===================================================================
RCS file: /home/cvs/OpenXM_contrib/gmp/Attic/gmp.texi,v
retrieving revision 1.1.1.3
retrieving revision 1.1.1.4
diff -u -p -r1.1.1.3 -r1.1.1.4
--- OpenXM_contrib/gmp/Attic/gmp.texi 2000/12/01 05:44:44 1.1.1.3
+++ OpenXM_contrib/gmp/Attic/gmp.texi 2003/08/25 16:06:01 1.1.1.4
@@ -9,57 +9,65 @@
@end iftex
@comment %**end of header
-@dircategory GNU libraries
-@direntry
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-@end direntry
+@copying
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version @value{VERSION}.
-@c smallbook
+Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002
+Free Software Foundation, Inc.
-@iftex
-@finalout
-@end iftex
+Permission is granted to copy, distribute and/or modify this document under
+the terms of the GNU Free Documentation License, Version 1.1 or any later
+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 Texinfo version 4 or up will be needed to process this into .info files.
+
+@c Texinfo version 4.2 or up will be needed to process this into .info files.
@c
-@c The edition number is in three places and the month/year in one, all
-@c taken from version.texi. version.texi is created when you configure with
-@c --enable-maintainer-mode, and is included in a distribution made with
-@c "make dist".
+@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.
-@ifnottex
-This file documents GNU MP, a library for arbitrary-precision arithmetic.
+@dircategory GNU libraries
+@direntry
+* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
+@end direntry
-Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000 Free
-Software Foundation, Inc.
+@c html
+@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
-this manual provided the copyright notice and this permission notice
-are preserved on all copies.
+@c smallbook
+@finalout
+@setchapternewpage on
-@ignore
-Permission is granted to process this file through TeX and print the
-results, provided the printed document carries copying permission
-notice identical to this one except for the removal of this paragraph
-(this paragraph not being relevant to the printed manual).
-@end ignore
-
-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.
+@ifnottex
+@node Top, Copying, (dir), (dir)
+@top GNU MP
@end ifnottex
-@setchapternewpage on
+@iftex
@titlepage
-@c use the new format for titles
-
@title GNU MP
@subtitle The GNU Multiple Precision Arithmetic Library
@subtitle Edition @value{EDITION}
@@ -79,66 +87,289 @@ by the Foundation.
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000
-Free Software Foundation, Inc.
+@end iftex
-@sp 2
+@insertcopying
+@ifnottex
+@sp 1
+@end ifnottex
-Published by the Free Software Foundation @*
-59 Temple Place - Suite 330 @*
-Boston, MA 02111-1307, USA @*
-
-Permission is granted to make and distribute verbatim copies of
-this manual provided the copyright notice and this permission notice
-are preserved on all copies.
-
-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.
+@iftex
@end titlepage
@headings double
+@end iftex
-@ifnottex
-@node Top, Copying, (dir), (dir)
+@c Don't bother with contents for html, the menus seem adequate.
+@ifnothtml
+@contents
+@end ifnothtml
-@top GNU MP
-
-This manual documents how to install and use the GNU multiple precision
-arithmetic library, version @value{VERSION}.
-
-@end ifnottex
-
@menu
* Copying:: GMP Copying Conditions (LGPL).
* Introduction to GMP:: Brief introduction to GNU MP.
* Installing GMP:: How to configure and compile the GMP library.
-* GMP Basics:: What every GMP user should now.
+* 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.
* Contributors:: Who brings your this library?
* References:: Some useful papers and books to read.
+* GNU Free Documentation License::
* Concept Index::
* Function Index::
@end menu
+
+@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
+
+@macro C {}
+,
+@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
+@ifnotinfo
+@macro nicode {S}
+@code{\S\}
+@end macro
+@end ifnotinfo
+
+@c @nisamp{S} is plain S in info, or @samp{S} elsewhere. This can be used
+@c when the quotes that @samp{} gives in info aren't wanted, but the
+@c fontification in tex or html is wanted.
+@ifinfo
+@macro nisamp {S}
+\S\
+@end macro
+@end ifinfo
+@ifnotinfo
+@macro nisamp {S}
+@samp{\S\}
+@end macro
+@end ifnotinfo
+
+@c Usage: @GMPtimes{}
+@c Give either \times or the word "times".
+@tex
+\gdef\GMPtimes{\times}
+@end tex
+@ifnottex
+@macro GMPtimes
+times
+@end macro
+@end ifnottex
+
+@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
@unnumbered GNU MP Copying Conditions
@cindex Copying conditions
@cindex Conditions for copying GNU MP
+@cindex License conditions
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
@@ -166,7 +397,9 @@ have is not what we distributed, so that any problems
will not reflect on our reputation.@refill
The precise conditions of the license for the GNU MP library are found in the
-Lesser 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 GMP, Installing GMP, Copying, Top
@@ -194,11 +427,11 @@ There is carefully optimized assembly code for these C
ARM,
DEC Alpha 21064, 21164, and 21264,
AMD 29000,
-AMD K6 and Athlon,
+AMD K6, K6-2 and Athlon,
Hitachi SuperH and SH-2,
HPPA 1.0, 1.1 and 2.0,
-Intel Pentium, Pentium Pro/Pentium II, generic x86,
-Intel i960,
+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,
@@ -206,25 +439,53 @@ IBM POWER,
MIPS R3000, R4000,
SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC,
DEC VAX,
-and Zilog Z8000.
-Some optimizations also for Clipper, IBM ROMP (RT), and Pyramid AP/XP.
+and
+Zilog Z8000.
+Some optimizations also for
+Cray vector systems,
+Clipper,
+IBM ROMP (RT),
+and
+Pyramid AP/XP.
-@cindex Mailing list
-There is a mailing list for GMP users. To join it, send a mail to
-@email{gmp-request@@swox.com} with the word @samp{subscribe} in the message
-@strong{body} (not in the subject line).
+@cindex Mailing lists
+There are two public mailing lists of interest. One for general questions and
+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 Home Pages at
-@uref{http://www.swox.com/gmp/}.
+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
@cindex About this manual
Everyone should read @ref{GMP Basics}. If you need to install the library
-yourself, you need to read @ref{Installing GMP}, 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
probably a good idea to glance through it.
@@ -235,8 +496,8 @@ probably a good idea to glance through it.
@chapter Installing GMP
@cindex Installing GMP
@cindex Configuring GMP
+@cindex Building GMP
-@noindent
GMP has an autoconf/automake/libtool based configuration system. On a
Unix-like system a basic build can be done with
@@ -259,10 +520,9 @@ And you can install (under @file{/usr/local} by defaul
make install
@end example
-@noindent
If you experience problems, please report them to @email{bug-gmp@@gnu.org}.
-(@xref{Reporting Bugs}, for information on what to include in useful bug
-reports.)
+See @ref{Reporting Bugs}, for information on what to include in useful bug
+reports.
@menu
* Build Options::
@@ -277,209 +537,470 @@ reports.)
@section Build Options
@cindex Build options
-@noindent
All the usual autoconf configure options are available, run @samp{./configure
---help} for a summary.
+--help} for a summary. The file @file{INSTALL.autoconf} has some generic
+installation information too.
@table @asis
@item Non-Unix Systems
-@samp{configure} needs various Unix-like tools installed. On an MS-DOS system
-cygwin or djgpp should work. 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.
+@samp{configure} requires various Unix-like tools. On an MS-DOS system DJGPP
+can be used, and on MS Windows Cygwin or MINGW can be used,
-@item Object Directory
+@display
+@uref{http://www.cygnus.com/cygwin}
+@uref{http://www.delorie.com/djgpp}
+@uref{http://www.mingw.org}
+@end display
-To compile in a separate object directory, @command{cd} to that directory, and
+Microsoft also publishes an Interix ``Services for Unix'' which can be used to
+build GMP on Windows (with a normal @samp{./configure}), but it's not free
+software.
+
+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 @samp{../src/gmp/configure}. 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 from a
-separate object directory. Use GNU @command{make} instead.
+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 are very slightly slower,
-having a small cost on each function call, but result in smaller executables
-and permit code sharing between separate running processes.
+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 @option{--target=CPU-VENDOR-OS}
+@item Native Compilation, @option{--build=CPU-VENDOR-OS}
-The build target can be specified in the usual way, for either native or cross
-compilation.
+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,
-If @option{--target} isn't given, @samp{./configure} builds for the host
-system as determined by @samp{./config.guess}. On some systems this can't
-distinguish between different CPUs in a family, and you should check the
-guess. Running @samp{./config.guess} on the target system will also show the
-relevant @samp{VENDOR-OS}, if you don't already know what it should be.
+@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 CPU targets have specific assembly code support. See
-@file{configure.in} for which @file{mpn} subdirectories get used by each.
+The following CPUs have specific support. See @file{configure.in} for details
+of what code and compiler options they select.
@itemize @bullet
@c Keep this formatting, it's easy to read and it can be grepped to
-@c automatically test that targets listed get through ./config.sub
+@c automatically test that CPUs listed get through ./config.sub
@item
Alpha:
-@samp{alpha},
-@samp{alphaev5},
-@samp{alphaev6}
+@nisamp{alpha},
+@nisamp{alphaev5},
+@nisamp{alphaev56},
+@nisamp{alphapca56},
+@nisamp{alphapca57},
+@nisamp{alphaev6},
+@nisamp{alphaev67},
+@nisamp{alphaev68}
@item
-Hitachi:
-@samp{sh},
-@samp{sh2}
+Cray:
+@nisamp{c90},
+@nisamp{j90},
+@nisamp{t90},
+@nisamp{sv1}
@item
HPPA:
-@samp{hppa1.0},
-@samp{hppa1.1},
-@samp{hppa2.0},
-@samp{hppa2.0w}
+@nisamp{hppa1.0},
+@nisamp{hppa1.1},
+@nisamp{hppa2.0},
+@nisamp{hppa2.0n},
+@nisamp{hppa2.0w}
@item
MIPS:
-@samp{mips},
-@samp{mips3},
+@nisamp{mips},
+@nisamp{mips3},
+@nisamp{mips64}
@item
Motorola:
-@samp{m68000},
-@samp{m68k},
-@samp{m88k},
-@samp{m88110}
+@nisamp{m68k},
+@nisamp{m68000},
+@nisamp{m68010},
+@nisamp{m68020},
+@nisamp{m68030},
+@nisamp{m68040},
+@nisamp{m68060},
+@nisamp{m68302},
+@nisamp{m68360},
+@nisamp{m88k},
+@nisamp{m88110}
@item
POWER:
-@samp{power1},
-@samp{power2},
-@samp{power2sc},
-@samp{powerpc},
-@samp{powerpc64}
+@nisamp{power},
+@nisamp{power1},
+@nisamp{power2},
+@nisamp{power2sc}
@item
+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},
+
+@item
SPARC:
-@samp{sparc},
-@samp{sparcv8},
-@samp{microsparc},
-@samp{supersparc},
-@samp{sparcv9},
-@samp{ultrasparc},
-@samp{sparc64}
+@nisamp{sparc},
+@nisamp{sparcv8},
+@nisamp{microsparc},
+@nisamp{supersparc},
+@nisamp{sparcv9},
+@nisamp{ultrasparc},
+@nisamp{ultrasparc2},
+@nisamp{ultrasparc2i},
+@nisamp{ultrasparc3},
+@nisamp{sparc64}
@item
80x86 family:
-@samp{i386},
-@samp{i486},
-@samp{i586},
-@samp{pentium},
-@samp{pentiummmx},
-@samp{pentiumpro},
-@samp{pentium2},
-@samp{pentium3},
-@samp{k6},
-@samp{k62},
-@samp{k63},
-@samp{athlon}
+@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}
@item
Other:
-@samp{a29k},
-@samp{arm},
-@samp{clipper},
-@samp{i960},
-@samp{ns32k},
-@samp{pyramid},
-@samp{vax},
-@samp{z8k}
+@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 use generic C code. If some of the assembly code causes
-problems, the generic C code can be selected with CPU @samp{none}.
+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}
-The C compiler used is chosen from among some likely candidates, with GCC
-normally preferred if it's present. The usual @samp{CC=whatever} can be
-passed to @samp{./configure} to choose something different.
+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 configurations specific compiler flags are set based on the target
-CPU and compiler, see @samp{CFLAGS} in the generated @file{Makefile}s. 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.
+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.
-Note that if @samp{CC} is set then @samp{CFLAGS} must also be set. This
-applies even if @samp{CC} is merely one of the choices GMP would make itself.
-This may change in a future release.
+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.
-@item @option{--disable-alloca}
+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=}
@cindex Stack overflow segfaults
@cindex @code{alloca}
-By default, GMP allocates temporary workspace using @code{alloca} if that
-function is available, or @code{malloc} if not. If you're working with large
-numbers and @code{alloca} overflows the available stack space, you can build
-with @option{--disable-alloca} to use @code{malloc} instead. @code{malloc}
-will probably be slightly slower than @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}.
-When not using @code{alloca}, it's actually the allocation function
-selected with @code{mp_set_memory_functions} that's used, this being
-@code{malloc} by default. @xref{Custom Allocation}.
+@itemize @bullet
+@item
+@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
-Depending on your system, the only indication of stack overflow might be a
-segmentation violation. It might be possible to increase available stack
-space with @command{limit}, @command{ulimit} or @code{setrlimit}, or under
-DJGPP with @command{stubedit} or @code{_stklen}.
+For convenience, the following choices are also available.
+@samp{--disable-alloca} is the same as @samp{--enable-alloca=no}.
-@item @option{--enable-fft}
+@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
-By default multiplications are done using Karatsuba and 3-way Toom-Cook
-algorithms, but a Fermat FFT can be enabled, for use on large to very large
-operands. Currently the FFT is recommended only for knowledgeable users who
-check the algorithm thresholds for their CPU.
+@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.
-@item @option{--enable-mpbsd}
+@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 Berkeley MP compatibility library (@file{libmp.a}) and header file
+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 mpn subroutines are provided, and, for a given
-CPU target, a search is made though a path to choose a version of each. For
-example @samp{sparcv8} has path @samp{"sparc32/v8 sparc32 generic"}, which
-means it looks first for v8 code, falls back on plain sparc32, and finally
-falls back on generic C. Knowledgeable users with special requirements can
-specify a path with @samp{MPN_PATH="dir list"}. This will normally be
-unnecessary because all sensible paths should be available under one or other
-CPU target.
+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
-@item Demonstration Programs
-@cindex Demonstration programs
-@cindex Example programs
+@example
+MPN_PATH="sparc32/v8 sparc32 generic"
+@end example
-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, @samp{make pexpr} and then @samp{./pexpr 68^975+10}.
+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 @file{gmp.ps} and/or @file{gmp.dvi}. Some
-supplementary notes can be found in the @file{doc} subdirectory.
+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
@@ -487,7 +1008,9 @@ supplementary notes can be found in the @file{doc} sub
@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
@@ -495,99 +1018,170 @@ types are. ISA (Instruction Set Architecture) refers
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 chooses the best
-ABI available for a given target system, and this generally gives
-significantly greater speed.
+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}.
-The burden is on application programs and cooperating libraries to ensure they
-match the ABI chosen by GMP. Fortunately this presents a difficulty only on a
-few systems, and if you have one of them then the performance gains are enough
-to make it worth the trouble.
+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,
-Some of what's described in this section may change in future releases of GMP.
+@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
+@item HPPA 2.0 (@samp{hppa2.0*})
-CPU target @samp{hppa2.0} uses the hppa2.0n 32-bit ABI, but either a 32-bit or
-64-bit limb.
+@table @asis
+@item @samp{ABI=2.0w}
-A 64-bit limb is available on HP-UX 10 or up when using @command{c89}. No
-@command{gcc} support is planned for 64-bit operations in this ABI.
-Applications must be compiled with the same options as GMP, which means
+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
-c89 +DA2.0 +e -D_LONG_LONG_LIMB
+cc +DD64
@end example
-A 32-bit limb is used in other cases, and no special compiler options are
-needed.
+@item @samp{ABI=2.0n}
-CPU target @samp{hppa2.0w} uses the hppa2.0w 64-bit ABI, which is available on
-HP-UX 11 or up when using @command{c89}. @command{gcc} support for this is in
-progress. Applications must be compiled for the same ABI, which means
+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
-c89 +DD64
+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 3 and 4 under IRIX 6
+@item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]})
-Targets @samp{mips*-*-irix6*} use the n32 ABI and a 64-bit limb. Applications
-must be compiled for the same ABI, which means either
+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
+@item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630})
-CPU target @samp{powerpc64} uses either the 32-bit ABI or the AIX 64-bit ABI.
-The latter is used on targets @samp{powerpc64-*-aix*} and applications must be
-compiled using either
+@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
-On other systems the 32-bit ABI is used, but with 64-bit limbs provided by
-@code{long long} in @command{gcc}. Applications must be compiled using
+@item @samp{ABI=32}
-@example
-gcc -D_LONG_LONG_LIMB
-@end example
+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
+@item Sparc V9 (@samp{sparcv9} and @samp{ultrasparc*})
-On a sparc v9 CPU, either the v8plus 32-bit ABI or v9 64-bit ABI is used.
-Targets @samp{ultrasparc*-*-solaris2.[7-9]}, @samp{sparcv9-*-solaris2.[7-9]}
-and @samp{sparc64-*-linux*} use the v9 ABI, if the compiler supports it.
-Other targets use the v8plus ABI (but with as much of the v9 ISA as possible
-in the circumstances). Note that Solaris prior to 2.7 doesn't save all
-registers properly, and hence uses the v8plus ABI.
+@table @asis
+@item @samp{ABI=64}
-For the v8plus ABI, applications can be compiled with either
+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
-For the v9 ABI, applications must be compiled with either
+@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
-gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
-cc -xarch=v9
+./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
@end example
-
-Don't be confused by the names of these options, they're called @samp{arch}
-but they effectively control the ABI.
@end table
@@ -601,31 +1195,47 @@ GMP should present no great difficulties for packaging
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 3 series
-will be upwardly binary compatible in each release, but may be adding
-additional function interfaces. 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 minor version of
-GMP.
+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{--target} 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.
+@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{--target} in a build so @samp{./config.guess} will detect
-the CPU. But a way to manually specify a @samp{--target} will be wanted for
-systems where @samp{./config.guess} is inexact.
+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
@@ -633,11 +1243,75 @@ systems where @samp{./config.guess} is inexact.
@c scratching their head. Fun with different ABIs on a system belongs in the
@c above section.
-@item AIX 4.3
+@item AIX 3 and 4
-Targets @samp{*-*-aix4.[3-9]*} have shared libraries disabled since they seem
-to fail on AIX 4.3.
+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
@@ -645,11 +1319,36 @@ unsuitable for @file{.asm} file processing. @samp{./c
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 Sparc V8
+@item Power CPU Types
-Using CPU target @samp{sparcv8} or @samp{supersparc} on relevant systems will
-give a significant performance increase over the V7 code.
+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}
@@ -657,28 +1356,34 @@ files, and instead @samp{./configure} will automatical
@command{/usr/5bin/m4}, which we believe is always available (if not then use
GNU m4).
-@item x86 Pentium and PentiumPro
+@item x86 CPU Types
-The Intel Pentium P5 code is good for its intended P5, but quite slow when run
-on Intel P6 class chips (PPro, P-II, P-III)@. @samp{i386} is a better choice
-if you're making binaries that must run on both.
+@samp{i386} selects generic code which will run reasonably well on all x86
+chips.
-@item x86 MMX and old GAS
+@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.
-Old versions of 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).
+@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.
-If the target CPU 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.
+@item x86 MMX and SSE2 Code
-@item x86 GCC 2.95.2 @samp{-march=pentiumpro}
+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.
-GCC 2.95.2 miscompiles @file{mpz/powm.c} when @samp{-march=pentiumpro} is
-used, so that option is omitted from the @env{CFLAGS} chosen for relevant
-CPUs. The problem is believed to be fixed in GCC 2.96.
+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
@@ -690,24 +1395,96 @@ CPUs. The problem is believed to be fixed in GCC 2.96
@c This section is more or less meant for known build problems that are not
@c otherwise worked around and require some sort of manual intervention.
-You might find more up-to-date information at @uref{http://www.swox.com/gmp/}.
+You might find more up-to-date information at @uref{http://swox.com/gmp/}.
@table @asis
+@item Compiler link options
+The version of libtool currently in use rather aggressively strips compiler
+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
-@item Generic C on a 64-bit system
+@example
+./configure CC=gcc-with-my-options
+@end example
-When making a generic C build using @samp{--target=none} on a 64-bit system
-(meaning where @code{unsigned long} is 64 bits), @code{BITS_PER_MP_LIMB},
-@code{BITS_PER_LONGINT} and @code{BYTES_PER_MP_LIMB} in
-@file{mpn/generic/gmp-mparam.h} need to be changed to 64 and 8. This will
-hopefully be automated in a future version of GMP.
+@item DJGPP
+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.
+
+@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
+LD_LIBRARY_PATH=/my/staging/area/usr/lib \
+ make install DESTDIR=/my/staging/area
+@end example
+
+@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 before you compile GMP. (NeXT may
-have fixed this in release 3.3 of their system.)
+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
@@ -720,79 +1497,109 @@ later).
Use the GNU assembler instead of the system assembler, since the latter has
serious bugs.
-@item Stripped Libraries
-@cindex Stripped libraries
+@item Solaris 2.6
-GNU binutils @samp{strip} should not be used on the static libraries
-@file{libgmp.a} and @file{libmp.a}, neither directly nor via @samp{make
-install-strip}. It can be used on the shared libraries @file{libgmp.so} and
-@file{libmp.so} though.
+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.
-Currently (binutils 2.10.0), @samp{strip} extracts archives into a single
-directory, but GMP contains multiple object files of the same name (eg. three
-versions of @file{init.o}), and they overwrite each other, leaving only the
-one that happens to be last.
+@item Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32
-If stripped static libraries are wanted, the suggested workaround is to build
-normally, strip the separate object files, and do another @samp{make all} to
-rebuild. Alternately @samp{CFLAGS} with @samp{-g} omitted can always be used
-if it's just debugging which is unwanted.
+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 SunOS 4 Native Tools
+@item Windows DLL test programs
-The setting for @code{GSYM_PREFIX} in @file{config.m4} may be incorrectly
-determined when using the native @command{grep}, leading at link-time to
-undefined symbols like @code{___gmpn_add_n}. To fix this, after running
-@samp{./configure}, change the relevant line in @file{config.m4} to
-@samp{define(, <_>)}.
-
-The @command{ranlib} command will need to be run manually when building a
-static library with the native @command{ar}. After @samp{make}, run
-@samp{ranlib .libs/libgmp.a}, and when using @option{--enable-mpbsd} run
-@samp{ranlib .libs/libmp.a} too.
-
-@item @file{version.c} compilation
-
-The current @samp{./configure} relies on certain features of @command{sed}
-that some old systems don't have. One symptom is @code{VERSION} not being set
-correctly in the generated @file{config.h}, leading to @file{version.c}
-failing to compile. Irix 5.3, MIPS RISC/OS and Ultrix 4.4 are believed to be
-affected. GNU @command{sed} is recommended, though it might be possible to
-build by editing @file{config.h} manually instead.
-
-@item VAX running Ultrix
-
-You need to build and install the GNU assembler before you compile GMP. The VAX
-assembly in GMP uses an instruction (@code{jsobgtr}) that cannot be assembled by
-the Ultrix assembler.
+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
@chapter GMP Basics
@cindex Basics
-@cindex @file{gmp.h}
-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.
-
@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
-* Nomenclature and Types:: Which data types are there?
-* Function Classes:: How the functions are organized.
-* GMP Variable Conventions:: Some rules and hints about variables.
-* GMP and Reentrancy:: What about reentrancy?
-* Useful Macros and Constants:: Convenient helpers.
-* Compatibility with older versions:: Compatibility issues.
-* Getting the Latest Version of GMP:: How to get the software.
+* 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 Nomenclature and Types, Function Classes, GMP Basics, GMP Basics
+@node Headers and Libraries, Nomenclature and Types, GMP Basics, GMP Basics
+@section Headers and Libraries
+@cindex Headers
+
+@cindex @file{gmp.h}
+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.
+
+@example
+#include
+@end example
+
+Note however that prototypes for GMP functions with @code{FILE *} parameters
+are only provided if @code{} is included too.
+
+@example
+#include
+#include
+@end example
+
+Likewise @code{} (or @code{}) is required for prototypes
+with @code{va_list} parameters, such as @code{gmp_vprintf}. And
+@code{} 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
@cindex Nomenclature
@cindex Types
@@ -833,12 +1640,12 @@ is @code{mpf_t}.
@tindex @code{mp_limb_t}
@noindent
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
-digit, only larger, and containing several digits.) Normally a limb contains
-32 or 64 bits. The C data type for a limb is @code{mp_limb_t}.
+machine word. (We chose this word because a limb of the human body is
+analogous to a digit, only larger, and containing several digits.) Normally a
+limb is 32 or 64 bits. The C data type for a limb is @code{mp_limb_t}.
-@node Function Classes, GMP Variable Conventions, Nomenclature and Types, GMP Basics
+@node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics
@section Function Classes
@cindex Function classes
@@ -847,48 +1654,45 @@ There are six classes of functions in the GMP library:
@enumerate
@item
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.
@item
Functions for rational number arithmetic, with names beginning with
-@code{mpq_}. The associated type is @code{mpq_t}. There are about 20
-functions in this class, but the functions in the previous class can be used
-for performing arithmetic on the numerator and denominator separately.
+@code{mpq_}. The associated type is @code{mpq_t}. There are about 40
+functions in this class, but the integer functions can be used for arithmetic
+on the numerator and denominator separately.
@item
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.
@item
-Functions compatible with Berkeley GMP, such as @code{itom}, @code{madd}, and
+Functions compatible with Berkeley MP, such as @code{itom}, @code{madd}, and
@code{mult}. The associated type is @code{MINT}.
@item
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
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
Miscellaneous functions. Functions for setting up custom allocation and
functions for generating random numbers.
@end enumerate
-@node GMP Variable Conventions, GMP and Reentrancy, Function Classes, GMP Basics
-@section GMP Variable Conventions
+@node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics
+@section Variable Conventions
@cindex Variable conventions
-@cindex Parameter conventions
@cindex Conventions for variables
-As a general rule, all GMP functions expect output arguments before input
-arguments. This notation is based on an analogy with the assignment operator.
-(The BSD MP compatibility functions disobey this rule, having the output
-argument(s) last.)
+GMP functions generally have output arguments before input arguments. This
+notation is by analogy with the assignment operator. The BSD MP compatibility
+functions are exceptions, having the output arguments last.
GMP lets you use the same variable for both input and output in one call. For
example, the main function for integer multiplication, @code{mpz_mul}, can be
@@ -905,39 +1709,57 @@ purpose. Which function to use depends on the type of
chapters on integer functions, rational number functions, and floating-point
functions for details.
-A variable should only be initialized once, or at least cleared out between
-each initialization. After a variable has been initialized, it may be
-assigned to any number of times.
+A variable should only be initialized once, or at least cleared between each
+initialization. After a variable has been initialized, it may be assigned to
+any number of times.
-For efficiency reasons, avoid initializing and clearing out a GMP variable in
-a loop. Instead, initialize it before entering the loop, and clear it out
-after the loop has exited.
+For efficiency reasons, avoid excessive initializing and clearing. In
+general, initialize near the start of a function and clear near the end. For
+example,
-GMP variables are small, containing only a couple of sizes, and pointers to
-allocated data. Once you have initialized a GMP variable, you don't need to
-worry about space allocation. All functions in GMP automatically allocate
-additional space when a variable does not already have enough. They do not,
-however, reduce the space when a smaller value is stored. Most of the time
-this policy is best, since it avoids frequent re-allocation.
+@example
+void
+foo (void)
+@{
+ 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
-When a variable of type @code{mpz_t} is used as a function parameter, it's
-effectively a call-by-reference, meaning anything the function does to it will
-be be done to the original in the caller. When a function is going to return
-an @code{mpz_t} result, it should provide a separate parameter or parameters
-that it sets, like the GMP library functions do. A @code{return} of an
-@code{mpz_t} doesn't return the object, only a pointer to it, and this is
-almost certainly not what you want. All this applies to @code{mpq_t} and
-@code{mpf_t} too.
-Here's an example function accepting an @code{mpz_t} parameter, doing a
-certain calculation, and returning a result.
+@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
-myfunction (mpz_t result, mpz_t param, unsigned long n)
+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);
@@ -949,70 +1771,108 @@ main (void)
mpz_t r, n;
mpz_init (r);
mpz_init_set_str (n, "123456", 0);
-
- myfunction (r, n, 20L);
- mpz_out_str (stdout, 10, r); printf ("\n");
-
+ foo (r, n, 20L);
+ gmp_printf ("%Zd\n", r);
return 0;
@}
@end example
-This example will work if @code{result} and @code{param} are the same
-variable, just like the library functions. But sometimes this is tricky to
-arrange, and an application might not want to bother for its own subroutines.
+@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.
-@code{mpz_t} is actually implemented as a one-element array of a certain
-structure type. This is why using it to declare a variable gives an object
-with the fields GMP needs, but then using it as a parameter passes a pointer
-to the object. Note that the actual contents of an @code{mpz_t} are for
-internal use only and you should not access them directly if you want your
-code to be compatible with future GMP releases.
+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.
-@node GMP and Reentrancy, Useful Macros and Constants, GMP Variable Conventions, GMP Basics
-@section GMP and Reentrancy
+@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
-The GMP code is reentrant and thread-safe, with some exceptions:
+GMP is reentrant and thread-safe, with some exceptions:
@itemize @bullet
@item
-The function @code{mpf_set_default_prec} saves the selected precision in
-a global variable.
+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
-The function @code{mp_set_memory_functions} uses several global
-variables for storing the selected memory allocation functions.
+@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, GMP will not be reentrant either.
+not reentrant, then GMP will not be reentrant either.
@item
-The old random number functions (@code{mpz_random}, etc) use a random number
-generator from the C library, usually @code{mrand48} or @code{random}. These
-routines are not reentrant, since they rely on global state.
-(However the newer random number functions that accept a
-@code{gmp_randstate_t} parameter are reentrant.)
+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
-If @code{alloca} is not available, or GMP is configured with
-@samp{--disable-alloca}, the library is not reentrant, due to the current
-implementation of @file{stack-alloc.c}. In the generated @file{config.h},
-@code{USE_STACK_ALLOC} set to 1 will mean not reentrant.
+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{} 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, GMP and Reentrancy, GMP Basics
+@node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics
@section Useful Macros and Constants
@cindex Useful macros and constants
@cindex Constants
@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.
@@ -1028,31 +1888,39 @@ For GMP i.j, these numbers will be i, j, and 0, respec
For GMP i.j.k, these numbers will be i, j, and k, respectively.
@end defmac
+@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
-@node Compatibility with older versions, Getting the Latest Version of GMP, Useful Macros and Constants, GMP Basics
+
+@node Compatibility with older versions, Demonstration Programs, Useful Macros and Constants, GMP Basics
@section Compatibility with older versions
@cindex Compatibility with older versions
@cindex Upward compatibility
-This version of GMP is upwardly binary compatible with versions 3.0 and 3.0.1,
-and upwardly compatible at the source level with versions 2.0, 2.0.1, and
-2.0.2, with the following exceptions.
+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.
@itemize @bullet
@item
-@code{mpn_gcd} had its source arguments swapped as of GMP 3.0 for consistency
+@code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency
with other @code{mpn} functions.
@item
@code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and
-3.0.1, but in 3.1 has reverted to the 2.0.x style.
-
+3.0.1, but in 3.1 reverted to the 2.x style.
@end itemize
There are a number of compatibility issues between GMP 1 and GMP 2 that of
-course also apply when porting applications from GMP 1 to GMP 3. Please
+course also apply when porting applications from GMP 1 to GMP 4. Please
see the GMP 2 manual for details.
+The Berkeley MP compatibility library (@pxref{BSD Compatible Functions}) is
+source and binary compatible with the standard @file{libmp}.
+
@c @enumerate
@c @item Integer division functions round the result differently. The obsolete
@c functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv},
@@ -1093,21 +1961,495 @@ see the GMP 2 manual for details.
@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
-@node Getting the Latest Version of GMP, , Compatibility with older versions, GMP Basics
-@section Getting the Latest Version of GMP
-@cindex Latest version of GMP
-@cindex Anonymous FTP of latest version
-@cindex FTP of latest version
+@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,
-The latest version of the GMP library is available at
-@uref{ftp://ftp.gnu.org/pub/gnu/gmp}. Many sites around the world mirror
-@samp{ftp.gnu.org}; please use a mirror site near you, see
-@uref{http://www.gnu.org/order/ftp.html}.
+@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 ])
+@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
@chapter Reporting Bugs
@@ -1116,9 +2458,12 @@ The latest version of the GMP library is available at
If you think you have found a bug in the GMP library, please investigate it
and report it. We have made this library available to you, and it is not too
-much to ask you to report the bugs you find. Before you report a bug, you may
-want to check @uref{http://www.swox.com/gmp/} for patches for this release.
+much to ask you to report the bugs you find.
+Before you report a bug, check it's not already addressed in @ref{Known Build
+Problems}, or perhaps @ref{Notes for Particular Systems}. You may also want
+to check @uref{http://swox.com/gmp/} for patches for this release.
+
Please include the following in any report,
@itemize @bullet
@@ -1138,7 +2483,7 @@ If you get a crash, include a stack backtrace from the
informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).
@item
-@strong{Please do not send core dumps, executables or @command{strace}s.}
+Please do not send core dumps, executables or @command{strace}s.
@item
The configuration options you used when building GMP, if any.
@@ -1151,7 +2496,8 @@ with @samp{gcc -v}, otherwise perhaps @samp{what `whic
The output from running @samp{uname -a}.
@item
-The output from running @samp{./config.guess}.
+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
@@ -1159,9 +2505,14 @@ If the bug is related to @samp{configure}, then the co
@item
If the bug is related to an @file{asm} file not assembling, then the contents
-of @file{config.m4}.
+of @file{config.m4} and the offending line or lines from the temporary
+@file{mpn/tmp-.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
compiler; the GMP code tends to explore interesting corners in compilers.
@@ -1199,6 +2550,7 @@ GMP integers are stored in objects of type @code{mpz_t
* Integer Logic and Bit Fiddling::
* I/O of Integers::
* Integer Random Numbers::
+* Integer Import and Export::
* Miscellaneous Integer Functions::
@end menu
@@ -1209,16 +2561,9 @@ GMP integers are stored in objects of type @code{mpz_t
@cindex Initialization functions
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
@{
mpz_t integ;
@@ -1233,37 +2578,87 @@ Here is an example of using @code{mpz_init}:
@}
@end example
-@noindent
As you can see, you can store new values any number of times, once an
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})
-Free the limb space occupied by @var{integer}. Make sure to call this
-function for all @code{mpz_t} variables when you are done with them.
+Free the space occupied by @var{integer}. Call this function for all
+@code{mpz_t} variables when you are done with them.
@end deftypefun
-@deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
-Change the limb space allocation to @var{new_alloc} limbs. This function is
-not normally called from user code, but it can be used to give memory back to
-the heap, or to increase the space of a variable to avoid repeated automatic
-re-allocation.
+@deftypefun void mpz_realloc2 (mpz_t @var{integer}, unsigned long @var{n})
+Change the space allocated for @var{integer} to @var{n} bits. The value in
+@var{integer} is preserved if it fits, or is set to 0 if not.
+
+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
-@deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, mp_size_t @var{fixed_num_bits})
-Allocate @strong{fixed} limb space for all @var{array_size} integers in
-@var{integer_array}. The fixed allocation for each integer in the array is
-enough to store @var{fixed_num_bits}. If the fixed space will be insufficient
-for storing the result of a subsequent calculation, the result is
-unpredictable.
+@deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
+This is a special type of initialization. @strong{Fixed} space of
+@var{fixed_num_bits} bits is allocated to each of the @var{array_size}
+integers in @var{integer_array}.
-This function is useful for decreasing the working set for some algorithms
-that use large integer arrays.
+The space will not be automatically increased, unlike the normal
+@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.
-Don't call @code{mpz_clear}!
+@itemize @bullet
+@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
+@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
@comment node-name, next, previous, up
@section Assignment Functions
@@ -1280,26 +2675,29 @@ 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_f (mpz_t @var{rop}, mpf_t @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
@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
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',
-hexadecimal is assumed, otherwise if the first character is `0', octal is
+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
assumed, otherwise decimal is assumed.
-This function returns 0 if the entire string up to the '\0' is a valid
-number in base @var{base}. Otherwise it returns @minus{}1.
+This function returns 0 if the entire string is a valid number in base
+@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. Please tell us your opinion about this
-change. Do you really want it to accept "3 14" as meaning 314 as it does
-now?]
+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
@deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
@@ -1364,52 +2762,62 @@ This section describes functions for converting GMP in
types. Functions for converting @emph{to} GMP integers are described in
@ref{Assigning Integers} and @ref{I/O of Integers}.
-@deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
-Return limb #@var{n} from @var{op}. This function allows for very efficient
-decomposition of a number in its limbs.
+@deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})
+Return the value of @var{op} as an @code{unsigned long}.
-The function @code{mpz_size} can be used to determine the useful range for
-@var{n}.
+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
-@deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})
-Return the least significant part from @var{op}. This function combined with
-@* @code{mpz_tdiv_q_2exp(@dots{}, @var{op}, CHAR_BIT*sizeof(unsigned long
-int))} can be used to decompose an integer into unsigned longs.
-@end deftypefun
-
@deftypefun {signed long int} mpz_get_si (mpz_t @var{op})
If @var{op} fits into a @code{signed long int} return the value of @var{op}.
Otherwise return the least significant part of @var{op}, with the same sign
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. To find out if the value will fit, use
the function @code{mpz_fits_slong_p}.
@end deftypefun
@deftypefun double mpz_get_d (mpz_t @var{op})
-Convert @var{op} to a double.
+Convert @var{op} to a @code{double}.
@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})
Convert @var{op} to a string of digits in base @var{base}. The base may vary
from 2 to 36.
-If @var{str} is @code{NULL}, space for the result string is allocated using the
-default allocation function.
+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 enough large
-for the result. To find out the right amount of space to provide for
-@var{str}, use @code{mpz_sizeinbase (@var{op}, @var{base}) + 2}. The two
-extra bytes are for a possible minus sign, and for the terminating null
-character.
+If @var{str} is not @code{NULL}, it should point to a block of storage large
+enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
++ 2}. The two extra bytes are for a possible minus sign, and the
+null-terminator.
-A pointer to the result string is returned. This pointer will will either
-equal @var{str}, or if that is @code{NULL}, will point to the allocated storage.
+A pointer to the result string is returned, being either the allocated block,
+or the given @var{str}.
@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.
+@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
@@ -1419,49 +2827,36 @@ equal @var{str}, or if that is @code{NULL}, will point
@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})
-@ifnottex
-Set @var{rop} to @var{op1} + @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1} + @var{op2}$.
-@end tex
+Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun
@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_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun
@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})
-@ifnottex
-Set @var{rop} to @var{op1} times @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1} \times @var{op2}$.
-@end tex
+Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun
-@deftypefun void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
-@ifnottex
-Add @var{op1} times @var{op2} to @var{rop}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{rop} + @var{op1} \times @var{op2}$.
-@end tex
+@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})
@cindex Bit shift left
-@ifnottex
-Set @var{rop} to @var{op1} times 2 raised to @var{op2}. This operation can
-also be defined as a left shift, @var{op2} steps.
-@end ifnottex
-@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
+Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
+@var{op2}}. This operation can also be defined as a left shift by @var{op2}
+bits.
@end deftypefun
@deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
@@ -1479,287 +2874,155 @@ Set @var{rop} to the absolute value of @var{op}.
@cindex Integer division functions
@cindex Division functions
-Division is undefined if the divisor is zero, and passing a zero divisor to the
-divide or modulo functions, as well passing a zero mod argument to the
-@code{mpz_powm} and @code{mpz_powm_ui} functions, will make these functions
-intentionally divide by zero. This lets the user handle arithmetic exceptions
-in these functions in the same manner as other arithmetic exceptions.
+Division is undefined if the divisor is zero. Passing a zero divisor to the
+division or modulo functions (including the modular powering functions
+@code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
+zero. This lets a program handle arithmetic exceptions in these functions the
+same way as for normal C @code{int} arithmetic.
-There are three main groups of division functions:
-@itemize @bullet
-@item
-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
-@ifnottex
-@minus{}infinity).
-@end ifnottex
-@tex
-$-\infty$
-@end tex
-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
-@ifnottex
-+infinity.
-@end ifnottex
-@tex
-$+\infty$
-@end tex
-The names of these routines start with @code{mpz_cdiv}. The @samp{c} in the
-name is short for @samp{ceil}.
-@end itemize
+@c Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
+@c between each, and seem to let tex do a better job of page breaks than an
+@c @sp 1 in the middle of one big set.
-For each rounding mode, there are a couple of variants. Here @samp{q} means
-that the quotient is computed, while @samp{r} means that the remainder is
-computed. Functions that compute both the quotient and remainder have
-@samp{qr} in the name.
-
-@deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
-Set @var{q} to [@var{n}/@var{d}], truncated towards 0.
-
-The function @code{mpz_tdiv_q_ui} returns the absolute value of the true
-remainder.
+@deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@maybepagebreak
+@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
+@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
+@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}})
+@deftypefunx {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
+@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
-@deftypefun void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{r} to (@var{n} - [@var{n}/@var{d}] * @var{d}), where the quotient is
-truncated towards 0. Unless @var{r} becomes zero, it will get the same sign as
-@var{n}.
-@end ifnottex
-@tex
-Set @var{r} to $(@var{n} - [@var{n}/@var{d}] \times @var{d})$, where the
-quotient is truncated towards 0. Unless @var{r} becomes zero, it will get the
-same sign as @var{n}.
-@end tex
-
-The function @code{mpz_tdiv_r_ui} returns the absolute value of the remainder.
-@end deftypefun
-
-@deftypefun void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to [@var{n}/@var{d}], truncated towards 0. Set @var{r} to (@var{n}
-- [@var{n}/@var{d}] * @var{d}). Unless @var{r} becomes zero, it will get the
-same sign as @var{n}. If @var{q} and @var{r} are the same variable, the
-results are undefined.
-@end ifnottex
-@tex
-Set @var{q} to [@var{n}/@var{d}], truncated towards 0. Set @var{r} to $(@var{n}
-- [@var{n}/@var{d}] \times @var{d})$. Unless @var{r} becomes zero, it will get the
-same sign as @var{n}. If @var{q} and @var{r} are the same variable, the
-results are undefined.
-@end tex
-
-The function @code{mpz_tdiv_qr_ui} returns the absolute value of the remainder.
-@end deftypefun
-
-@deftypefun {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
-Like @code{mpz_tdiv_r_ui}, but the remainder is not stored anywhere; its
-absolute value is just returned.
-@end deftypefun
-
@deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to @var{n}/@var{d}, rounded towards @minus{}infinity.
-@end ifnottex
-@tex
-Set @var{q} to $\lfloor@var{n}/@var{d}\rfloor$.
-@end tex
-
-The function @code{mpz_fdiv_q_ui} returns the remainder.
+@deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@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
-@deftypefun void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{r} to (@var{n} - @var{n}/@var{d} * @var{d}), where the quotient is
-rounded towards @minus{}infinity. Unless @var{r} becomes zero, it will get the
-same sign as @var{d}.
-@end ifnottex
-@tex
-Set @var{r} to $(@var{n} - \lfloor@var{n}/@var{d}\rfloor \times @var{d})$.
-Unless @var{r} becomes zero, it will get the same sign as @var{d}.
-@end tex
+@deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
+@maybepagebreak
+@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
-The function @code{mpz_fdiv_r_ui} returns the remainder.
-@end deftypefun
+@sp 1
+Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
+@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_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to @var{n}/@var{d}, rounded towards @minus{}infinity. Set @var{r}
-to (@var{n} - @var{n}/@var{d} * @var{d}). Unless @var{r} becomes zero, it
-will get the same sign as @var{d}. If @var{q} and @var{r} are the same
-variable, the results are undefined.
-@end ifnottex
-@tex
-Set @var{q} to $\lfloor@var{n}/@var{d}\rfloor$. Set @var{r} to $(@var{n} -
-\lfloor@var{n}/@var{d}\rfloor \times @var{d})$. Unless @var{r} becomes zero,
-it will get the same sign as @var{d}. If @var{q} and @var{r} are the same
-variable, the results are undefined.
-@end tex
+@itemize @bullet
+@item
+@code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
+have the opposite sign to @var{d}. The @code{c} stands for ``ceil''.
-The function @code{mpz_fdiv_qr_ui} returns the remainder.
-@end deftypefun
+@item
+@code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
+@var{r} will have the same sign as @var{d}. The @code{f} stands for
+``floor''.
-@deftypefun {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
-Like @code{mpz_fdiv_r_ui}, but the remainder is not stored anywhere; it is just
-returned.
-@end deftypefun
+@item
+@code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
+as @var{n}. The @code{t} stands for ``truncate''.
+@end itemize
-@deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to @var{n}/@var{d}, rounded towards +infinity.
-@end ifnottex
-@tex
-Set @var{q} to $\lceil@var{n}/@var{d}\rceil$.
-@end tex
+In all cases @var{q} and @var{r} will satisfy
+@m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and
+@var{r} will satisfy @math{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.
-The function @code{mpz_cdiv_q_ui} returns the negated remainder.
-@end deftypefun
+The @code{q} functions calculate only the quotient, the @code{r} functions
+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.
-@deftypefun void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{r} to (@var{n} - @var{n}/@var{d} * @var{d}), where the quotient is
-rounded towards +infinity. Unless @var{r} becomes zero, it will get the
-opposite sign as @var{d}.
-@end ifnottex
-@tex
-Set @var{r} to $(@var{n} - \lceil@var{n}/@var{d}\rceil \times @var{d})$. Unless
-@var{r} becomes zero, it will get the opposite sign as @var{d}.
-@end tex
+For the @code{ui} variants the return value is the remainder, and in fact
+returning the remainder is all the @code{div_ui} functions do. For
+@code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
+return value is the absolute value of the remainder.
-The function @code{mpz_cdiv_r_ui} returns the negated remainder.
+The @code{2exp} functions are right shifts and bit masks, but of course
+rounding the same as the other functions. For positive @var{n} both
+@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
-@deftypefun void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to @var{n}/@var{d}, rounded towards +infinity. Set @var{r}
-to (@var{n} - @var{n}/@var{d} * @var{d}). Unless @var{r} becomes zero, it
-will get the opposite sign as @var{d}. If @var{q} and @var{r} are the same
-variable, the results are undefined.
-@end ifnottex
-@tex
-Set @var{q} to $\lceil@var{n}/@var{d}\rceil$. Set @var{r} to $(@var{n} -
-\lceil@var{n}/@var{d}\rceil \times @var{d})$. Unless @var{r} becomes zero, it will
-get the opposite sign as @var{d}. If @var{q} and @var{r} are the same
-variable, the results are undefined.
-@end tex
-
-The function @code{mpz_cdiv_qr_ui} returns the negated remainder.
-@end deftypefun
-
-@deftypefun {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
-Like @code{mpz_tdiv_r_ui}, but the remainder is not stored anywhere; its
-negated value is just returned.
-@end deftypefun
-
@deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
-@deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-Set @var{r} to @var{n} @code{mod} @var{d}. The sign of the divisor is ignored;
-the result is always non-negative.
+@deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
+Set @var{r} to @var{n} @code{mod} @var{d}. The sign of the divisor is
+ignored; the result is always non-negative.
-The function @code{mpz_mod_ui} returns the remainder.
+@code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
+remainder as well as setting @var{r}. See @code{mpz_fdiv_ui} above if only
+the return value is wanted.
@end deftypefun
@deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long @var{d})
@cindex Exact division functions
-Set @var{q} to @var{n}/@var{d}. This function produces correct results only
+Set @var{q} to @var{n}/@var{d}. These functions produce correct results only
when it is known in advance that @var{d} divides @var{n}.
-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.
+These routines are much faster than the other division functions, and are the
+best choice when exact division is known to occur, for example reducing a
+rational to lowest terms.
@end deftypefun
-@deftypefun void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
-@cindex Bit shift right
-@ifnottex
-Set @var{q} to @var{n} divided by 2 raised to @var{d}. The quotient is truncated
-towards 0.
-@end ifnottex
-@tex
-Set @var{q} to $[{n}/2^{d}]$, truncated towards 0.
-@end tex
+@deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d})
+@deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, unsigned long int @var{d})
+@deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, unsigned long int @var{b})
+Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
+@code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.
@end deftypefun
-@deftypefun void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Divide @var{n} by (2 raised to @var{d}), rounding the quotient towards 0, and
-put the remainder in @var{r}.
-@end ifnottex
-@tex
-Set @var{r} to $n - [n / 2^{d}] \times 2^{d}$, where [ ] indicates rounding towards
-0.
-@end tex
-Unless it is zero, @var{r} will have the same sign as @var{n}.
+@deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d})
+@deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d})
+@deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, unsigned long int @var{b})
+Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
+case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.
@end deftypefun
-@deftypefun void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Set @var{q} to @var{n} divided by 2 raised to @var{d}, rounded towards
-@minus{}infinity.
-@end ifnottex
-@tex
-Set @var{q} to $\lfloor{n}/2^{d}\rfloor$.
-@end tex
-This operation can also be defined as arithmetic right shift @var{d} bit
-positions.
-@end deftypefun
-@deftypefun void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
-@ifnottex
-Divide @var{n} by (2 raised to @var{d}), rounding the quotient towards
-@minus{}infinity, and put the remainder in @var{r}.
-@end ifnottex
-@tex
-Set @var{r} to $n - \lfloor{n}/2^{d}\rfloor \times 2^{d}$.
-@end tex
-The sign of @var{r} will always be positive.
-This operation can also be defined as masking of the @var{d} least significant
-bits.
-@end deftypefun
-
-
@need 2000
@node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
@section Exponentiation Functions
@cindex Integer exponentiation functions
@cindex Exponentiation functions
+@cindex Powering functions
@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})
-@ifnottex
-Set @var{rop} to (@var{base} raised to @var{exp}) @code{mod} @var{mod}. If
-@var{exp} is negative, the result is undefined.
-@end ifnottex
-@tex
-Set @var{rop} to $base^{exp} \bmod mod$. If
-@var{exp} is negative, the result is undefined.
-@end tex
+Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
+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
@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})
-@ifnottex
-Set @var{rop} to @var{base} raised to @var{exp}. The case of 0^0 yields 1.
-@end ifnottex
-@tex
-Set @var{rop} to $base^{exp}$. The case of $0^0$ yields 1.
-@end tex
+Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}. The case
+@math{0^0} yields 1.
@end deftypefun
@@ -1770,58 +3033,41 @@ Set @var{rop} to $base^{exp}$. The case of $0^0$ yiel
@cindex Root extraction functions
@deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n})
-@ifnottex
-Set @var{rop} to the truncated integer part of the @var{n}th root of @var{op}.
-@end ifnottex
-@tex
-Set @var{rop} to $\lfloor\root n \of {op}\rfloor$, the truncated integer
-part of the @var{n}th root of @var{op}.
-@end tex
-Return non-zero if the computation was exact, i.e., if @var{op} is
-@var{rop} to the @var{n}th power.
+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})
-@ifnottex
-Set @var{rop} to the truncated integer part of the square root of @var{op}.
-@end ifnottex
-@tex
-Set @var{rop} to $\lfloor\sqrt{@var{op}}\rfloor$, the truncated integer part of
-the square root of @var{op}.
-@end tex
+Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor@C{},} the truncated
+integer part of the square root of @var{op}.
@end deftypefun
@deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
-@ifnottex
-Set @var{rop1} to the truncated integer part of the square root of @var{op},
-like @code{mpz_sqrt}. Set @var{rop2} to
-@var{op}@minus{}@var{rop1}*@var{rop1},
-@end ifnottex
-@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
-(i.e., zero if @var{op} is a perfect square).
+Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
+of the square root of @var{op}}, like @code{mpz_sqrt}. Set @var{rop2} to the
+remainder @m{(@var{op} - @var{rop1}^2),
+@var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
+perfect square.
If @var{rop1} and @var{rop2} are the same variable, the results are
undefined.
@end deftypefun
@deftypefun int mpz_perfect_power_p (mpz_t @var{op})
-@ifnottex
Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
-@var{a} and @var{b}, with @var{b} > 1, such that @var{op} equals a raised to
-b. Return zero otherwise.
-@end ifnottex
-@tex
-Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
-$a$ and $b$, with $b>1$, such that $@var{op}=a^b$. Return zero otherwise.
-@end tex
+@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})
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
@@ -1832,21 +3078,27 @@ Return non-zero if @var{op} is a perfect square, i.e.,
@deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
@cindex Prime testing functions
-If this function returns 0, @var{n} is definitely not prime. If it
-returns 1, then @var{n} is `probably' prime. If it returns 2, then
-@var{n} is surely prime. Reasonable values of reps vary from 5 to 10; a
-higher value lowers the probability for a non-prime to pass as a
-`probable' prime.
+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.
-The function uses Miller-Rabin's probabilistic test.
+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
-@deftypefun int mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op})
+@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, but for for
-practical purposes it's adequate, since the chance of a composite passing will
-be extremely small.
+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.
@@ -1865,7 +3117,7 @@ be extremely small.
@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}.
-The result is always positive even if either of or both input operands
+The result is always positive even if one or both input operands
are negative.
@end deftypefun
@@ -1881,52 +3133,67 @@ is non-zero.
@deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})
@cindex Extended GCD
-Compute @var{g}, @var{s}, and @var{t}, such that @var{a}@var{s} +
-@var{b}@var{t} = @var{g} = @code{gcd}(@var{a}, @var{b}). If @var{t} is
-@code{NULL}, that argument is not computed.
+Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
+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
@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})
@cindex Modular inverse functions
Compute the inverse of @var{op1} modulo @var{op2} and put the result in
-@var{rop}. Return non-zero if an inverse exists, zero otherwise. When the
-function returns zero, @var{rop} is undefined.
+@var{rop}. If the inverse exists, the return value is non-zero and @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
-@deftypefun int mpz_jacobi (mpz_t @var{op1}, mpz_t @var{op2})
-@deftypefunx int mpz_legendre (mpz_t @var{op1}, mpz_t @var{op2})
-@cindex Jabobi symbol functions
-Compute the Jacobi and Legendre symbols, respectively. @var{op2} should be
-odd and must be positive.
+@deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b})
+@cindex Jacobi symbol functions
+Calculate the Jacobi symbol @m{\left(a \over b\right),
+(@var{a}/@var{b})}. This is defined only for @var{b} odd.
@end deftypefun
-@deftypefun int mpz_si_kronecker (long @var{a}, mpz_t @var{b});
-@deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b});
-@deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b});
-@deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b});
+@deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p})
+Calculate the Legendre symbol @m{\left(a \over p\right),
+(@var{a}/@var{p})}. This is defined only for @var{p} an odd positive
+prime, and for such @var{p} it's identical to the Jacobi symbol.
+@end deftypefun
+
+@deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b})
+@deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b})
+@deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b})
+@deftypefunx int mpz_si_kronecker (long @var{a}, mpz_t @var{b})
+@deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b})
@cindex Kronecker symbol functions
-@tex
-Calculate the value of the Kronecker/Jacobi symbol $\left(a \over b\right)$,
-with the Kronecker extension $\left(a \over 2\right) = \left(2 \over a\right)$
-when $a$ odd, or $\left(a \over 2\right) = 0$ when $a$ even.
-@end tex
-@ifnottex
-Calculate the value of the Kronecker/Jacobi symbol (@var{a}/@var{b}), with the
-Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
-@end ifnottex
-All values of @var{a} and @var{b} give a well-defined result. See Henri
-Cohen, section 1.4.2, for more information (@pxref{References}). See also the
-example program @file{demos/qcn.c} which uses @code{mpz_kronecker_ui}.
+Calculate the Jacobi symbol @m{\left(a \over b\right),
+(@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
+2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or
+@m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even.
+
+When @var{b} is odd the Jacobi symbol and Kronecker symbol are
+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
@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}. Return the multiplicity of @var{f} in @var{op}.
+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})
@@ -1937,93 +3204,87 @@ Set @var{rop} to @var{op}!, the factorial of @var{op}.
@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
-@ifnottex
-@var{n} over @var{k}
-@end ifnottex
-@tex
-$\left({n}\atop{k}\right)$
-@end tex
-and store the result in @var{rop}. Negative values of @var{n} are supported
-by @code{mpz_bin_ui}, using the identity
-@ifnottex
-bin(-n,k) = (-1)^k * bin(n+k-1,k)
-@end ifnottex
-@tex
-$\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right)$
-@end tex
-(see Knuth volume 1 section 1.2.6 part G).
+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{rop}, unsigned long int @var{n})
+@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
-Compute the @var{n}th Fibonacci number and store the result in @var{rop}.
+@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
-@deftypefun int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
-@ifnottex
-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 ifnottex
-@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 deftypefun
-
-@deftypefn Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})
+@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})
-@ifnottex
-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 ifnottex
-@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
+@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 @math{@var{op1} >
+@var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
+@math{@var{op1} < @var{op2}}.
-These functions are actually implemented as macros. They evaluate their
-arguments multiple times.
+Note that @code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate
+their arguments more than once.
@end deftypefn
-
-@deftypefun int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2})
-@deftypefunx int mpz_cmpabs_ui (mpz_t @var{op1}, unsigned long int @var{op2})
-@ifnottex
+@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 @var{op1} > @var{op2}, zero if @var{op1} = @var{op2}, and a negative
-value if @var{op1} < @var{op2}.
-@end ifnottex
-@tex
-Compare the absolute values of @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 deftypefun
+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})
-@ifnottex
-Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
-@end ifnottex
-@tex
-Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
-@end tex
+@cindex Sign tests
+@cindex Integer sign tests
+Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
+@math{-1} if @math{@var{op} < 0}.
-This function is actually implemented as a macro. It evaluates its
-arguments multiple times.
+This function is actually implemented as a macro. It evaluates its argument
+multiple times.
@end deftypefn
+
@node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions
@comment node-name, next, previous, up
@section Logical and Bit Manipulation Functions
@@ -2031,8 +3292,9 @@ arguments multiple times.
@cindex Bit manipulation functions
@cindex Integer bit manipulation functions
-These functions behave as if two's complement arithmetic were used (although
-sign-magnitude is used by the actual implementation).
+These functions behave as if twos complement arithmetic were used (although
+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})
Set @var{rop} to @var{op1} logical-and @var{op2}.
@@ -2051,29 +3313,33 @@ Set @var{rop} to the one's complement of @var{op}.
@end deftypefun
@deftypefun {unsigned long int} mpz_popcount (mpz_t @var{op})
-For non-negative numbers, return the population count of @var{op}. For
-negative numbers, return the largest possible value (@var{MAX_ULONG}).
+If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is
+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
@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
-between the two operands. Otherwise, return the largest possible value
-(@var{MAX_ULONG}).
-
-It is possible to extend this function to return a useful value when the
-operands are both negative, but the current implementation returns
-@var{MAX_ULONG} in this case. @strong{Do not depend on this behavior, since
-it will change in a future release.}
+If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return
+the hamming distance between the two operands, which is the number of bit
+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
+different is infinite, and the return value is @var{MAX_ULONG}, the largest
+possible @code{unsigned long}.
@end deftypefun
@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
-bits, until the first clear bit is found. Return the index of the found bit.
-@end deftypefun
+@deftypefunx {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})
+Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
+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})
-Scan @var{op}, starting with bit @var{starting_bit}, towards more significant
-bits, until the first set bit is found. Return the index of the found bit.
+If the bit at @var{starting_bit} is already what's sought, then
+@var{starting_bit} is returned.
+
+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
@deftypefun void mpz_setbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
@@ -2085,7 +3351,7 @@ Clear bit @var{bit_index} in @var{rop}.
@end deftypefun
@deftypefun int mpz_tstbit (mpz_t @var{op}, unsigned long int @var{bit_index})
-Check bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
+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
@@ -2149,7 +3415,7 @@ machines.
@need 2000
-@node Integer Random Numbers, 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
@section Random Number Functions
@cindex Integer random number functions
@@ -2161,31 +3427,18 @@ parameter that is read and modified. Please see the @
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
-@ifnottex
-0 to 2^@var{n} @minus{} 1,
-@end ifnottex
-@tex
-0 to $2^n-1$,
-@end tex
-inclusive.
+@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
-@ifnottex
-@var{n} @minus{} 1, inclusive.
-@end ifnottex
-@tex
-$n-1$, inclusive.
-@end tex
+@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})
@@ -2197,13 +3450,7 @@ Generate a random integer with long strings of zeros a
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
-@ifnottex
-0 to 2^@var{n} @minus{} 1,
-@end ifnottex
-@tex
-0 to $2^n-1$,
-@end tex
-inclusive.
+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})
@@ -2230,8 +3477,86 @@ This function is obsolete. Use @code{mpz_rrandomb} in
@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 Random Numbers, Integer Functions
+@node Miscellaneous Integer Functions, , Integer Import and Export, Integer Functions
@comment node-name, next, previous, up
@section Miscellaneous Functions
@cindex Miscellaneous integer functions
@@ -2251,7 +3576,7 @@ short int}, or @code{signed short int}, respectively.
@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 arguments more than once.
+yes, zero if no. These macros evaluate their argument more than once.
@end deftypefn
@deftypefun size_t mpz_size (mpz_t @var{op})
@@ -2262,13 +3587,15 @@ the returned value will be zero.
@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}.
-The base may vary from 2 to 36. The returned value will be exact or 1 too
-big. If @var{base} is a power of 2, the returned value will always be exact.
+The base may vary from 2 to 36. The sign of @var{op} is ignored, just the
+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
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
-minus sign and one for the terminating '\0').
+minus sign and one for the null-terminator).
@end deftypefun
@@ -2289,8 +3616,7 @@ Zero has the unique representation 0/1.
Pure assignment functions do not canonicalize the assigned variable. It is
the responsibility of the user to canonicalize the assigned variable before
-any arithmetic operations are performed on that variable. @strong{Note that
-this is an incompatible change from version 1 of the library.}
+any arithmetic operations are performed on that variable.
@deftypefun void mpq_canonicalize (mpq_t @var{op})
Remove any factors that are common to the numerator and denominator of
@@ -2299,14 +3625,14 @@ Remove any factors that are common to the numerator an
@menu
* Initializing Rationals::
+* Rational Conversions::
* Rational Arithmetic::
* Comparing Rationals::
* Applying Integer Functions::
* I/O of Rationals::
-* Miscellaneous Rational Functions::
@end menu
-@node Initializing Rationals, Rational Arithmetic, Rational Number Functions, Rational Number Functions
+@node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
@comment node-name, next, previous, up
@section Initialization and Assignment Functions
@cindex Initialization and assignment functions
@@ -2335,13 +3661,74 @@ Set the value of @var{rop} to @var{op1}/@var{op2}. No
@code{mpq_canonicalize} before any operations are performed on @var{rop}.
@end deftypefun
+@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
-@node Rational Arithmetic, Comparing Rationals, Initializing Rationals, Rational Number Functions
+@need 2000
+@node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
@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
@cindex Rational arithmetic functions
@cindex Arithmetic functions
@@ -2355,23 +3742,32 @@ Set @var{difference} to @var{minuend} @minus{} @var{su
@end deftypefun
@deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})
-@ifnottex
-Set @var{product} to @var{multiplier} times @var{multiplicand}.
-@end ifnottex
-@tex
-Set @var{product} to $@var{multiplier} \times @var{multiplicand}$.
-@end tex
+Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
@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})
@cindex Division functions
Set @var{quotient} to @var{dividend}/@var{divisor}.
@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})
Set @var{negated_operand} to @minus{}@var{operand}.
@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})
Set @var{inverted_number} to 1/@var{number}. If the new denominator is
zero, this routine will divide by zero.
@@ -2384,47 +3780,32 @@ zero, this routine will divide by zero.
@cindex Comparison functions
@deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})
-@ifnottex
-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 ifnottex
-@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
+Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} >
+@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
+@math{@var{op1} < @var{op2}}.
To determine if two rationals are equal, @code{mpq_equal} is faster than
@code{mpq_cmp}.
@end deftypefun
@deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
-@ifnottex
+@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
-@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 ifnottex
-@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
+@math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
+@var{num2}/@var{den2}}, and a negative value if @math{@var{op1} <
+@var{num2}/@var{den2}}.
-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
-arguments multiple times.
+These functions are implemented as a macros and evaluate their arguments
+multiple times.
@end deftypefn
@deftypefn Macro int mpq_sgn (mpq_t @var{op})
-@ifnottex
-Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
-@end ifnottex
-@tex
-Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
-@end tex
+@cindex Sign tests
+@cindex Rational sign tests
+Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
+@math{-1} if @math{@var{op} < 0}.
This function is actually implemented as a macro. It evaluates its
arguments multiple times.
@@ -2443,20 +3824,33 @@ function is much faster.
@cindex Numerator and denominator
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
-to apply any @code{mpz} function on the numerator or denominator of a rational
-number. If these macros are used to assign to the rational number,
-@code{mpq_canonicalize} normally need to be called afterwards.
+functions for either input or output. The following functions give direct
+access to the numerator and denominator of an @code{mpq_t}.
+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})
@deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})
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.
@end deftypefn
+@deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
+@deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
+@deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
+@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
+equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
+@code{mpq_denref}. Direct use of @code{mpq_numref} or @code{mpq_denref} is
+recommended instead of these functions.
+@end deftypefun
+
@need 2000
-@node I/O of Rationals, Miscellaneous Rational Functions, Applying Integer Functions, Rational Number Functions
+@node I/O of Rationals, , Applying Integer Functions, Rational Number Functions
@comment node-name, next, previous, up
@section Input and Output Functions
@cindex Rational input and output functions
@@ -2464,15 +3858,14 @@ The @code{mpz} functions can be used on the result of
@cindex Output functions
@cindex I/O functions
-Functions that perform input from a stdio stream, and functions that output to
-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
-@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's a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.
+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 @code{stdout},
+respectively.
+
@deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op})
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
@@ -2481,104 +3874,81 @@ Output @var{op} on stdio stream @var{stream}, as a str
Return the number of bytes written, or if an error occurred, return 0.
@end deftypefun
+@deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
+Read a string of digits from @var{stream} and convert them to a rational in
+@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.
-@need 2000
-@node Miscellaneous Rational Functions, , I/O of Rationals, Rational Number Functions
-@comment node-name, next, previous, up
-@section Miscellaneous Functions
-@cindex Rational miscellaneous functions
-@cindex Miscellaneous rational functions
+The input can be a fraction like @samp{17/63} or just an integer like
+@samp{123}. Reading stops at the first character not in this form, and white
+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 double mpq_get_d (mpq_t @var{op})
-Convert @var{op} to a double.
+The @var{base} can be between 2 and 36, or can be 0 in which case the leading
+characters of the string determine the base, @samp{0x} or @samp{0X} for
+hexadecimal, @samp{0} for octal, or decimal otherwise. The leading characters
+are examined separately for the numerator and denominator of a fraction, so
+for instance @samp{0x10/11} is 16/11, whereas @samp{0x10/0x11} is 16/17.
@end deftypefun
-@deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{d})
-Set @var{rop} to the value of d, without rounding.
-@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})
-Copy @var{numerator} to the numerator of @var{rational}. When this risks to
-make the numerator and denominator of @var{rational} have common factors, you
-have to pass @var{rational} to @code{mpq_canonicalize} before any operations
-are performed on @var{rational}.
-
-This function is equivalent to
-@code{mpz_set (mpq_numref (@var{rational}), @var{numerator})}.
-@end deftypefun
-
-@deftypefun void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
-Copy @var{denominator} to the denominator of @var{rational}. When this risks
-to make the numerator and denominator of @var{rational} have common factors,
-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
-copying the sign to the numerator. That is no longer done.}
-
-This function is equivalent to
-@code{mpz_set (mpq_denref (@var{rational}), @var{denominators})}.
-@end deftypefun
-
-@deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
-Copy the numerator of @var{rational} to the integer @var{numerator}, to
-prepare for integer operations on the numerator.
-
-This function is equivalent to
-@code{mpz_set (@var{numerator}, mpq_numref (@var{rational}))}.
-@end deftypefun
-
-@deftypefun void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
-Copy the denominator of @var{rational} to the integer @var{denominator}, to
-prepare for integer operations on the denominator.
-
-This function is equivalent to
-@code{mpz_set (@var{denominator}, mpq_denref (@var{rational}))}.
-@end deftypefun
-
-
@node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
@comment node-name, next, previous, up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions
+@cindex User-defined precision
+@cindex Precision of floats
-This chapter describes the GMP functions for performing floating point
-arithmetic. These functions start with the prefix @code{mpf_}.
+GMP floating point numbers are stored in objects of type @code{mpf_t} and
+functions operating on them have an @code{mpf_} prefix.
-GMP floating point numbers are stored in objects 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 GMP floating-point functions have an interface that is similar to the GMP
-integer functions. The function prefix for floating-point operations is
-@code{mpf_}.
+The exponent of each float is a fixed precision, one machine word on most
+systems. In the current implementation the exponent is a count of limbs, so
+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
-motivated a difference between this function class and other GMP function
-classes: the inherent inexactness of floating point arithmetic. The user has
-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.
+Each variable keeps a size for the mantissa data actually in use. This means
+that if a float is exactly represented in only a few bits then only those bits
+will be used in a calculation, even if the selected precision is high.
-@cindex User-defined precision
-@cindex Precision of floats
-The precision of a calculation is defined as follows: Compute the requested
-operation exactly (with ``infinite precision''), and truncate the result to
-the destination variable precision. Even if the user has asked for a very
-high precision, GMP 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.
+All calculations are performed to the precision of the destination variable.
+Each function is defined to calculate with ``infinite precision'' followed by
+a truncation to the destination precision, but of course the work done is only
+what's needed to determine a result under that definition.
-The GMP floating-point functions are @emph{not} intended as a smooth extension
-to the IEEE P754 arithmetic. Specifically, the results obtained on one
-computer often differs from the results obtained on a computer with a
-different word size.
+The precision selected for a variable is a minimum value, GMP may increase it
+a little to facilitate efficient calculation. Currently this means rounding
+up to a whole limb, and then sometimes having a further partial limb,
+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
* Initializing Floats::
* Assigning Floats::
@@ -2602,6 +3972,10 @@ subsequent calls to @code{mpf_init} will use this prec
initialized variables are unaffected.
@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
it. The functions @code{mpf_init} and @code{mpf_init2} are used for that
purpose.
@@ -2629,8 +4003,8 @@ Here is an example on how to initialize floating-point
@example
@{
mpf_t x, y;
- mpf_init (x); /* use default precision */
- mpf_init2 (y, 256); /* precision @emph{at least} 256 bits */
+ mpf_init (x); /* use default precision */
+ mpf_init2 (y, 256); /* precision @emph{at least} 256 bits */
@dots{}
/* Unless the program is about to exit, do ... */
mpf_clear (x);
@@ -2643,23 +4017,42 @@ calculation. A typical use would be for adjusting the
iterative algorithms like Newton-Raphson, making the computation precision
closely match the actual accurate part of the numbers.
-@deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
-Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.
-Since changing the precision involves calls to @code{realloc}, this routine
-should not be called in a tight loop.
+@deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})
+Return the current precision of @var{op}, in bits.
@end deftypefun
-@deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})
-Return the precision actually used for assignments of @var{op}.
+@deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
+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
@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
-is a low-level function that does not change the allocation. The @var{prec}
-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
-ultimately reset to exactly the value returned by @code{mpf_get_prec} before
-the first call to @code{mpf_set_prec_raw}.
+Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
+without changing the memory allocated.
+
+@var{prec} must be no more than the allocated precision for @var{rop}, that
+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
@@ -2687,7 +4080,8 @@ 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}.
@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
-@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
@minus{}2. Negative values are used to specify that the exponent is in
@@ -2703,14 +4097,15 @@ the mantissa, but not in other places, such as after a
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 "3 14" as meaning 314 as it does now?]
+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
-in base @var{base}. Otherwise it returns @minus{}1.
+This function returns 0 if the entire string is a valid number in base
+@var{base}. Otherwise it returns @minus{}1.
@end deftypefun
@deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
-Swap the values @var{rop1} and @var{rop2} efficiently.
+Swap @var{rop1} and @var{rop2} efficiently. Both the values and the
+precisions of the two variables are swapped.
@end deftypefun
@@ -2758,36 +4153,54 @@ set by @code{mpf_set_default_prec}.
@cindex Conversion functions
@deftypefun double mpf_get_d (mpf_t @var{op})
-Convert @var{op} to a double.
+Convert @var{op} to a @code{double}.
@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})
-Convert @var{op} to a string of digits in base @var{base}. The base may vary
-from 2 to 36. Generate at most @var{n_digits} significant digits, or if
-@var{n_digits} is 0, the maximum number of digits accurately representable by
-@var{op}.
+@deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, mpf_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}. This is similar to the standard C function
+@code{frexp}.
+@end deftypefun
-If @var{str} is @code{NULL}, space for the mantissa is allocated using the default
-allocation function.
+@deftypefun long mpf_get_si (mpf_t @var{op})
+@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 @code{NULL}, it should point to a block of storage enough large
-for the mantissa, i.e., @var{n_digits} + 2. The two extra bytes are for a
-possible minus sign, and for the terminating null character.
+See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p}
+(@pxref{Miscellaneous Float Functions}).
+@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
-from the precision of @var{op} will be generated. Note that the space
-requirements for @var{str} in this case will be impossible for the user to
-predetermine. Therefore, you need to pass @code{NULL} for the string argument
-whenever @var{n_digits} is 0.
+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
+@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
-to the left of the first digit. For example, the number 3.1416 would be
-returned as "31416" in the string and 1 written at @var{expptr}.
+to the left of the first digit. The applicable exponent is written through
+the @var{expptr} pointer. For example, the number 3.1416 would be returned as
+string @nicode{"31416"} and exponent 1.
-A pointer to the result string is returned. This pointer will will either
-equal @var{str}, or if that is @code{NULL}, will point to the allocated storage.
+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
@@ -2799,12 +4212,7 @@ equal @var{str}, or if that is @code{NULL}, will point
@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})
-@ifnottex
-Set @var{rop} to @var{op1} + @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1} + @var{op2}$.
-@end tex
+Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun
@deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@@ -2815,17 +4223,12 @@ 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})
@deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
-@ifnottex
-Set @var{rop} to @var{op1} times @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1} \times @var{op2}$.
-@end tex
+Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun
-Division is undefined if the divisor is zero, and passing a zero divisor to
-the divide functions will make these functions intentionally divide by zero.
-This lets the user handle arithmetic exceptions in these functions in the same
+Division is undefined if the divisor is zero, and passing a zero divisor to the
+divide functions will make these functions intentionally divide by zero. This
+lets the user handle arithmetic exceptions in these functions in the same
manner as other arithmetic exceptions.
@deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@@ -2838,22 +4241,13 @@ Set @var{rop} to @var{op1}/@var{op2}.
@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})
@cindex Root extraction functions
-@ifnottex
-Set @var{rop} to the square root of @var{op}.
-@end ifnottex
-@tex
-Set @var{rop} to $\sqrt{@var{op}}$.
-@end tex
+Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
@end deftypefun
@deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Exponentiation functions
-@ifnottex
-Set @var{rop} to @var{op1} raised to the power @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1}^{op2}$.
-@end tex
+@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})
@@ -2865,21 +4259,13 @@ Set @var{rop} to the absolute value of @var{op}.
@end deftypefun
@deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
-@ifnottex
-Set @var{rop} to @var{op1} times 2 raised to @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1} \times 2^{op2}$.
-@end tex
+Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
+@var{op2}}.
@end deftypefun
@deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
-@ifnottex
-Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.
-@end ifnottex
-@tex
-Set @var{rop} to $@var{op1}/2^{op2}$.
-@end tex
+Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
+@var{op2}}.
@end deftypefun
@node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
@@ -2889,41 +4275,37 @@ Set @var{rop} to $@var{op1}/2^{op2}$.
@cindex Comparison functions
@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_si (mpf_t @var{op1}, signed long int @var{op2})
-@ifnottex
-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 ifnottex
-@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
+Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} >
+@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
+@math{@var{op1} < @var{op2}}.
@end deftypefun
@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
-equal, zero otherwise. I.e., test of @var{op1} and @var{op2} are
-approximately equal.
+equal, zero otherwise. I.e., test of @var{op1} and @var{op2} are approximately
+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
@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
-result in @var{rop}.
+result in @var{rop}. This is @math{@GMPabs{@var{op1}-@var{op2}}/@var{op1}}.
@end deftypefun
@deftypefn Macro int mpf_sgn (mpf_t @var{op})
-@ifnottex
-Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
-@end ifnottex
-@tex
-Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
-@end tex
+@cindex Sign tests
+@cindex Float sign tests
+Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
+@math{-1} if @math{@var{op} < 0}.
-This function is actually implemented as a macro. It evaluates its
-arguments multiple times.
+This function is actually implemented as a macro. It evaluates its arguments
+multiple times.
@end deftypefn
@node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
@@ -2935,8 +4317,8 @@ arguments multiple times.
@cindex I/O functions
Functions that perform input from a stdio stream, and functions that output to
-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
+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
@code{stdout}, respectively.
When using any of these functions, it is a good idea to include @file{stdio.h}
@@ -2944,26 +4326,28 @@ before @file{gmp.h}, since that will allow @file{gmp.h
for these functions.
@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
-base @var{base}. The base may vary from 2 to 36. Print at most
-@var{n_digits} significant digits, or if @var{n_digits} is 0, the maximum
-number of digits accurately representable by @var{op}.
+Print @var{op} to @var{stream}, as a string of digits. Return the number of
+bytes written, or if an error occurred, return 0.
-In addition to the significant digits, a leading @samp{0.} and a
-trailing exponent, in the form @samp{eNNN}, are printed. If @var{base}
-is greater than 10, @samp{@@} will be used instead of @samp{e} as
-exponent delimiter.
+The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
+which may vary from 2 to 36. An exponent then printed, separated by an
+@samp{e}, or if @var{base} is greater than 10 then by an @samp{@@}. The
+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
@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 float in @var{rop}. The string is of the 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 in the specified base. The
-exponent is either in the specified base or, if @var{base} is negative, in
-decimal.
+Read a string in base @var{base} from @var{stream}, and put the read float in
+@var{rop}. The string is of the 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 in the specified base. The exponent is
+either in the specified base or, if @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
@minus{}2. Negative values are used to specify that the exponent is in
@@ -2998,31 +4382,43 @@ Return the number of bytes read, or if an error occurr
@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.
+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 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 0 <=
-@var{rop} < 1, with @var{nbits} significant bits in the mantissa.
+@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.
+@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{max_exp})
+@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
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
-testing functions and algorithms, since this kind of random numbers have
-proven to be more likely to trigger corner-case bugs. Negative random numbers
-are generated when @var{max_size} is negative.
+testing functions and algorithms, since this kind of random numbers have proven
+to be more likely to trigger corner-case bugs. Negative random numbers are
+generated when @var{max_size} is negative.
@end deftypefun
-
-
@c @deftypefun size_t mpf_size (mpf_t @var{op})
@c Return the size of @var{op} measured in number of limbs. If @var{op} is
@c zero, the returned value will be zero. (@xref{Nomenclature}, for an
@@ -3032,13 +4428,14 @@ are generated when @var{max_size} is negative.
@c releases.}
@c @end deftypefun
+
@node Low-level Functions, Random Number Functions, Floating-point Functions, Top
@comment node-name, next, previous, up
@chapter Low-level Functions
@cindex Low-level functions
-This chapter describes low-level GMP functions, used to implement the high-level
-GMP functions, but also intended for time-critical user code.
+This chapter describes low-level GMP functions, used to implement the
+high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix @code{mpn_}.
@@ -3056,380 +4453,360 @@ limb count. A destination operand is specified by jus
responsibility of the caller to ensure that the destination has enough space
for storing the result.
-With this way of specifying operands, it is possible to perform computations
-on subranges of an argument, and store the result into a subrange of a
+With this way of specifying operands, it is possible to perform computations on
+subranges of an argument, and store the result into a subrange of a
destination.
-A common requirement for all functions is that each source area needs at
-least 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.
+A common requirement for all functions is that each source area needs at least
+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 are the base for the implementation of the
@code{mpz_}, @code{mpf_}, and @code{mpq_} functions.
-This example adds the number beginning at @var{s1p} and the number
-beginning at @var{s2p} and writes the sum at @var{destp}. All areas
-have @var{size} limbs.
+This example adds the number beginning at @var{s1p} and the number beginning at
+@var{s2p} and writes the sum at @var{destp}. All areas have @var{n} limbs.
@example
-cy = mpn_add_n (destp, s1p, s2p, size)
+cy = mpn_add_n (destp, s1p, s2p, n)
@end example
@noindent
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,
-@{s1p, s1size@}.
+@{@var{s1p}, @var{s1n}@}.
-@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{size})
-Add @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@}, and
-write the @var{size} least significant limbs of the result to @var{rp}.
-Return carry, either 0 or 1.
+@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{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n}
+least significant limbs of the result to @var{rp}. Return carry, either 0 or
+1.
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
-of a variable to itself (i.e., @var{s1p} equals @var{s2p}, use
-@code{mpn_lshift} with a count of 1 for optimal speed.
+for addition, since it is written in assembly for most CPUs. For addition of
+a variable to itself (i.e., @var{s1p} equals @var{s2p}, use @code{mpn_lshift}
+with a count of 1 for optimal speed.
@end deftypefun
-@deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
-Add @{@var{s1p}, @var{size}@} and @var{s2limb}, and write the
-@var{size} least significant limbs of the result to @var{rp}. Return
-carry, either 0 or 1.
+@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{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least
+significant limbs of the result to @var{rp}. Return carry, either 0 or 1.
@end deftypefun
-@deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
-Add @{@var{s1p}, @var{s1size}@} and @{@var{s2p},
-@var{s2size}@}, and write the @var{s1size} least significant limbs of
-the result to @var{rp}. Return carry, either 0 or 1.
+@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{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
+@var{s1n} least significant limbs of the result to @var{rp}. Return carry,
+either 0 or 1.
-This function requires that @var{s1size} is greater than or equal to
-@var{s2size}.
+This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun
-@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{size})
-Subtract @{@var{s2p}, @var{s2size}@} from @{@var{s1p},
-@var{size}@}, and write the @var{size} least significant limbs of the result
-to @var{rp}. Return borrow, either 0 or 1.
+@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{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the
+@var{n} least significant limbs of the result to @var{rp}. Return borrow,
+either 0 or 1.
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
-@deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
-Subtract @var{s2limb} from @{@var{s1p}, @var{size}@}, and write the
-@var{size} least significant limbs of the result to @var{rp}. Return
-borrow, either 0 or 1.
+@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{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least
+significant limbs of the result to @var{rp}. Return borrow, either 0 or 1.
@end deftypefun
-@deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
-Subtract @{@var{s2p}, @var{s2size}@} from @{@var{s1p},
-@var{s1size}@}, and write the @var{s1size} least significant limbs of
-the result to @var{rp}. Return borrow, either 0 or 1.
+@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{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
+@var{s1n} least significant limbs of the result to @var{rp}. Return borrow,
+either 0 or 1.
-This function requires that @var{s1size} is greater than or equal to
-@var{s2size}.
+This function requires that @var{s1n} is greater than or equal to
+@var{s2n}.
@end deftypefun
-@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{size})
-Multiply @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@},
-and write the @strong{entire} result to @var{rp}.
+@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{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
+2*@var{n}-limb result to @var{rp}.
-The destination has to have space for 2*@var{size} limbs, even if the
-significant result might be one limb smaller.
+The destination has to have space for 2*@var{n} limbs, even if the product's
+most significant limb is zero.
@end deftypefun
-@deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and write the
-@var{size} least significant limbs of the product to @var{rp}. Return
-the most significant limb of the product.
+@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{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
+significant limbs of the product to @var{rp}. Return the most significant
+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
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{s2limb} is a power of 2; use
-@code{mpn_lshift} with a count equal to the logarithm of @var{s2limb}
-instead, for optimal speed.
+Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
+with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
@end deftypefun
-@deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and add the
-@var{size} least significant limbs of the product to @{@var{rp},
-@var{size}@} and write the result to @var{rp}. Return
-the most significant limb of the product, plus carry-out from the addition.
+@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{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least
+significant limbs of the product to @{@var{rp}, @var{n}@} and write the result
+to @var{rp}. Return the most significant limb of the product, plus carry-out
+from the addition.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written in assembly
-for most targets.
+for most CPUs.
@end deftypefun
-@deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
-Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and subtract the
-@var{size} least significant limbs of the product from @{@var{rp},
-@var{size}@} and write the result to @var{rp}. Return the most
-significant limb of the product, minus borrow-out from the subtraction.
+@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{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n}
+least significant limbs of the product from @{@var{rp}, @var{n}@} and write the
+result to @var{rp}. Return the most significant limb of the product, minus
+borrow-out from the subtraction.
This is a low-level function that is a building block for general
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
-@deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
-Multiply @{@var{s1p}, @var{s1size}@} and @{@var{s2p},
-@var{s2size}@}, and write the result to @var{rp}. Return the most
-significant limb of the result.
+@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{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
+result to @var{rp}. Return the most significant limb of the result.
-The destination has to have space for @var{s1size} + @var{s2size}
-limbs, even if the result might be one limb smaller.
+The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
+result might be one limb smaller.
-This function requires that @var{s1size} is greater than or equal to
-@var{s2size}. The destination must be distinct from either input operands.
+This function requires that @var{s1n} is greater than or equal to
+@var{s2n}. The destination must be distinct from both input operands.
@end deftypefun
@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{np}, @var{nn}@} by @{@var{dp}, @var{dn}@}. Write the quotient
-at @var{qp} and the remainder at @var{rp}.
+Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
+at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
+@var{dn}@}. The quotient is rounded towards 0.
-The quotient written at @var{qp} will be @var{nn} @minus{} @var{dn} + 1 limbs.
-The remainder written at @var{rp} will be @var{dn} limbs.
-
-It is required that @var{nn} is greater than or equal to @var{dn}. The
-@var{qxn} operand must be zero.
-
-The quotient is rounded towards 0.
-
-No overlap between arguments is permitted.
+No overlap is permitted between arguments. @var{nn} must be greater than or
+equal to @var{dn}. The most significant limb of @var{dp} must be non-zero.
+The @var{qxn} operand must be zero.
+@comment FIXME: Relax overlap requirements!
@end deftypefun
-@deftypefun mp_limb_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})
-[This function is obsolete. Please call @code{mpn_tdiv_qr} instead for
-best performance.]
+@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})
+[This function is obsolete. Please call @code{mpn_tdiv_qr} instead for best
+performance.]
-Divide @{@var{rs2p}, @var{rs2size}@} by @{@var{s3p}, @var{s3size}@}, and write
-the 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{s3size} limbs long (i.e., as many limbs as the divisor).
+Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
+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{xsize} fraction limbs are developed,
-and stored after the integral limbs. For most usages, @var{xsize} will be
-zero.
+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{rs2size} is greater than or equal to @var{s3size}.
-It is required that the most significant bit of the divisor is set.
+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{s3size} as @var{r1p}.
-Aside from that special case, no overlap between arguments is permitted.
+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.
-The area at @var{r1p} needs to be @var{rs2size} @minus{} @var{s3size} +
-@var{xsize} limbs large.
+The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
+limbs large.
@end deftypefun
-@deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{xsize}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2size}, mp_limb_t @var{s3limb})
-@deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2size}}, @w{mp_limb_t @var{s3limb}})
-Divide @{@var{s2p}, @var{s2size}@} by @var{s3limb}, and write the quotient
-at @var{r1p}. Return the remainder.
+@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})
+@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}})
+Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
+@var{r1p}. Return the remainder.
-The integer quotient is written to @{@var{r1p}+@var{xsize}, @var{s2size}@} and
-in addition @var{xsize} fraction limbs are developed and written to
-@{@var{r1p}, @var{xsize}@}. Either or both @var{s2size} and @var{xsize} can
-be zero. For most usages, @var{xsize} will be zero.
+The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in
+addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
+@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 an @var{xsize} of 0.
+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
separate, not partially overlapping.
@end deftypefn
-@deftypefun mp_limb_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})
-@strong{This interface is obsolete. It will disappear from future releases.
-Use @code{mpn_divrem} in its stead.}
+@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})
+[This function is obsolete. Please call @code{mpn_tdiv_qr} instead for best
+performance.]
@end deftypefun
-@deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{size}})
-@deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{size}}, mp_limb_t @var{carry})
-Divide @{@var{sp}, @var{size}@} by 3, expecting it to divide exactly, and
-writing the result to @{@var{rp}, @var{size}@}. 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.
+@deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
+@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})
+Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
+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.
@code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
return value from a previous call, so a large calculation can be done piece by
-piece. @code{mpn_divexact_by3} is simply a macro calling
+piece from low to high. @code{mpn_divexact_by3} is simply a macro calling
@code{mpn_divexact_by3c} with a 0 carry parameter.
These routines use a multiply-by-inverse and will be faster than
@code{mpn_divrem_1} on CPUs with fast multiplication but slow division.
The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i},
-and return value @math{c} satisfy
-@tex
-$c b^n + a - i = 3q$,
-@end tex
-@ifnottex
-@math{c*b^n + a-i = 3*q},
-@end ifnottex
-where @math{b} is the size of a limb
-@tex
-($2^{32}$ or $2^{64}$).
-@end tex
-@ifnottex
-(@math{2^32} or @math{2^64}).
-@end ifnottex
-@math{c} is always 0, 1 or 2, and the initial carry must also be 0, 1 or 2
-(these are both borrows really). When @math{c=0}, clearly @math{q=(a-i)/3}.
-When
-@tex
-$c \neq 0$,
-the remainder $(a-i) \, mod \, 3$
-@end tex
-@ifnottex
-@math{c!=0}, the remainder @math{(a-i) mod 3}
-@end ifnottex
-is given by @math{3-c}, because
-@tex
-$b \equiv 1 \, mod \, 3$.
-@end tex
-@ifnottex
-@math{b @equiv{} 1 mod 3}.
-@end ifnottex
+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{s1size}, mp_limb_t @var{s2limb})
-Divide @{@var{s1p}, @var{s1size}@} by @var{s2limb}, and return the remainder.
-@var{s1size} can be zero.
+@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
-@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})
-@strong{This interface is obsolete. It will disappear from future releases.
-Use @code{mpn_mod_1} in its stead.}
-@end deftypefun
+@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})
+This function puts the low
+@math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of @var{q} =
+@{@var{s1p}, @var{s1n}@}/@{@var{s2p}, @var{s2n}@} mod @m{2^d,2^@var{d}} at
+@var{rp}, and returns the high @var{d} mod @code{mp_bits_per_limb} bits of
+@var{q}.
-@deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, 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})
-The function puts the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of
-@var{q} =
-@{@var{s1p}, @var{s1size}@}/@{@var{s2p}, @var{s2size}@}
-mod 2^@var{d}
-at @var{rp},
-and returns the high @var{d} mod @var{BITS_PER_MP_LIMB} bits of @var{q}.
+@{@var{s1p}, @var{s1n}@} - @var{q} * @{@var{s2p}, @var{s2n}@} mod @m{2
+\GMPraise{@var{s1n}*@code{mp\_bits\_per\_limb}},
+2^(@var{s1n}*@nicode{mp\_bits\_per\_limb})} is placed at @var{s1p}. Since the
+low @math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of this
+difference are zero, it is possible to overwrite the low limbs at @var{s1p}
+with this difference, provided @math{@var{rp} @le{} @var{s1p}}.
-@{@var{s1p}, @var{s1size}@} - @var{q} * @{@var{s2p}, @var{s2size}@}
-mod 2^(@var{s1size}*@var{BITS_PER_MP_LIMB})
-is placed at @var{s1p}.
-Since the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of
-this difference are zero, it is possible to overwrite the low limbs at
-@var{s1p} with this difference,
-provided @var{rp} <= @var{s1p}.
+This function requires that @math{@var{s1n} * @nicode{mp\_bits\_per\_limb}
+@ge{} @var{D}}, and that @{@var{s2p}, @var{s2n}@} is odd.
-This function requires that @var{s1size} * @var{BITS_PER_MP_LIMB} >= @var{D},
-and that @{@var{s2p}, @var{s2size}@} is odd.
-
-@strong{This interface is preliminary. It might change incompatibly in
-future revisions.}
+@strong{This interface is preliminary. It might change incompatibly in future
+revisions.}
@end deftypefun
-@deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})
-Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the left, and
-write the @var{src_size} least significant limbs of the result to
-@var{rp}. @var{count} might be in the range 1 to n @minus{} 1, on an
-n-bit machine. The bits shifted out to the left are returned.
+@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{sp}, @var{n}@} left by @var{count} bits, and write the result to
+@{@var{rp}, @var{n}@}. The bits shifted out at the left are returned in the
+least significant @var{count} bits of the return value (the rest of the return
+value is zero).
-Overlapping of the destination space and the source space is allowed in this
-function, provided @var{rp} >= @var{src_ptr}.
+@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The
+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
-@deftypefun mp_limp_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})
-Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the right, and
-write the @var{src_size} most significant limbs of the result to
-@var{rp}. @var{count} might be in the range 1 to n @minus{} 1, on an
-n-bit machine. The bits shifted out to the right are returned.
+@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{sp}, @var{n}@} right by @var{count} bits, and write the result to
+@{@var{rp}, @var{n}@}. The bits shifted out at the right are returned in the
+most significant @var{count} bits of the return value (the rest of the return
+value is zero).
-Overlapping of the destination space and the source space is allowed in this
-function, provided @var{rp} <= @var{src_ptr}.
+@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The
+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
-@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{size})
-Compare @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@} and
-return a positive value if s1 > src2, 0 of they are equal, and a negative
-value if s1 < src2.
+@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
+Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
+positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a
+negative value if @math{@var{s1} < @var{s2}}.
@end deftypefun
-@deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
-Puts at @var{rp} the greatest common divisor of @{@var{s1p},
-@var{s1size}@} and @{@var{s2p}, @var{s2size}@}; both source
-operands are destroyed by the operation. The size in limbs of the greatest
-common divisor is returned.
+@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})
+Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{s1p},
+@var{s1n}@} and @{@var{s2p}, @var{s2n}@}. The result can be up to @var{s2n}
+limbs, the return value is the actual number produced. Both source operands
+are destroyed.
-@{@var{s1p}, @var{s1size}@} must have at least as many bits as
-@{@var{s2p}, @var{s2size}@}, and @{@var{s2p}, @var{s2size}@} must be odd.
+@{@var{s1p}, @var{s1n}@} must have at least as many bits as @{@var{s2p},
+@var{s2n}@}. @{@var{s2p}, @var{s2n}@} must be odd. Both operands must have
+non-zero most significant limbs. No overlap is permitted between @{@var{s1p},
+@var{s1n}@} and @{@var{s2p}, @var{s2n}@}.
@end deftypefun
-@deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb})
-Return the greatest common divisor of @{@var{s1p}, @var{s1size}@}
-and @var{s2limb}, where @var{s2limb} (as well as @var{s1size})
-must be different from 0.
+@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{s1p}, @var{s1n}@} and
+@var{s2limb}. Both operands must be non-zero.
@end deftypefun
-@deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, mp_size_t *@var{r2size}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
-Compute the greatest common divisor of @{@var{s1p}, @var{s1size}@} and
-@{@var{s2p}, @var{s2size}@}. Store the gcd at @var{r1p} and return its size
-in limbs. Write the first cofactor at @var{r2p} and store its size in
-*@var{r2size}. If the cofactor is negative, *@var{r2size} is negative and
-@var{r2p} is the absolute value of the cofactor.
+@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})
+Calculate the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
+@{@var{s2p}, @var{s2n}@}. Store the gcd at @{@var{r1p}, @var{retval}@} and
+the first cofactor at @{@var{r2p}, *@var{r2n}@}, with *@var{r2n} negative if
+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.
-@{@var{s1p}, @var{s1size}@} must be greater than or equal to @{@var{s2p},
-@var{s2size}@}. Neither operand may equal 0. Both source operands are
-destroyed, plus one limb past the end of each, ie. @{@var{s1p},
-@var{s1size}+1@} and @{@var{s2p}, @var{s2size}+1@}.
+@math{@{@var{s1p}, @var{s1n}@} @ge{} @{@var{s2p}, @var{s2n}@}} is required,
+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
-@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})
-Compute the square root of @{@var{sp}, @var{size}@} and put the result at
-@var{r1p}. Write the remainder at @var{r2p}, unless @var{r2p} is @code{NULL}.
+@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{n}@} and put the result at
+@{@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 @code{NULL} or non-@code{NULL}.
-Iff the operand was a perfect square, the return value will be 0.
+The most significant limb of @{@var{sp}, @var{n}@} must be non-zero. The
+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
-@var{r2p} and @var{sp} have to be identical or completely separate, not
-partially overlapping.
+If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
+case the return value is zero or non-zero according to whether the remainder
+would have been zero or non-zero.
-@ifnottex
-The area at @var{r1p} needs to have space for ceil(@var{size}/2) limbs.
-@end ifnottex
-@tex
-The area at @var{r1p} needs to have space for $\lceil@var{size}/2\rceil$ limbs.
-@end tex
-The area at @var{r2p} needs to be @var{size} limbs large.
+A return value of zero indicates a perfect square. See also
+@code{mpz_perfect_square_p}.
@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})
-Convert @{@var{s1p}, @var{s1size}@} to a raw unsigned char array in base
-@var{base}. The string is not in ASCII; to convert it to printable format,
-add the ASCII codes for @samp{0} or @samp{A}, depending on the base and
-range. There may be leading zeros in the string.
+@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{s1n}@} to a raw unsigned char array at @var{str} in
+base @var{base}, and return the number of characters produced. There may be
+leading zeros in the string. The string is not in ASCII; to convert it to
+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
-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
-@deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{r1p}, const char *@var{str}, size_t @var{strsize}, int @var{base})
-Convert the raw unsigned char array at @var{str} of length @var{strsize} to
-a limb array @{@var{s1p}, @var{s1size}@}. The base of @var{str} is
-@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 bytes @{@var{str},@var{strsize}@} in the given @var{base} to limbs at
+@var{rp}.
-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
@deftypefun {unsigned long int} mpn_scan0 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
@@ -3446,10 +4823,10 @@ It is required that there be a set bit within the area
beyond bit position @var{bit}, so that the function has something to return.
@end deftypefun
-@deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1size})
-@deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1size})
-Generate a random number of length @var{r1size} and store it at @var{r1p}.
-The most significant limb is always non-zero. @code{mpn_random} generates
+@deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
+@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.
@@ -3457,176 +4834,1294 @@ zeros and ones in the binary representation.
routines.
@end deftypefun
-@deftypefun {unsigned long int} mpn_popcount (const mp_limb_t *@var{s1p}, unsigned long int @var{size})
-Count the number of set bits in @{@var{s1p}, @var{size}@}.
+@deftypefun {unsigned long int} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
+Count the number of set bits in @{@var{s1p}, @var{n}@}.
@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
-@{@var{s2p}, @var{size}@}.
+@deftypefun {unsigned long int} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
+Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p},
+@var{n}@}.
@end deftypefun
-@deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{size})
-Return non-zero iff @{@var{s1p}, @var{size}@} is a perfect square.
+@deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
+Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square.
@end deftypefun
-@node Random Number Functions, BSD Compatible Functions, Low-level Functions, Top
-@chapter Random Number Functions
-@cindex Random number functions
+@sp 1
+@section Nails
+@cindex Nails
-There are two groups of random number functions in GNU MP; older
-functions that call C library random number generators, rely on a global
-state, and aren't very random; and newer functions that don't have these
-problems. The newer functions are self-contained, they accept a random
-state parameter that supplants global state, and generate good random
-numbers.
+@strong{Everything in this section is highly experimental and may disappear or
+be subject to incompatible changes in a future version of GMP.}
-The random state parameter is of the type @code{gmp_randstate_t}. It must be
-initialized by a call to one of the @code{gmp_randinit} functions (@ref{Random
-State Initialization}). The initial seed is set using one of the
-@code{gmp_randseed} functions (@ref{Random State Initialization}).
+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.
-The size of the seed determines the number of different sequences of
-random numbers that is possible to generate. The ``quality'' of the
-seed is the randomness of a given seed compared to the previous seed
-used and affects the randomness of separate number sequences.
+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.
-The algorithm for assigning seed is critical if the generated random numbers
-are to be used for important applications, such as generating cryptographic
-keys.
+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}.
-The traditional method is to use the current system time for seeding. One has
-to be careful when using the current time though. If the application seeds the
-random functions very often, say several times per second, and the resolution
-of the system clock is comparatively low, like one second, the same sequence of
-numbers will be generated until the system clock ticks. Furthermore, the
-current system time is quite easy to guess, so a system depending on any
-unpredictability of the random number sequence should absolutely not use that
-as its only source for a seed value.
+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.
-On some systems there is a special device, often called @code{/dev/random},
-which provides a source of somewhat random numbers more usable as seed.
+For the higher level routines, meaning @code{mpz} etc, a nail build should be
+fully source and binary compatible with a non-nail build.
-The functions actually generating random functions are documented under
-``Miscellaneous Functions'' in their respective function class:
-@ref{Miscellaneous Integer Functions}, @ref{Miscellaneous Float Functions}.
+@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:: How to initialize a random state.
+* Random State Initialization::
+* Random State Seeding::
@end menu
-@node Random State Initialization, , Random Number Functions, Random Number Functions
+@node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
@section Random State Initialization
@cindex Random number state
-See @ref{Random Number Functions} for a discussion on how to choose the
-initial seed value passed to these functions.
+@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
-@deftypefun void gmp_randinit (gmp_randstate_t @var{state}, gmp_randalg_t @var{alg}, ...)
-Initialize random state variable @var{state}.
+@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}})
+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}}.
-@var{alg} denotes what algorithm to use for random number generation.
-Use one of
-@itemize @minus
-@item GMP_RAND_ALG_LC --- Linear congruential.
+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.
-A fast generator defined by @math{X = (aX + c) mod m}.
+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
-A third argument @var{size} of type unsigned long int is required. @var{size}
-is the size of the largest good quality random number to be generated,
-expressed in number of bits. If the random generation functions are asked for
-a bigger random number than indicated by this parameter, two or more numbers
-of @var{size} bits will be generated and concatenated, resulting in a ``bad''
-random number. This can be used to generate big random numbers relatively
-cheap if the quality of randomness isn't of great importance.
+@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}}.
-a, c, and m are picked from a table where the modulus (m) is a power of 2 and
-the multiplier is congruent to 5 (mod 8). The choice is based on the
-@var{size} parameter. The maximum @var{size} supported by this algorithm is
-128. If you need bigger random numbers, use your own scheme and call one of
-the other @code{gmp_randinit} functions.
+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
-@item GMP_RAND_ALG_BBS --- Blum, Blum, and Shub.
+@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
-@end itemize
-If @var{alg} is 0 or GMP_RAND_ALG_DEFAULT, the default algorithm is used. The
-default algorithm is typically a fast algorithm like the linear congruential
-and requires a third @var{size} argument (see GMP_RAND_ALG_LC).
+@deftypefun void gmp_randclear (gmp_randstate_t @var{state})
+Free all memory occupied by @var{state}.
+@end deftypefun
-When you're done with a @var{state} variable, call @code{gmp_randclear}
-to deallocate any memory allocated by this function.
-@code{gmp_randinit} may set the following bits in @var{gmp_errno}:
-@c FIXME: gmp_errno is printed in uppercase. That's wrong.
-@itemize
-@item GMP_ERROR_UNSUPPORTED_ARGUMENT --- @var{alg} is unsupported
-@item GMP_ERROR_INVALID_ARGUMENT --- @var{size} is too big
-@end itemize
+@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
-@ignore
-@deftypefun void gmp_randinit_lc (gmp_randstate_t @var{state}, mpz_t @var{a},
-unsigned long int @var{c}, mpz_t @var{m})
+@node Formatted Output, Formatted Input, Random Number Functions, Top
+@chapter Formatted Output
+@cindex Formatted output
+@cindex @code{printf} formatted output
-Initialize random state variable @var{state} with given linear congruential
-scheme.
+@menu
+* Formatted Output Strings::
+* Formatted Output Functions::
+* C++ Formatted Output::
+@end menu
-Parameters @var{a}, @var{c}, and @var{m} are the multiplier, adder, and modulus
-for the linear congruential scheme to use, respectively.
+@node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
+@section Format Strings
-When you're done with a @var{state} variable, call @code{gmp_randclear}
-to deallocate any memory allocated by this function.
+@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
-@end ignore
-@deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a},
-unsigned long int @var{c}, unsigned long int @var{m2exp})
+@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
-Initialize random state variable @var{state} with given linear congruential
-scheme.
+@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.
-Parameters @var{a}, @var{c}, and @var{m2exp} are the multiplier, adder, and
-modulus for the linear congruential scheme to use, respectively. The modulus
-is expressed as a power of 2, so that
-@ifnottex
-@var{m} = 2^@var{m2exp}.
-@end ifnottex
-@tex
-$m = 2^{m2exp}$.
-@end tex
+No overlap is permitted between the space at @var{buf} and the string
+@var{fmt}.
-The least significant bits of a random number generated by the linear
-congruential algorithm where the modulus is a power of two are not very random.
-Therefore, the lower half of a random number generated by an LC scheme
-initialized with this function is discarded. Thus, the size of a random number
-is @var{m2exp} / 2 (rounded upwards) bits when this function has been used for
-initializing the random state.
+These functions are not recommended, since there's no protection against
+exceeding the space available at @var{buf}.
+@end deftypefun
-When you're done with a @var{state} variable, call @code{gmp_randclear}
-to deallocate any memory allocated by this function.
+@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 void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed})
-@deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, unsigned long int @var{seed})
+@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.
-Set the initial seed value.
+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
-Parameter @var{seed} is the initial random seed. The function
-@code{gmp_randseed_ui} takes the @var{seed} as an unsigned long int rather
-than as an mpz_t.
+@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
-@deftypefun void gmp_randclear (gmp_randstate_t @var{state})
-Free all memory occupied by @var{state}. Make sure to call this
-function for all @code{gmp_randstate_t} variables when you are done with
-them.
+
+@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{}.
+
+@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
-@node BSD Compatible Functions, Custom Allocation, Random Number Functions, Top
+@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{}.
+
+@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
+@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
+@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
+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
+void fun (T f, T g)
+@{
+ fun2 (f, f+g); // Bad
+@}
+
+template
+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
@chapter Berkeley MP Compatible Functions
@cindex Berkeley MP compatible functions
@@ -3648,11 +6143,11 @@ Apart from the incomplete set of functions, the interf
with @code{pow} in @file{libm.a}.
@cindex @file{mp.h}
-Include the header @file{mp.h} to get the definition of the necessary types
-and functions. If you are on a BSD derived system, make sure to include GNU
+Include the header @file{mp.h} to get the definition of the necessary types and
+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 your program.
-This means that you probably need to give the -I option to the compiler,
-where is the directory where you have GNU @file{mp.h}.
+This means that you probably need to give the @samp{-I} option to the
+compiler, where @samp{} is the directory where you have GNU @file{mp.h}.
@deftypefun {MINT *} itom (signed short int @var{initial_value})
Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
@@ -3662,8 +6157,8 @@ Initialize the integer to @var{initial_value}. Return
@deftypefun {MINT *} xtom (char *@var{initial_value})
Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
-Initialize the integer from @var{initial_value}, a hexadecimal, '\0'-terminate
-C string. Return a pointer to the @code{MINT} object.
+Initialize the integer from @var{initial_value}, a hexadecimal,
+null-terminated C string. Return a pointer to the @code{MINT} object.
@end deftypefun
@deftypefun void move (MINT *@var{src}, MINT *@var{dest})
@@ -3681,8 +6176,7 @@ Subtract @var{src_2} from @var{src_1} and put the diff
@end deftypefun
@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
-@var{destination}.
+Multiply @var{src_1} and @var{src_2} and put the product in @var{destination}.
@end deftypefun
@deftypefun void mdiv (MINT *@var{dividend}, MINT *@var{divisor}, MINT *@var{quotient}, MINT *@var{remainder})
@@ -3695,17 +6189,11 @@ Some implementations of these functions work different
negative arguments.
@end deftypefun
-@deftypefun void msqrt (MINT *@var{operand}, MINT *@var{root}, MINT *@var{remainder})
-@ifnottex
-Set @var{root} to the truncated integer part of the square root of
-@var{operand}. Set @var{remainder} to
-@var{operand}@minus{}@var{root}*@var{root},
-@end ifnottex
-@tex
-Set @var{root} to $\lfloor\sqrt{@var{operand}}\rfloor$, like
-@code{mpz_sqrt}. Set @var{remainder} to $(operand - root^2)$,
-@end tex
-(i.e., zero if @var{operand} is a perfect square).
+@deftypefun void msqrt (MINT *@var{op}, MINT *@var{root}, MINT *@var{remainder})
+Set @var{root} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
+of the square root of @var{op}}, like @code{mpz_sqrt}. Set @var{remainder} to
+@m{(@var{op} - @var{root}^2), @var{op}@minus{}@var{root}*@var{root}}, i.e.
+zero if @var{op} is a perfect square.
If @var{root} and @var{remainder} are the same variable, the results are
undefined.
@@ -3719,15 +6207,14 @@ Set @var{dest} to (@var{base} raised to @var{exp}) mod
Set @var{dest} to @var{base} raised to @var{exp}.
@end deftypefun
-@deftypefun void gcd (MINT *@var{operand1}, MINT *@var{operand2}, MINT *@var{res})
-Set @var{res} to the greatest common divisor of @var{operand1} and
-@var{operand2}.
+@deftypefun void gcd (MINT *@var{op1}, MINT *@var{op2}, MINT *@var{res})
+Set @var{res} to the greatest common divisor of @var{op1} and @var{op2}.
@end deftypefun
-@deftypefun int mcmp (MINT *@var{operand1}, MINT *@var{operand2})
-Compare @var{operand1} and @var{operand2}. Return a positive value if
-@var{operand1} > @var{operand2}, zero if @var{operand1} =
-@var{operand2}, and a negative value if @var{operand1} < @var{operand2}.
+@deftypefun int mcmp (MINT *@var{op1}, MINT *@var{op2})
+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 deftypefun
@deftypefun void min (MINT *@var{dest})
@@ -3739,78 +6226,2803 @@ Input a decimal string from @code{stdin}, and put the
Output @var{src} to @code{stdout}, as a decimal string. Also output a newline.
@end deftypefun
-@deftypefun {char *} mtox (MINT *@var{operand})
-Convert @var{operand} to a hexadecimal string, and return a pointer to the
-string. The returned string is allocated using the default memory allocation
-function, @code{malloc} by default.
+@deftypefun {char *} mtox (MINT *@var{op})
+Convert @var{op} to a hexadecimal string, and return a pointer to the string.
+The returned string is allocated using the default memory allocation function,
+@code{malloc} by default. It will be @code{strlen(str)+1} bytes, that being
+exactly enough for the string and null-terminator.
@end deftypefun
-@deftypefun void mfree (MINT *@var{operand})
-De-allocate, the space used by @var{operand}. @strong{This function should
-only be passed a value returned by @code{itom} or @code{xtom}.}
+@deftypefun void mfree (MINT *@var{op})
+De-allocate, the space used by @var{op}. @strong{This function should only be
+passed a value returned by @code{itom} or @code{xtom}.}
@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
@chapter Custom Allocation
@cindex Custom allocation
@cindex Memory allocation
@cindex Allocation of memory
-By default, GMP uses @code{malloc}, @code{realloc} and @code{free} for memory
-allocation. If @code{malloc} or @code{realloc} fails, GMP prints a message to
-the standard error output and terminates execution.
+By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory
+allocation, and if they fail GMP prints a message to the standard error output
+and terminates the program.
-Some applications might want to allocate memory in other ways, or might not
-want a fatal error when there is no more memory available. To accomplish
-this, you can specify alternative memory allocation functions.
+Alternate functions can be specified to allocate memory in a different way or
+to have a different error action on running out of memory.
-This can be done in the Berkeley compatibility library as well as the main GMP
-library.
+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))
Replace the current allocation functions from the arguments. If an argument
-is @code{NULL}, the corresponding default function is retained.
+is @code{NULL}, the corresponding default function is used.
-@strong{Be sure to call this function only when there are no active GMP
-objects allocated using the previous memory functions! Usually, that means
-that you have to call this function before any other GMP function.}
+These functions will be used for all memory allocation done by GMP, apart from
+temporary space from @code{alloca} if that function is available and GMP is
+configured to use it (@pxref{Build Options}).
+
+@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
-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})
-This function should return a pointer to newly allocated space with at least
-@var{alloc_size} storage units.
+Return a pointer to newly allocated space with at least @var{alloc_size}
+bytes.
@end deftypefun
@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
-@var{new_size} storage units, after copying at least the first @var{old_size}
-storage units from @var{ptr}. It should also de-allocate the space at
-@var{ptr}.
+Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be
+@var{new_size} bytes.
-You can assume that the space at @var{ptr} was formerly returned from
-@code{allocate_function} or @code{reallocate_function}, for a request for
-@var{old_size} storage units.
+The block may be moved if necessary or if desired, and in that case the
+smaller of @var{old_size} and @var{new_size} bytes must be copied to the new
+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
@deftypefun void deallocate_function (void *@var{ptr}, size_t @var{size})
De-allocate the space pointed to by @var{ptr}.
-You can assume that the space at @var{ptr} was formerly returned from
-@code{allocate_function} or @code{reallocate_function}, for a request for
-@var{size} storage units.
+@var{ptr} is never @code{NULL}, it's always a previously allocated block of
+@var{size} bytes.
@end deftypefun
-(A @dfn{storage unit} is the unit in which the @code{sizeof} operator returns
-the size of an object, normally an 8 bit byte.)
+A @dfn{byte} here means the unit used by the @code{sizeof} operator.
+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 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 >
+operator+(const mpf_class &f, const mpf_class &g)
+@{
+ return __gmp_expr
+ <__gmp_binary_expr >(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
+mpf_class & mpf_class::operator=(const __gmp_expr &expr)
+@{
+ expr.eval(this->get_mpf_t(), this->precision());
+ return *this;
+@}
+
+template
+void __gmp_expr<__gmp_binary_expr >::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
+__gmp_expr
+<__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> >
+operator+(const __gmp_expr &expr1, const __gmp_expr &expr2)
+@{
+ return __gmp_expr
+ <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> >
+ (expr1, expr2);
+@}
+@end example
+
+And the corresponding specializations of @code{__gmp_expr::eval}:
+
+@example
+template
+void __gmp_expr
+<__gmp_binary_expr<__gmp_expr, __gmp_expr, 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
-@unnumbered Contributors
+@appendix Contributors
@cindex Contributors
Torbjorn Granlund wrote the original GMP library and is still developing and
@@ -3843,18 +9055,18 @@ releases.
Joachim Hollman was involved in the design of the @code{mpf} interface, and in
the @code{mpz} design revisions for version 2.
-Bennet Yee contributed the functions @code{mpz_jacobi} and @code{mpz_legendre}.
+Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
+@code{mpz_legendre}.
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 part
by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
System SOlving).
-GNU MP 2 was finished and released by SWOX AB (formerly known as TMG
-Datakonsult), Swedenborgsgatan 23, SE-118 27 STOCKHOLM, SWEDEN, in
-cooperation with the IDA Center for Computing Sciences, USA.
+GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the
+IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England, suggested clever
improvements for population count.
@@ -3867,22 +9079,29 @@ Torsten Ekedahl of the Mathematical department of Stoc
significant inspiration during several phases of the GMP development. His
mathematical expertise helped improve several algorithms.
-Paul Zimmermann wrote the Burnikel-Ziegler division code, the REDC code, the
-REDC-based mpz_powm code, and the FFT multiply code. The ECMNET project Paul
-is organizing has been a driving force behind many of the optimization of GMP
-3.
+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 wrote a lot of very high quality x86 code, optimized for most CPU
-variants. He also made countless other valuable contributions.
+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.
-GNU MP 3.1 was finished and released by Torbjorn Granlund and Kevin Ryde.
+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.
@@ -3890,65 +9109,146 @@ USA.
contributed to GMP but are not listed above, please tell @email{tege@@swox.com}
about the omission!)
-@node References, Concept Index, Contributors, Top
+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
-@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
+@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
-Donald E. Knuth, "The Art of Computer Programming", vol 2,
-"Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1988.
+Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate
+Texts in Mathematics number 138, Springer-Verlag, 1993.
+@texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen}
@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.
@item
-Richard M. Stallman, "Using and Porting GCC", Free Software Foundation, 1999,
+Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of
+Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}
+
+@item
+Richard M. Stallman, ``Using and Porting GCC'', Free Software Foundation, 1999,
available online @uref{http://www.gnu.org/software/gcc/onlinedocs/}, and in
-the GCC package @uref{ftp://ftp.gnu.org/pub/gnu/gcc/}.
+the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/}
+@end itemize
+@section Papers
+
+@itemize @bullet
@item
-Peter L. Montgomery, "Modular Multiplication Without Trial Division", in
-Mathematics of Computation, volume 44, number 170, April 1985.
+Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'',
+Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{}
+@uref{http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022}
@item
-Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
-Integers using Multiplication", in Proceedings of the SIGPLAN
-PLDI'94 Conference, June 1994. Available online, @*
-@uref{ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz} (and .psl.gz too).
+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,
-"An algorithm for exact division",
+``An algorithm for exact division'',
Journal of Symbolic Computation,
-v. 15, 1993, pp. 169-180.
-Research report version available online @*
+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
-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,
-v. 21 (March), 1995, pp. 111-122.
+volume 21, number 1, March 1995, pp. 111-122.
@item
-Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
-Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @*
-@uref{http://www.mpi-sb.mpg.de/~ziegler/TechRep.ps.gz}.
+Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805,
+November 1999, @uref{http://www.inria.fr/RRRT/RR-3805.html}
@item
-Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of
-Applied Cryptography", @uref{http://cacr.math.uwaterloo.ca/hac/}.
+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
-Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate
-Texts in Mathematics number 138, Springer-Verlag, 1993. Errata available
-online @* @uref{http://www.math.u-bordeaux.fr/~cohen}
+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
-@node Concept Index, Function Index, References, 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
@unnumbered Concept Index
@printindex cp
@@ -3958,10 +9258,9 @@ online @* @uref{http://www.math.u-bordeaux.fr/~cohen}
@unnumbered Function and Type Index
@printindex fn
-
-@contents
@bye
@c Local variables:
@c fill-column: 78
+@c compile-command: "make gmp.info"
@c End: