.
   
   Each @file{mpn} subdirectory is for an ISA family.  Generally 32-bit and
   64-bit variants in a family cannot share code and will have separate
   directories.  Within a family further subdirectories may exist for CPU
   variants.
   
   
   @node Assembler Basics, Assembler Carry Propagation, Assembler Code Organisation, Assembler Coding
   @subsection Assembler Basics
   
   @code{mpn_addmul_1} and @code{mpn_submul_1} are the most important routines
   for overall GMP performance.  All multiplications and divisions come down to
   repeated calls to these.  @code{mpn_add_n}, @code{mpn_sub_n},
   @code{mpn_lshift} and @code{mpn_rshift} are next most important.
   
   On some CPUs assembler versions of the internal functions
   @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} give significant speedups,
   mainly through avoiding function call overheads.  They can also potentially
   make better use of a wide superscalar processor.
   
   The restrictions on overlaps between sources and destinations
   (@pxref{Low-level Functions}) are designed to facilitate a variety of
   implementations.  For example, knowing @code{mpn_add_n} won't have partly
   overlapping sources and destination means reading can be done far ahead of
   writing on superscalar processors, and loops can be vectorized on a vector
   processor, depending on the carry handling.
   
   
   @node Assembler Carry Propagation, Assembler Cache Handling, Assembler Basics, Assembler Coding
   @subsection Carry Propagation
   
   The problem that presents most challenges in GMP is propagating carries from
   one limb to the next.  In functions like @code{mpn_addmul_1} and
   @code{mpn_add_n}, carries are the only dependencies between limb operations.
   
   On processors with carry flags, a straightforward CISC style @code{adc} is
   generally best.  AMD K6 @code{mpn_addmul_1} however is an example of an
   unusual set of circumstances where a branch works out better.
   
   On RISC processors generally an add and compare for overflow is used.  This
   sort of thing can be seen in @file{mpn/generic/aors_n.c}.  Some carry
   propagation schemes require 4 instructions, meaning at least 4 cycles per
   limb, but other schemes may use just 1 or 2.  On wide superscalar processors
   performance may be completely determined by the number of dependent
   instructions between carry-in and carry-out for each limb.
   
   On vector processors good use can be made of the fact that a carry bit only
   very rarely propagates more than one limb.  When adding a single bit to a
   limb, there's only a carry out if that limb was @code{0xFF...FF} which on
   random data will be only 1 in @m{2\GMPraise{@code{mp\_bits\_per\_limb}},
   2^mp_bits_per_limb}.  @file{mpn/cray/add_n.c} is an example of this, it adds
   all limbs in parallel, adds one set of carry bits in parallel and then only
   rarely needs to fall through to a loop propagating further carries.
   
   On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
   for the RISC style idioms that are necessary to handle carry bits in
   C.  Often conditional jumps are generated where @code{adc} or @code{sbb} forms
   would be better.  And so unfortunately almost any loop involving carry bits
   needs to be coded in assembler for best results.
   
   
   @node Assembler Cache Handling, Assembler Floating Point, Assembler Carry Propagation, Assembler Coding
   @subsection Cache Handling
   
   GMP aims to perform well both on operands that fit entirely in L1 cache and
   those which don't.
   
   Basic routines like @code{mpn_add_n} or @code{mpn_lshift} are often used on
   large operands, so L2 and main memory performance is important for them.
   @code{mpn_mul_1} and @code{mpn_addmul_1} are mostly used for multiply and
   square basecases, so L1 performance matters most for them, unless assembler
   versions of @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} exist, in
   which case the remaining uses are mostly for larger operands.
   
   For L2 or main memory operands, memory access times will almost certainly be
   more than the calculation time.  The aim therefore is to maximize memory
   throughput, by starting a load of the next cache line which processing the
   contents of the previous one.  Clearly this is only possible if the chip has a
   lock-up free cache or some sort of prefetch instruction.  Most current chips
   have both these features.
   
   Prefetching sources combines well with loop unrolling, since a prefetch can be
   initiated once per unrolled loop (or more than once if the loop covers more
   than one cache line).
   
   On CPUs without write-allocate caches, prefetching destinations will ensure
   individual stores don't go further down the cache hierarchy, limiting
   bandwidth.  Of course for calculations which are slow anyway, like
   @code{mpn_divrem_1}, write-throughs might be fine.
   
   The distance ahead to prefetch will be determined by memory latency versus
   throughput.  The aim of course is to have data arriving continuously, at peak
   throughput.  Some CPUs have limits on the number of fetches or prefetches in
   progress.
   
   If a special prefetch instruction doesn't exist then a plain load can be used,
   but in that case care must be taken not to attempt to read past the end of an
   operand, since that might produce a segmentation violation.
   
   Some CPUs or systems have hardware that detects sequential memory accesses and
   initiates suitable cache movements automatically, making life easy.
   
   
   @node Assembler Floating Point, Assembler SIMD Instructions, Assembler Cache Handling, Assembler Coding
   @subsection Floating Point
   
   Floating point arithmetic is used in GMP for multiplications on CPUs with poor
   integer multipliers.  It's mostly useful for @code{mpn_mul_1},
   @code{mpn_addmul_1} and @code{mpn_submul_1} on 64-bit machines, and
   @code{mpn_mul_basecase} on both 32-bit and 64-bit machines.
   
   With IEEE 53-bit double precision floats, integer multiplications producing up
   to 53 bits will give exact results.  Breaking a 64@cross{}64 multiplication
   into eight 16@cross{}@math{32@rightarrow{}48} bit pieces is convenient.  With
   some care though six 21@cross{}@math{32@rightarrow{}53} bit products can be
   used, if one of the lower two 21-bit pieces also uses the sign bit.
   
   For the @code{mpn_mul_1} family of functions on a 64-bit machine, the
   invariant single limb is split at the start, into 3 or 4 pieces.  Inside the
   loop, the bignum operand is split into 32-bit pieces.  Fast conversion of
   these unsigned 32-bit pieces to floating point is highly machine-dependent.
   In some cases, reading the data into the integer unit, zero-extending to
   64-bits, then transferring to the floating point unit back via memory is the
   only option.
   
   Converting partial products back to 64-bit limbs is usually best done as a
   signed conversion.  Since all values are smaller than @m{2^{53},2^53}, signed
   and unsigned are the same, but most processors lack unsigned conversions.
   
   @sp 2
   
   Here is a diagram showing 16@cross{}32 bit products for an @code{mpn_mul_1} or
   @code{mpn_addmul_1} with a 64-bit limb.  The single limb operand V is split
   into four 16-bit parts.  The multi-limb operand U is split in the loop into
   two 32-bit parts.
   
   @tex
   \global\newdimen\GMPbits      \global\GMPbits=0.18em
   \def\GMPbox#1#2#3{%
     \hbox{%
       \hbox to 128\GMPbits{\hfil
         \vbox{%
           \hrule
           \hbox to 48\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
           \hrule}%
         \hskip #1\GMPbits}%
       \raise \GMPboxdepth \hbox{\hskip 2em #3}}}
   %
   \GMPdisplay{%
     \vbox{%
       \hbox{%
         \hbox to 128\GMPbits {\hfil
           \vbox{%
             \hrule
             \hbox to 64\GMPbits{%
               \GMPvrule \hfil$v48$\hfil
               \vrule    \hfil$v32$\hfil
               \vrule    \hfil$v16$\hfil
               \vrule    \hfil$v00$\hfil
               \vrule}
             \hrule}}%
          \raise \GMPboxdepth \hbox{\hskip 2em V Operand}}
       \vskip 0.5ex
       \hbox{%
         \hbox to 128\GMPbits {\hfil
           \raise \GMPboxdepth \hbox{$\times$\hskip 1.5em}%
           \vbox{%
             \hrule
             \hbox to 64\GMPbits {%
               \GMPvrule \hfil$u32$\hfil
               \vrule \hfil$u00$\hfil
               \vrule}%
             \hrule}}%
          \raise \GMPboxdepth \hbox{\hskip 2em U Operand (one limb)}}%
       \vskip 0.5ex
       \hbox{\vbox to 2ex{\hrule width 128\GMPbits}}%
       \GMPbox{0}{u00 \times v00}{$p00$\hskip 1.5em 48-bit products}%
       \vskip 0.5ex
       \GMPbox{16}{u00 \times v16}{$p16$}
       \vskip 0.5ex
       \GMPbox{32}{u00 \times v32}{$p32$}
       \vskip 0.5ex
       \GMPbox{48}{u00 \times v48}{$p48$}
       \vskip 0.5ex
       \GMPbox{32}{u32 \times v00}{$r32$}
       \vskip 0.5ex
       \GMPbox{48}{u32 \times v16}{$r48$}
       \vskip 0.5ex
       \GMPbox{64}{u32 \times v32}{$r64$}
       \vskip 0.5ex
       \GMPbox{80}{u32 \times v48}{$r80$}
   }}
   @end tex
   @ifnottex
   @example
   @group
                   +---+---+---+---+
                   |v48|v32|v16|v00|    V operand
                   +---+---+---+---+
   
                   +-------+---+---+
               x   |  u32  |  u00  |    U operand (one limb)
                   +---------------+
   
   ---------------------------------
   
                       +-----------+
                       | u00 x v00 |    p00    48-bit products
                       +-----------+
                   +-----------+
                   | u00 x v16 |        p16
                   +-----------+
               +-----------+
               | u00 x v32 |            p32
               +-----------+
           +-----------+
           | u00 x v48 |                p48
           +-----------+
               +-----------+
               | u32 x v00 |            r32
               +-----------+
           +-----------+
           | u32 x v16 |                r48
           +-----------+
       +-----------+
       | u32 x v32 |                    r64
       +-----------+
   +-----------+
   | u32 x v48 |                        r80
   +-----------+
   @end group
   @end example
   @end ifnottex
   
   @math{p32} and @math{r32} can be summed using floating-point addition, and
   likewise @math{p48} and @math{r48}.  @math{p00} and @math{p16} can be summed
   with @math{r64} and @math{r80} from the previous iteration.
   
   For each loop then, four 49-bit quantities are transfered to the integer unit,
   aligned as follows,
   
   @tex
   % GMPbox here should be 49 bits wide, but use 51 to better show p16+r80'
   % crossing into the upper 64 bits.
   \def\GMPbox#1#2#3{%
     \hbox{%
       \hbox to 128\GMPbits {%
         \hfil
         \vbox{%
           \hrule
           \hbox to 51\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
           \hrule}%
         \hskip #1\GMPbits}%
       \raise \GMPboxdepth \hbox{\hskip 1.5em $#3$\hfil}%
   }}
   \newbox\b \setbox\b\hbox{64 bits}%
   \newdimen\bw \bw=\wd\b \advance\bw by 2em
   \newdimen\x \x=128\GMPbits
   \advance\x by -2\bw
   \divide\x by4
   \GMPdisplay{%
     \vbox{%
       \hbox to 128\GMPbits {%
         \GMPvrule
         \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
         \hfil 64 bits\hfil
         \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
         \vrule
         \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
         \hfil 64 bits\hfil
         \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
         \vrule}%
       \vskip 0.7ex
       \GMPbox{0}{p00+r64'}{i00}
       \vskip 0.5ex
       \GMPbox{16}{p16+r80'}{i16}
       \vskip 0.5ex
       \GMPbox{32}{p32+r32}{i32}
       \vskip 0.5ex
       \GMPbox{48}{p48+r48}{i48}
   }}
   @end tex
   @ifnottex
   @example
   @group
   |-----64bits----|-----64bits----|
                      +------------+
                      | p00 + r64' |    i00
                      +------------+
                  +------------+
                  | p16 + r80' |        i16
                  +------------+
              +------------+
              | p32 + r32  |            i32
              +------------+
          +------------+
          | p48 + r48  |                i48
          +------------+
   @end group
   @end example
   @end ifnottex
   
   The challenge then is to sum these efficiently and add in a carry limb,
   generating a low 64-bit result limb and a high 33-bit carry limb (@math{i48}
   extends 33 bits into the high half).
   
   
   @node Assembler SIMD Instructions, Assembler Software Pipelining, Assembler Floating Point, Assembler Coding
   @subsection SIMD Instructions
   
   The single-instruction multiple-data support in current microprocessors is
   aimed at signal processing algorithms where each data point can be treated
   more or less independently.  There's generally not much support for
   propagating the sort of carries that arise in GMP.
   
   SIMD multiplications of say four 16@cross{}16 bit multiplies only do as much
   work as one 32@cross{}32 from GMP's point of view, and need some shifts and
   adds besides.  But of course if say the SIMD form is fully pipelined and uses
   less instruction decoding then it may still be worthwhile.
   
   On the 80x86 chips, MMX has so far found a use in @code{mpn_rshift} and
   @code{mpn_lshift} since it allows 64-bit operations, and is used in a special
   case for 16-bit multipliers in the P55 @code{mpn_mul_1}.  3DNow and SSE
   haven't found a use so far.
   
   
   @node Assembler Software Pipelining, Assembler Loop Unrolling, Assembler SIMD Instructions, Assembler Coding
   @subsection Software Pipelining
   
   Software pipelining consists of scheduling instructions around the branch
   point in a loop.  For example a loop taking a checksum of an array of limbs
   might have a load and an add, but the load wouldn't be for that add, rather
   for the one next time around the loop.  Each load then is effectively
   scheduled back in the previous iteration, allowing latency to be hidden.
   
   Naturally this is wanted only when doing things like loads or multiplies that
   take a few cycles to complete, and only where a CPU has multiple functional
   units so that other work can be done while waiting.
   
   A pipeline with several stages will have a data value in progress at each
   stage and each loop iteration moves them along one stage.  This is like
   juggling.
   
   Within the loop some moves between registers may be necessary to have the
   right values in the right places for each iteration.  Loop unrolling can help
   this, with each unrolled block able to use different registers for different
   values, even if some shuffling is still needed just before going back to the
   top of the loop.
   
   
   @node Assembler Loop Unrolling,  , Assembler Software Pipelining, Assembler Coding
   @subsection Loop Unrolling
   
   Loop unrolling consists of replicating code so that several limbs are
   processed in each loop.  At a minimum this reduces loop overheads by a
   corresponding factor, but it can also allow better register usage, for example
   alternately using one register combination and then another.  Judicious use of
   @command{m4} macros can help avoid lots of duplication in the source code.
   
   Unrolling is commonly done to a power of 2 multiple so the number of unrolled
   loops and the number of remaining limbs can be calculated with a shift and
   mask.  But other multiples can be used too, just by subtracting each @var{n}
   limbs processed from a counter and waiting for less than @var{n} remaining (or
   offsetting the counter by @var{n} so it goes negative when there's less than
   @var{n} remaining).
   
   The limbs not a multiple of the unrolling can be handled in various ways, for
   example
   
   @itemize @bullet
   @item
   A simple loop at the end (or the start) to process the excess.  Care will be
   wanted that it isn't too much slower than the unrolled part.
   
   @item
   A set of binary tests, for example after an 8-limb unrolling, test for 4 more
   limbs to process, then a further 2 more or not, and finally 1 more or not.
   This will probably take more code space than a simple loop.
   
   @item
   A @code{switch} statement, providing separate code for each possible excess,
   for example an 8-limb unrolling would have separate code for 0 remaining, 1
   remaining, etc, up to 7 remaining.  This might take a lot of code, but may be
   the best way to optimize all cases in combination with a deep pipelined loop.
   
   @item
   A computed jump into the middle of the loop, thus making the first iteration
   handle the excess.  This should make times smoothly increase with size, which
   is attractive, but setups for the jump and adjustments for pointers can be
   tricky and could become quite difficult in combination with deep pipelining.
   @end itemize
   
   One way to write the setups and finishups for a pipelined unrolled loop is
   simply to duplicate the loop at the start and the end, then delete
   instructions at the start which have no valid antecedents, and delete
   instructions at the end whose results are unwanted.  Sizes not a multiple of
   the unrolling can then be handled as desired.
   
   
   @node Internals, Contributors, Algorithms, Top
   @chapter Internals
   
   @strong{This chapter is provided only for informational purposes and the
   various internals described here may change in future GMP releases.
   Applications expecting to be compatible with future releases should use only
   the documented interfaces described in previous chapters.}
   
   @menu
   * Integer Internals::
   * Rational Internals::
   * Float Internals::
   * Raw Output Internals::
   * C++ Interface Internals::
   @end menu
   
   @node Integer Internals, Rational Internals, Internals, Internals
   @section Integer Internals
   
   @code{mpz_t} variables represent integers using sign and magnitude, in space
   dynamically allocated and reallocated.  The fields are as follows.
   
   @table @asis
   @item @code{_mp_size}
   The number of limbs, or the negative of that when representing a negative
   integer.  Zero is represented by @code{_mp_size} set to zero, in which case
   the @code{_mp_d} data is unused.
   
   @item @code{_mp_d}
   A pointer to an array of limbs which is the magnitude.  These are stored
   ``little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the
   least significant limb and @code{_mp_d[ABS(_mp_size)-1]} is the most
   significant.  Whenever @code{_mp_size} is non-zero, the most significant limb
   is non-zero.
   
   Currently there's always at least one limb allocated, so for instance
   @code{mpz_set_ui} never needs to reallocate, and @code{mpz_get_ui} can fetch
   @code{_mp_d[0]} unconditionally (though its value is then only wanted if
   @code{_mp_size} is non-zero).
   
   @item @code{_mp_alloc}
   @code{_mp_alloc} is the number of limbs currently allocated at @code{_mp_d},
   and naturally @code{_mp_alloc >= ABS(_mp_size)}.  When an @code{mpz} routine
   is about to (or might be about to) increase @code{_mp_size}, it checks
   @code{_mp_alloc} to see whether there's enough space, and reallocates if not.
   @code{MPZ_REALLOC} is generally used for this.
   @end table
   
   The various bitwise logical functions like @code{mpz_and} behave as if
   negative values were twos complement.  But sign and magnitude is always used
   internally, and necessary adjustments are made during the calculations.
   Sometimes this isn't pretty, but sign and magnitude are best for other
   routines.
   
   Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these
   have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory
   allocation functions.  Care is taken to ensure that these are big enough that
   no reallocation is necessary (since it would have unpredictable consequences).
   
   
   @node Rational Internals, Float Internals, Integer Internals, Internals
   @section Rational Internals
   
   @code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and
   denominator (@pxref{Integer Internals}).
   
   The canonical form adopted is denominator positive (and non-zero), no common
   factors between numerator and denominator, and zero uniquely represented as
   0/1.
   
   It's believed that casting out common factors at each stage of a calculation
   is best in general.  A GCD is an @math{O(N^2)} operation so it's better to do
   a few small ones immediately than to delay and have to do a big one later.
   Knowing the numerator and denominator have no common factors can be used for
   example in @code{mpq_mul} to make only two cross GCDs necessary, not four.
   
   This general approach to common factors is badly sub-optimal in the presence
   of simple factorizations or little prospect for cancellation, but GMP has no
   way to know when this will occur.  As per @ref{Efficiency}, that's left to
   applications.  The @code{mpq_t} framework might still suit, with
   @code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and
   denominator, or of course @code{mpz_t} variables can be used directly.
   
   
   @node Float Internals, Raw Output Internals, Rational Internals, Internals
   @section Float Internals
   
   Efficient calculation is the primary aim of GMP floats and the use of whole
   limbs and simple rounding facilitates this.
   
   @code{mpf_t} floats have a variable precision mantissa and a single machine
   word signed exponent.  The mantissa is represented using sign and magnitude.
   
   @c FIXME: The arrow heads don't join to the lines exactly.
   @tex
   \global\newdimen\GMPboxwidth \GMPboxwidth=5em
   \global\newdimen\GMPboxheight \GMPboxheight=3ex
   \def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
   \GMPdisplay{%
   \vbox{%
     \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb}
     \vskip 0.7ex
     \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
     \hbox {
       \hbox to 3\GMPboxwidth {%
         \setbox 0 = \hbox{@code{\_mp\_exp}}%
         \dimen0=3\GMPboxwidth
         \advance\dimen0 by -\wd0
         \divide\dimen0 by 2
         \advance\dimen0 by -1em
         \setbox1 = \hbox{$\rightarrow$}%
         \dimen1=\dimen0
         \advance\dimen1 by -\wd1
         \GMPcentreline{\dimen0}%
         \hfil
         \box0%
         \hfil
         \GMPcentreline{\dimen1{}}%
         \box1}
       \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}}
     \vskip 0.5ex
     \vbox {%
       \hrule
       \hbox{%
         \vrule height 2ex depth 1ex
         \hbox to \GMPboxwidth {}%
         \vrule
         \hbox to \GMPboxwidth {}%
         \vrule
         \hbox to \GMPboxwidth {}%
         \vrule
         \hbox to \GMPboxwidth {}%
         \vrule
         \hbox to \GMPboxwidth {}%
         \vrule}
       \hrule
     }
     \hbox {%
       \hbox to 0.8 pt {}
       \hbox to 3\GMPboxwidth {%
         \hfil $\cdot$} \hbox {$\leftarrow$ radix point\hfil}}
     \hbox to 5\GMPboxwidth{%
       \setbox 0 = \hbox{@code{\_mp\_size}}%
       \dimen0 = 5\GMPboxwidth
       \advance\dimen0 by -\wd0
       \divide\dimen0 by 2
       \advance\dimen0 by -1em
       \dimen1 = \dimen0
       \setbox1 = \hbox{$\leftarrow$}%
       \setbox2 = \hbox{$\rightarrow$}%
       \advance\dimen0 by -\wd1
       \advance\dimen1 by -\wd2
       \hbox to 0.3 em {}%
       \box1
       \GMPcentreline{\dimen0}%
       \hfil
       \box0
       \hfil
       \GMPcentreline{\dimen1}%
       \box2}
   }}
   @end tex
   @ifnottex
   @example
      most                   least
   significant            significant
      limb                   limb
   
                               _mp_d
    |---- _mp_exp --->           |
     _____ _____ _____ _____ _____
    |_____|_____|_____|_____|_____|
                      . <------------ radix point
   
     <-------- _mp_size --------->
   @sp 1
   @end example
   @end ifnottex
   
   @noindent
   The fields are as follows.
   
   @table @asis
   @item @code{_mp_size}
   The number of limbs currently in use, or the negative of that when
   representing a negative value.  Zero is represented by @code{_mp_size} and
   @code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is
   unused.  (In the future @code{_mp_exp} might be undefined when representing
   zero.)
   
   @item @code{_mp_prec}
   The precision of the mantissa, in limbs.  In any calculation the aim is to
   produce @code{_mp_prec} limbs of result (the most significant being non-zero).
   
   @item @code{_mp_d}
   A pointer to the array of limbs which is the absolute value of the mantissa.
   These are stored ``little endian'' as per the @code{mpn} functions, so
   @code{_mp_d[0]} is the least significant limb and
   @code{_mp_d[ABS(_mp_size)-1]} the most significant.
   
   The most significant limb is always non-zero, but there are no other
   restrictions on its value, in particular the highest 1 bit can be anywhere
   within the limb.
   
   @code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being
   for convenience (see below).  There are no reallocations during a calculation,
   only in a change of precision with @code{mpf_set_prec}.
   
   @item @code{_mp_exp}
   The exponent, in limbs, determining the location of the implied radix point.
   Zero means the radix point is just above the most significant limb.  Positive
   values mean a radix point offset towards the lower limbs and hence a value
   @math{@ge{} 1}, as for example in the diagram above.  Negative exponents mean
   a radix point further above the highest limb.
   
   Naturally the exponent can be any value, it doesn't have to fall within the
   limbs as the diagram shows, it can be a long way above or a long way below.
   Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data
   are treated as zero.
   @end table
   
   @sp 1
   @noindent
   The following various points should be noted.
   
   @table @asis
   @item Low Zeros
   The least significant limbs @code{_mp_d[0]} etc can be zero, though such low
   zeros can always be ignored.  Routines likely to produce low zeros check and
   avoid them to save time in subsequent calculations, but for most routines
   they're quite unlikely and aren't checked.
   
   @item Mantissa Size Range
   The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if
   the value can be represented in less.  This means low precision values or
   small integers stored in a high precision @code{mpf_t} can still be operated
   on efficiently.
   
   @code{_mp_size} can also be greater than @code{_mp_prec}.  Firstly a value is
   allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d},
   and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves
   @code{_mp_size} unchanged and so the size can be arbitrarily bigger than
   @code{_mp_prec}.
   
   @item Rounding
   All rounding is done on limb boundaries.  Calculating @code{_mp_prec} limbs
   with the high non-zero will ensure the application requested minimum precision
   is obtained.
   
   The use of simple ``trunc'' rounding towards zero is efficient, since there's
   no need to examine extra limbs and increment or decrement.
   
   @item Bit Shifts
   Since the exponent is in limbs, there are no bit shifts in basic operations
   like @code{mpf_add} and @code{mpf_mul}.  When differing exponents are
   encountered all that's needed is to adjust pointers to line up the relevant
   limbs.
   
   Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts,
   but the choice is between an exponent in limbs which requires shifts there, or
   one in bits which requires them almost everywhere else.
   
   @item Use of @code{_mp_prec+1} Limbs
   The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just
   @code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its
   operation.  @code{mpf_add} for instance will do an @code{mpn_add} of
   @code{_mp_prec} limbs.  If there's no carry then that's the result, but if
   there is a carry then it's stored in the extra limb of space and
   @code{_mp_size} becomes @code{_mp_prec+1}.
   
   Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not
   needed for the intended precision, only the @code{_mp_prec} high limbs.  But
   zeroing it out or moving the rest down is unnecessary.  Subsequent routines
   reading the value will simply take the high limbs they need, and this will be
   @code{_mp_prec} if their target has that same precision.  This is no more than
   a pointer adjustment, and must be checked anyway since the destination
   precision can be different from the sources.
   
   Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs
   if available.  This ensures that a variable which has @code{_mp_size} equal to
   @code{_mp_prec+1} will get its full exact value copied.  Strictly speaking
   this is unnecessary since only @code{_mp_prec} limbs are needed for the
   application's requested precision, but it's considered that an @code{mpf_set}
   from one variable into another of the same precision ought to produce an exact
   copy.
   
   @item Application Precisions
   @code{__GMPF_BITS_TO_PREC} converts an application requested precision to an
   @code{_mp_prec}.  The value in bits is rounded up to a whole limb then an
   extra limb is added since the most significant limb of @code{_mp_d} is only
   non-zero and therefore might contain only one bit.
   
   @code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra
   limb from @code{_mp_prec} before converting to bits.  The net effect of
   reading back with @code{mpf_get_prec} is simply the precision rounded up to a
   multiple of @code{mp_bits_per_limb}.
   
   Note that the extra limb added here for the high only being non-zero is in
   addition to the extra limb allocated to @code{_mp_d}.  For example with a
   32-bit limb, an application request for 250 bits will be rounded up to 8
   limbs, then an extra added for the high being only non-zero, giving an
   @code{_mp_prec} of 9.  @code{_mp_d} then gets 10 limbs allocated.  Reading
   back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and
   multiply by 32, giving 256 bits.
   
   Strictly speaking, the fact the high limb has at least one bit means that a
   float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
   for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice
   multiple of the limb size.
   @end table
   
   
   @node Raw Output Internals, C++ Interface Internals, Float Internals, Internals
   @section Raw Output Internals
   
   @noindent
   @code{mpz_out_raw} uses the following format.
   
   @tex
   \global\newdimen\GMPboxwidth \GMPboxwidth=5em
   \global\newdimen\GMPboxheight \GMPboxheight=3ex
   \def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
   \GMPdisplay{%
   \vbox{%
     \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
     \vbox {%
       \hrule
       \hbox{%
         \vrule height 2.5ex depth 1.5ex
         \hbox to \GMPboxwidth {\hfil size\hfil}%
         \vrule
         \hbox to 3\GMPboxwidth {\hfil data bytes\hfil}%
         \vrule}
       \hrule}
   }}
   @end tex
   @ifnottex
   @example
   +------+------------------------+
   | size |       data bytes       |
   +------+------------------------+
   @end example
   @end ifnottex
   
   The size is 4 bytes written most significant byte first, being the number of
   subsequent data bytes, or the twos complement negative of that when a negative
   integer is represented.  The data bytes are the absolute value of the integer,
   written most significant byte first.
   
   The most significant data byte is always non-zero, so the output is the same
   on all systems, irrespective of limb size.
   
   In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
   of the limb size.  @code{mpz_inp_raw} will still accept this, for
   compatibility.
   
   The use of ``big endian'' for both the size and data fields is deliberate, it
   makes the data easy to read in a hex dump of a file.  Unfortunately it also
   means that the limb data must be reversed when reading or writing, so neither
   a big endian nor little endian system can just read and write @code{_mp_d}.
   
   
   @node C++ Interface Internals,  , Raw Output Internals, Internals
   @section C++ Interface Internals
   
   A system of expression templates is used to ensure something like @code{a=b+c}
   turns into a simple call to @code{mpz_add} etc.  For @code{mpf_class} and
   @code{mpfr_class} the scheme also ensures the precision of the final
   destination is used for any temporaries within a statement like
   @code{f=w*x+y*z}.  These are important features which a naive implementation
   cannot provide.
   
   A simplified description of the scheme follows.  The true scheme is
   complicated by the fact that expressions have different return types.  For
   detailed information, refer to the source code.
   
   To perform an operation, say, addition, we first define a ``function object''
   evaluating it,
   
   @example
   struct __gmp_binary_plus
   @{
     static void eval(mpf_t f, mpf_t g, mpf_t h) @{ mpf_add(f, g, h); @}
   @};
   @end example
   
   @noindent
   And an ``additive expression'' object,
   
   @example
   __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
   operator+(const mpf_class &f, const mpf_class &g)
   @{
     return __gmp_expr
       <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
   @}
   @end example
   
   The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<...>>} is used to
   encapsulate any possible kind of expression into a single template type.  In
   fact even @code{mpf_class} etc are @code{typedef} specializations of
   @code{__gmp_expr}.
   
   Next we define assignment of @code{__gmp_expr} to @code{mpf_class}.
   
   @example
   template <class T>
   mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
   @{
     expr.eval(this->get_mpf_t(), this->precision());
     return *this;
   @}
   
   template <class Op>
   void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
   (mpf_t f, unsigned long int precision)
   @{
     Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
   @}
   @end example
   
   where @code{expr.val1} and @code{expr.val2} are references to the expression's
   operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the
   @code{__gmp_expr}).
   
   This way, the expression is actually evaluated only at the time of assignment,
   when the required precision (that of @code{f}) is known.  Furthermore the
   target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly
   with @code{f} as the output argument.
   
   Compound expressions are handled by defining operators taking subexpressions
   as their arguments, like this:
   
   @example
   template <class T, class U>
   __gmp_expr
   <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
   operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
   @{
     return __gmp_expr
       <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
       (expr1, expr2);
   @}
   @end example
   
   And the corresponding specializations of @code{__gmp_expr::eval}:
   
   @example
   template <class T, class U, class Op>
   void __gmp_expr
   <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
   (mpf_t f, unsigned long int precision)
   @{
     // declare two temporaries
     mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
     Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
   @}
   @end example
   
   The expression is thus recursively evaluated to any level of complexity and
   all subexpressions are evaluated to the precision of @code{f}.
   
   
   @node Contributors, References, Internals, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @unnumbered Contributors  @appendix Contributors
   @cindex Contributors
   
 I would like to thank Gunnar Sjoedin and Hans Riesel for their help with  Torbjorn Granlund wrote the original GMP library and is still developing and
 mathematical problems, Richard Stallman for his help with design issues and  maintaining it.  Several other individuals and organizations have contributed
 for revising the first version of this manual, Brian Beuning and Doug Lea for  to GMP in various ways.  Here is a list in chronological order:
 their testing of early versions of the library.  
   
   Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early
   versions of the library.
   
   Richard Stallman contributed to the interface design and revised the first
   version of this manual.
   
   Brian Beuning and Doug Lea helped with testing of early versions of the
   library and made creative suggestions.
   
 John Amanatides of York University in Canada contributed the function  John Amanatides of York University in Canada contributed the function
 @code{mpz_probab_prime_p}.  @code{mpz_probab_prime_p}.
   
Line 2621  Ken Weber (Kent State University, Universidade Federal
Line 9048  Ken Weber (Kent State University, Universidade Federal
 contributed @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and  contributed @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and
 @code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2.  @code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2.
   
 Per Bothner of Cygnus Support helped to set up MP to use Cygnus' configure.  Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure.
 He has also made valuable suggestions and tested numerous intermediary  He has also made valuable suggestions and tested numerous intermediary
 releases.  releases.
   
 Joachim Hollman was involved in the design of the @code{mpf} interface, and in  Joachim Hollman was involved in the design of the @code{mpf} interface, and in
 the @code{mpz} design revisions for version 2.  the @code{mpz} design revisions for version 2.
   
 Bennet Yee contributed the functions @code{mpz_jacobi} and  Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
 @code{mpz_legendre}.  @code{mpz_legendre}.
   
 Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and  Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
 @file{mpn/m68k/rshift.S}.  @file{mpn/m68k/rshift.S} (now in @file{.asm} form).
   
 The development of floating point functions of GNU MP 2, were supported in  The development of floating point functions of GNU MP 2, were supported in part
 part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO  by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
 (POlynomial System SOlving).  System SOlving).
   
 GNU MP 2 was finished and released by TMG Datakonsult, Sodermannagatan 5, 116  GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the
 23 STOCKHOLM, SWEDEN, in cooperation with the IDA Center for Computing  IDA Center for Computing Sciences, USA.
 Sciences, USA.  
   
   Robert Harley of Inria, France and David Seal of ARM, England, suggested clever
   improvements for population count.
   
 @node References, , Contributors, Top  Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
   multiplication functions for GMP 3.  He also contributed the ARM assembly
   code.
   
   Torsten Ekedahl of the Mathematical department of Stockholm University provided
   significant inspiration during several phases of the GMP development.  His
   mathematical expertise helped improve several algorithms.
   
   Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the
   REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square
   root.  The ECMNET project Paul is organizing was a driving force behind many
   of the optimizations in GMP 3.
   
   Linus Nordberg wrote the new configure system based on autoconf and
   implemented the new random functions.
   
   Kent Boortz made the Macintosh port.
   
   Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros,
   parameter tuning, speed measuring, the configure system, function inlining,
   divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number
   functions, printf and scanf functions, perl interface, demo expression parser,
   the algorithms chapter in the manual, @file{gmpasm-mode.el}, and various
   miscellaneous improvements elsewhere.
   
   Steve Root helped write the optimized alpha 21264 assembly code.
   
   Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++
   @code{istream} input routines.
   
   GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde.
   Torbjorn's work was partially funded by the IDA Center for Computing Sciences,
   USA.
   
   (This list is chronological, not ordered after significance.  If you have
   contributed to GMP but are not listed above, please tell @email{tege@@swox.com}
   about the omission!)
   
   Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system
   environment.
   
   @node References, GNU Free Documentation License, Contributors, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @unnumbered References  @appendix References
   @cindex References
   
   @c  FIXME: In tex, the @uref's are unhyphenated, which is good for clarity,
   @c  but being long words they upset paragraph formatting (the preceding line
   @c  can get badly stretched).  Would like an conditional @* style line break
   @c  if the uref is too long to fit on the last line of the paragraph, but it's
   @c  not clear how to do that.  For now explicit @texlinebreak{}s are used on
   @c  paragraphs that come out bad.
   
   @section Books
   
 @itemize @bullet  @itemize @bullet
   @item
   Jonathan M. Borwein and Peter B. Borwein, ``Pi and the AGM: A Study in
   Analytic Number Theory and Computational Complexity'', Wiley, John & Sons,
   1998.
   
 @item  @item
 Donald E. Knuth, "The Art of Computer Programming", vol 2,  Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate
 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.  Texts in Mathematics number 138, Springer-Verlag, 1993.
   @texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen}
   
 @item  @item
 John D. Lipson, "Elements of Algebra and Algebraic Computing",  Donald E. Knuth, ``The Art of Computer Programming'', volume 2,
   ``Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998.
   @texlinebreak{} @uref{http://www-cs-faculty.stanford.edu/~knuth/taocp.html}
   
   @item
   John D. Lipson, ``Elements of Algebra and Algebraic Computing'',
 The Benjamin Cummings Publishing Company Inc, 1981.  The Benjamin Cummings Publishing Company Inc, 1981.
   
 @item  @item
 Richard M. Stallman, "Using and Porting GCC", Free Software Foundation,  Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of
 1995.  Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}
   
 @item  @item
 Peter L. Montgomery, "Modular Multiplication Without Trial Division", in  Richard M. Stallman, ``Using and Porting GCC'', Free Software Foundation, 1999,
 Mathematics of Computation, volume 44, number 170, April 1985.  available online @uref{http://www.gnu.org/software/gcc/onlinedocs/}, and in
   the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/}
   @end itemize
   
   @section Papers
   
   @itemize @bullet
 @item  @item
 Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant  Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'',
 Integers using Multiplication", in Proceedings of the SIGPLAN  Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{}
 PLDI'94 Conference, June 1994.  @uref{http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022}
   
 @item  @item
   Torbjorn Granlund and Peter L. Montgomery, ``Division by Invariant Integers
   using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June
   1994.  Also available @uref{ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz}
   (and .psl.gz).
   
   @item
   Peter L. Montgomery, ``Modular Multiplication Without Trial Division'', in
   Mathematics of Computation, volume 44, number 170, April 1985.
   
   @item
 Tudor Jebelean,  Tudor Jebelean,
 "An algorithm for exact division",  ``An algorithm for exact division'',
 Journal of Symbolic Computation,  Journal of Symbolic Computation,
 v. 15, 1993, pp. 169-180.  volume 15, 1993, pp. 169-180.
   Research report version available @texlinebreak{}
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz}
   
 @item  @item
 Kenneth Weber, "The accelerated integer GCD algorithm",  Tudor Jebelean, ``Exact Division with Karatsuba Complexity - Extended
   Abstract'', RISC-Linz technical report 96-31, @texlinebreak{}
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz}
   
   @item
   Tudor Jebelean, ``Practical Integer Division with Karatsuba Complexity'',
   ISSAC 97, pp. 339-341.  Technical report available @texlinebreak{}
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz}
   
   @item
   Tudor Jebelean, ``A Generalization of the Binary GCD Algorithm'', ISSAC 93,
   pp. 111-116.  Technical report version available @texlinebreak{}
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz}
   
   @item
   Tudor Jebelean, ``A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD
   of Long Integers'', Journal of Symbolic Computation, volume 19, 1995,
   pp. 145-157.  Technical report version also available @texlinebreak{}
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz}
   
   @item
   Werner Krandick and Tudor Jebelean, ``Bidirectional Exact Integer Division'',
   Journal of Symbolic Computation, volume 21, 1996, pp. 441-455.  Early
   technical report version also available
   @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz}
   
   @item
   R. Moenck and A. Borodin, ``Fast Modular Transforms via Division'',
   Proceedings of the 13th Annual IEEE Symposium on Switching and Automata
   Theory, October 1972, pp. 90-96.  Reprinted as ``Fast Modular Transforms'',
   Journal of Computer and System Sciences, volume 8, number 3, June 1974,
   pp. 366-386.
   
   @item
   Arnold Sch@"onhage and Volker Strassen, ``Schnelle Multiplikation grosser
   Zahlen'', Computing 7, 1971, pp. 281-292.
   
   @item
   Kenneth Weber, ``The accelerated integer GCD algorithm'',
 ACM Transactions on Mathematical Software,  ACM Transactions on Mathematical Software,
 v. 21 (March), 1995, pp. 111-122.  volume 21, number 1, March 1995, pp. 111-122.
   
   @item
   Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805,
   November 1999, @uref{http://www.inria.fr/RRRT/RR-3805.html}
   
   @item
   Paul Zimmermann, ``A Proof of GMP Fast Division and Square Root
   Implementations'', @texlinebreak{}
   @uref{http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz}
   
   @item
   Dan Zuras, ``On Squaring and Multiplying Large Integers'', ARITH-11: IEEE
   Symposium on Computer Arithmetic, 1993, pp. 260 to 271.  Reprinted as ``More
   on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers,
   volume 43, number 8, August 1994, pp. 899-908.
 @end itemize  @end itemize
   
 @node Concept Index, , , Top  
   @node GNU Free Documentation License, Concept Index, References, Top
   @appendix GNU Free Documentation License
   @cindex GNU Free Documentation License
   @include fdl.texi
   
   
   @node Concept Index, Function Index, GNU Free Documentation License, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @unnumbered Concept Index  @unnumbered Concept Index
 @printindex cp  @printindex cp
   
 @node Function Index, , , Top  @node Function Index,  , Concept Index, Top
 @comment  node-name,  next,  previous,  up  @comment  node-name,  next,  previous,  up
 @unnumbered Function and Type Index  @unnumbered Function and Type Index
 @printindex fn  @printindex fn
   
   
 @contents  
 @bye  @bye
   
   @c Local variables:
   @c fill-column: 78
   @c compile-command: "make gmp.info"
   @c End:
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