=================================================================== RCS file: /home/cvs/OpenXM_contrib/gmp/Attic/gmp.info-1,v retrieving revision 1.1 retrieving revision 1.1.1.2 diff -u -p -r1.1 -r1.1.1.2 --- OpenXM_contrib/gmp/Attic/gmp.info-1 2000/01/10 15:35:21 1.1 +++ OpenXM_contrib/gmp/Attic/gmp.info-1 2000/09/09 14:12:18 1.1.1.2 @@ -1,15 +1,15 @@ -This is Info file gmp.info, produced by Makeinfo-1.64 from the input -file gmp.texi. +This is gmp.info, produced by makeinfo version 4.0 from gmp.texi. +INFO-DIR-SECTION GNU libraries START-INFO-DIR-ENTRY -* gmp: (gmp.info). GNU Multiple Precision Arithmetic Library. +* gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY This file documents GNU MP, a library for arbitrary-precision arithmetic. - Copyright (C) 1991, 1993, 1994, 1995, 1996 Free Software Foundation, -Inc. + Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000 +Free Software Foundation, Inc. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are @@ -32,29 +32,30 @@ GNU MP ****** This manual documents how to install and use the GNU multiple -precision arithmetic library, version 2.0.2. +precision arithmetic library, version 3.1. * Menu: -* Copying:: GMP Copying Conditions (LGPL). -* Introduction to MP:: Brief introduction to GNU MP. -* Installing MP:: How to configure and compile the MP library. -* MP Basics:: What every MP user should now. -* 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. -* BSD Compatible Functions:: All functions found in BSD MP. -* Custom Allocation:: How to customize the internal allocation. +* 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. +* 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. +* BSD Compatible Functions:: All functions found in BSD MP. +* Custom Allocation:: How to customize the internal allocation. -* Contributors:: -* References:: +* Contributors:: Who brings your this library? +* References:: Some useful papers and books to read. * Concept Index:: * Function Index::  -File: gmp.info, Node: Copying, Next: Introduction to MP, Prev: Top, Up: Top +File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top GNU MP Copying Conditions ************************* @@ -89,7 +90,7 @@ found in the Library General Public License that accom code.  -File: gmp.info, Node: Introduction to MP, Next: Installing MP, Prev: Copying, Up: Top +File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top Introduction to GNU MP ********************** @@ -101,185 +102,476 @@ that need higher precision than is directly supported types. Many applications use just a few hundred bits of precision; but some -applications may need thousands or even millions of bits. MP is +applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum. - The speed of MP is achieved by using fullwords as the basic + The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance). - There is carefully optimized assembly code for these CPUs: DEC -Alpha, Amd 29000, HPPA 1.0 and 1.1, Intel Pentium and generic x86, -Intel i960, Motorola MC68000, MC68020, MC88100, and MC88110, -Motorola/IBM PowerPC, National NS32000, IBM POWER, MIPS R3000, R4000, -SPARCv7, SuperSPARC, generic SPARCv8, and DEC VAX. Some optimizations -also for ARM, Clipper, IBM ROMP (RT), and Pyramid AP/XP. + There is carefully optimized assembly code for these CPUs: ARM, DEC +Alpha 21064, 21164, and 21264, AMD 29000, AMD K6 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, Motorola MC68000, MC68020, +MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, +IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, +UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for +Clipper, IBM ROMP (RT), and Pyramid AP/XP. - This version of MP is released under a more liberal license than -previous versions. It is now permitted to link MP to non-free -programs, as long as MP source code is provided when distributing the -non-free program. + There is a mailing list for GMP users. To join it, send a mail to + with the word `subscribe' in the message *body* +(not in the subject line). + For up-to-date information on GMP, please see the GMP Home Pages at +`http://www.swox.com/gmp/'. + How to use this Manual ====================== - Everyone should read *Note MP Basics::. If you need to install the -library yourself, you need to read *Note Installing MP::, too. + Everyone should read *Note GMP Basics::. If you need to install the +library yourself, you need to read *Note Installing GMP::, too. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.  -File: gmp.info, Node: Installing MP, Next: MP Basics, Prev: Introduction to MP, Up: Top +File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top -Installing MP -************* +Installing GMP +************** - To build MP, you first have to configure it for your CPU and -operating system. You need a C compiler, preferably GCC, but any -reasonable compiler should work. And you need a standard Unix `make' -program, plus some other standard Unix utility programs. +GMP has an autoconf/automake/libtool based configuration system. On a +Unix-like system a basic build can be done with - (If you're on an MS-DOS machine, your can build MP using `make.bat'. -It requires that djgpp is installed. It does not require -configuration, nor is `make' needed; `make.bat' both configures and -builds the library.) + ./configure + make - Here are the steps needed to install the library on Unix systems: +Some self-tests can be run with - 1. In most cases, `./configure --target=cpu-vendor-os', should work - both for native and cross-compilation. If you get error messages, - your machine might not be supported. + make check - If you want to compile in a separate object directory, cd to that - directory, and prefix the configure command with the path to the - MP source directory. Not all `make' programs have the necessary - features to support this. In particular, SunOS and Slowaris - `make' have bugs that makes them unable to build from a separate - object directory. Use GNU `make' instead. +And you can install (under `/usr/local' by default) with - In addition to the standard cpu-vendor-os tuples, MP recognizes - sparc8 and supersparc as valid CPU names. Specifying these CPU - names for relevant systems will improve performance significantly. + make install +If you experience problems, please report them to . +(*Note Reporting Bugs::, for information on what to include in useful +bug reports.) + +* Menu: + +* Build Options:: +* ABI and ISA:: +* Notes for Package Builds:: +* Notes for Particular Systems:: +* Known Build Problems:: + + +File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP + +Build Options +============= + +All the usual autoconf configure options are available, run `./configure +--help' for a summary. + +Non-Unix Systems + `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 `configure', certainly all the code is there, + but unfortunately you'll be on your own. + +Object Directory + To compile in a separate object directory, `cd' to that directory, + and prefix the configure command with the path to the GMP source + directory. For example `../src/gmp/configure'. Not all `make' + programs have the necessary features (`VPATH') to support this. + In particular, SunOS and Slowaris `make' have bugs that make them + unable to build from a separate object directory. Use GNU `make' + instead. + +`--disable-shared', `--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. + +`--target=CPU-VENDOR-OS' + The build target can be specified in the usual way, for either + native or cross compilation. + + If `--target' isn't given, `./configure' builds for the host + system as determined by `./config.guess'. On some systems this + can't distinguish between different CPUs in a family, and you + should check the guess. Running `./config.guess' on the target + system will also show the relevant `VENDOR-OS', if you don't + already know what it should be. + In general, if you want a library that runs as fast as possible, - you should make sure you configure MP for the exact CPU type your - system uses. + 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. - If you have `gcc' in your `PATH', it will be used by default. To - override this, pass `-with-gcc=no' to `configure'. + The following CPU targets have specific assembly code support. See + `configure.in' for which `mpn' subdirectories get used by each. - 2. `make' + * Alpha: `alpha', `alphaev5', `alphaev6' - This will compile MP, and create a library archive file `libgmp.a' - in the working directory. + * Hitachi: `sh', `sh2' - 3. `make check' + * HPPA: `hppa1.0', `hppa1.1', `hppa2.0', `hppa2.0w' - This will make sure MP was built correctly. If you get error - messages, please report this to `bug-gmp@prep.ai.mit.edu'. (*Note - Reporting Bugs::, for information on what to include in useful bug - reports.) + * MIPS: `mips', `mips3', - 4. `make install' + * Motorola: `m68000', `m68k', `m88k', `m88110' - This will copy the file `gmp.h' and `libgmp.a', as well as the info - files, to `/usr/local' (or if you passed the `--prefix' option to - `configure', to the directory given as argument to `--prefix'). + * POWER: `power1', `power2', `power2sc', `powerpc', `powerpc64' -If you wish to build and install the BSD MP compatible functions, use -`make libmp.a' and `make install-bsdmp'. + * SPARC: `sparc', `sparcv8', `microsparc', `supersparc', + `sparcv9', `ultrasparc', `sparc64' - There are some other useful make targets: + * 80x86 family: `i386', `i486', `i586', `pentium', `pentiummmx', + `pentiumpro', `pentium2', `pentium3', `k6', `k62', `k63', + `athlon' - * `doc' + * Other: `a29k', `arm', `clipper', `i960', `ns32k', `pyramid', + `vax', `z8k' - Create a DVI version of the manual, in `gmp.dvi' and a set of info - files, in `gmp.info', `gmp.info-1', `gmp.info-2', etc. + CPUs not listed use generic C code. If some of the assembly code + causes problems, the generic C code can be selected with CPU + `none'. - * `ps' +`CC', `CFLAGS' + The C compiler used is chosen from among some likely candidates, + with GCC normally preferred if it's present. The usual + `CC=whatever' can be passed to `./configure' to choose something + different. - Create a Postscript version of the manual, in `gmp.ps'. + For some configurations specific compiler flags are set based on + the target CPU and compiler, for others `CFLAGS="-whatever"' can + be used to set the best flags. - * `html' + If `CC' is set then `CFLAGS' must also be set. This applies even + if `CC' is merely one of the choices GMP would make itself. This + may change in a future release. - Create a HTML version of the manual, in `gmp.html'. +`--disable-alloca' + By default, GMP allocates temporary workspace using `alloca' if + that function is available, or `malloc' if not. If you're working + with large numbers and `alloca' overflows the available stack + space, you can build with `--disable-alloca' to use `malloc' + instead. `malloc' will probably be slightly slower than `alloca'. - * `clean' + When not using `alloca', it's actually the allocation function + selected with `mp_set_memory_functions' that's used, this being + `malloc' by default. *Note Custom Allocation::. - Delete all object files and archive files, but not the - configuration files. + 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 `limit', `ulimit' or + `setrlimit', or under DJGPP with `stubedit' or `_stklen'. - * `distclean' +`--enable-fft' + 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. - Delete all files not included in the distribution. +`--enable-mpbsd' + The Berkeley MP compatibility library (`libmp.a') and header file + (`mp.h') are built and installed only if `--enable-mpbsd' is used. + *Note BSD Compatible Functions::. - * `uninstall' +`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 `sparcv8' has path `"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 `MPN_PATH="dir list"'. This will normally be unnecessary + because all sensible paths should be available under one or other + CPU target. - Delete all files copied by `make install'. +Demonstration Programs + The `demos' subdirectory has some sample programs using GMP. These + aren't built or installed, but there's a `Makefile' with rules for + them. For instance, `make pexpr' and then `./pexpr 68^975+10'. +Documentation + The document you're now reading is `gmp.texi'. The usual automake + targets are available to make `gmp.ps' and/or `gmp.dvi'. Some + supplementary notes can be found in the `doc' subdirectory. + + +File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP + +ABI and ISA +=========== + + ABI (Application Binary Interface) refers to the calling conventions +between functions, meaning what registers are used and what sizes the +various C data types are. ISA (Instruction Set Architecture) refers to +the instructions and registers a CPU has available. + + Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI +defined, the latter for compatibility with older CPUs in the family. +GMP chooses the best ABI available for a given target system, and this +generally gives significantly greater speed. + + 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. + + Some of what's described in this section may change in future +releases of GMP. + +HPPA 2.0 + CPU target `hppa2.0' uses the hppa2.0n 32-bit ABI, but either a + 32-bit or 64-bit limb. + + A 64-bit limb is available on HP-UX 10 or up when using `c89'. No + `gcc' support is planned for 64-bit operations in this ABI. + Applications must be compiled with the same options as GMP, which + means + + c89 +DA2.0 +e -D_LONG_LONG_LIMB + + A 32-bit limb is used in other cases, and no special compiler + options are needed. + + CPU target `hppa2.0w' uses the hppa2.0w 64-bit ABI, which is + available on HP-UX 11 or up when using `c89'. `gcc' support for + this is in progress. Applications must be compiled for the same + ABI, which means + + c89 +DD64 + +MIPS 3 and 4 under IRIX 6 + Targets `mips*-*-irix6*' use the n32 ABI and a 64-bit limb. + Applications must be compiled for the same ABI, which means either + + gcc -mabi=n32 + cc -n32 + +PowerPC 64 + CPU target `powerpc64' uses either the 32-bit ABI or the AIX + 64-bit ABI. The latter is used on targets `powerpc64-*-aix*' and + applications must be compiled using either + + gcc -maix64 + xlc -q64 + + On other systems the 32-bit ABI is used, but with 64-bit limbs + provided by `long long' in `gcc'. Applications must be compiled + using + + gcc -D_LONG_LONG_LIMB + +Sparc V9 + On a sparc v9 CPU, either the v8plus 32-bit ABI or v9 64-bit ABI + is used. Targets `ultrasparc*-*-solaris2.[7-9]', + `sparcv9-*-solaris2.[7-9]' and `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. + + For the v8plus ABI, applications can be compiled with either + + gcc -mv8plus + cc -xarch=v8plus + + For the v9 ABI, applications must be compiled with either + + gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 + cc -xarch=v9 + + Don't be confused by the names of these options, they're called + `arch' but they effectively control the ABI. + + +File: gmp.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing GMP + +Notes for Package Builds +======================== + + GMP should present no great difficulties for packaging in a binary +distribution. + + Libtool is used to build the library and `-version-info' is set +appropriately, having started from `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. + + When building a package for a CPU family, care should be taken to use +`--target' to choose the least common denominator among the CPUs which +might use the package. For example this might necessitate `i386' for +x86s, or plain `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 `--target' in a build so +`./config.guess' will detect the CPU. But a way to manually specify a +`--target' will be wanted for systems where `./config.guess' is inexact. + + +File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP + +Notes for Particular Systems +============================ + +AIX 4.3 + Targets `*-*-aix4.[3-9]*' have shared libraries disabled since + they seem to fail on AIX 4.3. + +OpenBSD 2.6 + `m4' in this release of OpenBSD has a bug in `eval' that makes it + unsuitable for `.asm' file processing. `./configure' will detect + the problem and either abort or choose another m4 in the `PATH'. + The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4. + +Sparc V8 + Using CPU target `sparcv8' or `supersparc' on relevant systems will + give a significant performance increase over the V7 code. + +SunOS 4 + `/usr/bin/m4' lacks various features needed to process `.asm' + files, and instead `./configure' will automatically use + `/usr/5bin/m4', which we believe is always available (if not then + use GNU m4). + +x86 Pentium and PentiumPro + 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). `i386' + is a better choice if you're making binaries that must run on both. + +x86 MMX and old GAS + 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). + + 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. + +x86 GCC 2.95.2 `-march=pentiumpro' + GCC 2.95.2 miscompiles `mpz/powm.c' when `-march=pentiumpro' is + used, so that option is omitted from the `CFLAGS' chosen for + relevant CPUs. The problem is believed to be fixed in GCC 2.96. + + +File: gmp.info, Node: Known Build Problems, Prev: Notes for Particular Systems, Up: Installing GMP + Known Build Problems ==================== - GCC 2.7.2 (as well as 2.6.3) for the RS/6000 and PowerPC can not be -used to compile MP, due to a bug in GCC. If you want to use GCC for -these machines, you need to apply the patch below to GCC, or use a -later version of the compiler. + You might find more up-to-date information at +`http://www.swox.com/gmp/'. - If you are on a Sequent Symmetry, use the GNU assembler instead of -the system's assembler, since the latter has serious bugs. +Generic C on a 64-bit system + When making a generic C build using `--target=none' on a 64-bit + system (meaning where `unsigned long' is 64 bits), + `BITS_PER_MP_LIMB', `BITS_PER_LONGINT' and `BYTES_PER_MP_LIMB' in + `mpn/generic/gmp-mparam.h' need to be changed to 64 and 8. This + will hopefully be automated in a future version of GMP. - The system compiler on NeXT is a massacred and old gcc, even if the -compiler calls itself `cc'. This compiler cannot be used to build MP. -You need to get a real gcc, and install that before you compile MP. -(NeXT might have fixed this in newer releases of their system.) +NeXT prior to 3.3 + The system compiler on old versions of NeXT was a massacred and + old GCC, even if it called itself `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.) - The system C compiler under SunOS 4 has a bug that makes it -miscompile mpq/get_d.c. This will make `make check' fail. +POWER and PowerPC + Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP + on POWER or PowerPC. If you want to use GCC for these machines, + get GCC 2.7.2.1 (or later). - Please report other problems to `bug-gmp@prep.ai.mit.edu'. *Note -Reporting Bugs::. +Sequent Symmetry + Use the GNU assembler instead of the system assembler, since the + latter has serious bugs. - Patch to apply to GCC 2.6.3 and 2.7.2: +Stripped Libraries + GNU binutils `strip' should not be used on the static libraries + `libgmp.a' and `libmp.a', neither directly nor via `make + install-strip'. It can be used on the shared libraries + `libgmp.so' and `libmp.so' though. - *** config/rs6000/rs6000.md Sun Feb 11 08:22:11 1996 - --- config/rs6000/rs6000.md.new Sun Feb 18 03:33:37 1996 - *************** - *** 920,926 **** - (set (match_operand:SI 0 "gpc_reg_operand" "=r") - (not:SI (match_dup 1)))] - "" - ! "nor. %0,%2,%1" - [(set_attr "type" "compare")]) - - (define_insn "" - --- 920,926 ---- - (set (match_operand:SI 0 "gpc_reg_operand" "=r") - (not:SI (match_dup 1)))] - "" - ! "nor. %0,%1,%1" - [(set_attr "type" "compare")]) - - (define_insn "" + Currently (binutils 2.10.0), `strip' extracts archives into a + single directory, but GMP contains multiple object files of the + same name (eg. three versions of `init.o'), and they overwrite + each other, leaving only the one that happens to be last. + If stripped static libraries are wanted, the suggested workaround + is to build normally, strip the separate object files, and do + another `make all' to rebuild. Alternately `CFLAGS' with `-g' + omitted can always be used if it's just debugging which is + unwanted. + +SunOS 4 Native Tools + The setting for `GSYM_PREFIX' in `config.m4' may be incorrectly + determined when using the native `grep', leading at link-time to + undefined symbols like `___gmpn_add_n'. To fix this, after running + `./configure', change the relevant line in `config.m4' to + `define(, <_>)'. + + The `ranlib' command will need to be run manually when building a + static library with the native `ar'. After `make', run `ranlib + .libs/libgmp.a', and when using `--enable-mpbsd' run `ranlib + .libs/libmp.a' too. + +VAX running Ultrix + You need to build and install the GNU assembler before you compile + GMP. The VAX assembly in GMP uses an instruction (`jsobgtr') that + cannot be assembled by the Ultrix assembler. +  -File: gmp.info, Node: MP Basics, Next: Reporting Bugs, Prev: Installing MP, Up: Top +File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top -MP Basics -********* +GMP Basics +********** - All declarations needed to use MP are collected in the include file + All declarations needed to use GMP are collected in the include file `gmp.h'. It is designed to work with both C and C++ compilers. + *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. + + +File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: GMP Basics, Up: GMP Basics + Nomenclature and Types ====================== In this manual, "integer" usually means a multiple precision integer, as -defined by the MP library. The C data type for such integers is +defined by the GMP library. The C data type for such integers is `mpz_t'. Here are some examples of how to declare such integers: mpz_t sum; @@ -294,7 +586,7 @@ for these fractions is `mpq_t'. For example: mpq_t quotient; "Floating point number" or "Float" for short, is an arbitrary precision -mantissa with an limited precision exponent. The C data type for such +mantissa with a limited precision exponent. The C data type for such objects is `mpf_t'. A "limb" means the part of a multi-precision number that fits in a @@ -303,10 +595,13 @@ analogous to a digit, only larger, and containing seve Normally a limb contains 32 or 64 bits. The C data type for a limb is `mp_limb_t'. + +File: gmp.info, Node: Function Classes, Next: GMP Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics + Function Classes ================ - There are six classes of functions in the MP library: + There are six classes of functions in the GMP library: 1. Functions for signed integer arithmetic, with names beginning with `mpz_'. The associated type is `mpz_t'. There are about 100 @@ -322,7 +617,7 @@ Function Classes `mpf_'. The associated type is `mpf_t'. There are about 50 functions is this class. - 4. Functions compatible with Berkeley MP, such as `itom', `madd', and + 4. Functions compatible with Berkeley GMP, such as `itom', `madd', and `mult'. The associated type is `MINT'. 5. Fast low-level functions that operate on natural numbers. These @@ -334,22 +629,26 @@ Function Classes The associated type is array of `mp_limb_t'. 6. Miscellaneous functions. Functions for setting up custom - allocation. + allocation and functions for generating random numbers. -MP Variable Conventions -======================= + +File: gmp.info, Node: GMP Variable Conventions, Next: GMP and Reentrancy, Prev: Function Classes, Up: GMP Basics - As a general rule, all MP functions expect output arguments before +GMP Variable Conventions +======================== + + 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.) - MP allows you to use the same variable for both input and output in -the same expression. For example, the main function for integer -multiplication, `mpz_mul', can be used like this: `mpz_mul (x, x, x)'. -This computes the square of X and puts the result back in X. + GMP lets you use the same variable for both input and output in one +call. For example, the main function for integer multiplication, +`mpz_mul', can be used to square `x' and put the result back in `x' with - Before you can assign to an MP variable, you need to initialize it + mpz_mul (x, x, x); + + Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type @@ -360,17 +659,93 @@ functions, and floating-point functions for details. between each initialization. After a variable has been initialized, it may be assigned to any number of times. - For efficiency reasons, avoid to initialize and clear out a variable -in loops. Instead, initialize it before entering the loop, and clear -it out after the loop has exited. + 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. - You don't need to be concerned about allocating additional space for -MP variables. All functions in MP automatically allocate additional -space when a variable does not already have enough space. They do not, -however, reduce the space when a smaller number is stored in the -object. Most of the time, this policy is best, since it avoids -frequent re-allocation. + 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. + When a variable of type `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 `mpz_t' result, it should provide a separate +parameter or parameters that it sets, like the GMP library functions +do. A `return' of an `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 `mpq_t' and `mpf_t' too. + + Here's an example function accepting an `mpz_t' parameter, doing a +certain calculation, and returning a result. + + void + myfunction (mpz_t result, mpz_t param, unsigned long n) + { + unsigned long i; + + mpz_mul_ui (result, param, n); + for (i = 1; i < n; i++) + mpz_add_ui (result, result, i*7); + } + + int + main (void) + { + mpz_t r, n; + mpz_init (r); + mpz_init_set_str (n, "123456", 0); + + myfunction (r, n, 20L); + mpz_out_str (stdout, 10, r); printf ("\n"); + + return 0; + } + + This example will work if `result' and `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. + + `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 +`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. + + +File: gmp.info, Node: GMP and Reentrancy, Next: Useful Macros and Constants, Prev: GMP Variable Conventions, Up: GMP Basics + +GMP and Reentrancy +================== + + The GMP code is reentrant and thread-safe, with some exceptions: + + * The function `mpf_set_default_prec' saves the selected precision in + a global variable. + + * The function `mp_set_memory_functions' uses several global + variables for storing the selected memory allocation functions. + + * If the memory allocation functions set by a call to + `mp_set_memory_functions' (or `malloc' and friends by default) are + not reentrant, GMP will not be reentrant either. + + * The old random number functions (`mpz_random', etc) use a random + number generator from the C library, usually `mrand48' or + `random'. These routines are not reentrant, since they rely on + global state. (However the newer random number functions that + accept a `gmp_randstate_t' parameter are reentrant.) + + +File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: GMP and Reentrancy, Up: GMP Basics + Useful Macros and Constants =========================== @@ -379,87 +754,96 @@ Useful Macros and Constants - Macro: __GNU_MP_VERSION - Macro: __GNU_MP_VERSION_MINOR - The major and minor MP version, respectively, as integers. + - Macro: __GNU_MP_VERSION_PATCHLEVEL + The major and minor GMP version, and patch level, respectively, as + integers. For GMP i.j, these numbers will be i, j, and 0, + respectively. For GMP i.j.k, these numbers will be i, j, and k, + respectively. -Compatibility with Version 1.x -============================== + +File: gmp.info, Node: Compatibility with older versions, Next: Getting the Latest Version of GMP, Prev: Useful Macros and Constants, Up: GMP Basics - This version of MP is upward compatible with previous versions of -MP, with a few exceptions. +Compatibility with older versions +================================= - 1. Integer division functions round the result differently. The old - functions (`mpz_div', `mpz_divmod', `mpz_mdiv', `mpz_mdivmod', - etc) now all use floor rounding (i.e., they round the quotient to - -infinity). There are a lot of new functions for integer - division, giving the user better control over the rounding. + 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. - 2. The function `mpz_mod' now compute the true *mod* function. + * `mpn_gcd' had its source arguments swapped as of GMP 3.0 for + consistency with other `mpn' functions. - 3. The functions `mpz_powm' and `mpz_powm_ui' now use *mod* for - reduction. + * `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. - 4. The assignment functions for rational numbers do no longer - canonicalize their results. In the case a non-canonical result - could arise from an assignment, the user need to insert an - explicit call to `mpq_canonicalize'. This change was made for - efficiency. - 5. Output generated by `mpz_out_raw' in this release cannot be read - by `mpz_inp_raw' in previous releases. This change was made for - making the file format truly portable between machines with - different word sizes. + 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 see the GMP 2 manual for details. - 6. Several `mpn' functions have changed. But they were intentionally - undocumented in previous releases. + +File: gmp.info, Node: Getting the Latest Version of GMP, Prev: Compatibility with older versions, Up: GMP Basics - 7. The functions `mpz_cmp_ui', `mpz_cmp_si', and `mpq_cmp_ui' are now - implementated as macros, and thereby sometimes evaluate their - arguments multiple times. +Getting the Latest Version of GMP +================================= - 8. The functions `mpz_pow_ui' and `mpz_ui_pow_ui' now yield 1 for - 0^0. (In version 1, they yielded 0.) + The latest version of the GMP library is available at +`ftp://ftp.gnu.org/pub/gnu/gmp'. Many sites around the world mirror +`ftp.gnu.org'; please use a mirror site near you, see +`http://www.gnu.org/order/ftp.html'. - -Getting the Latest Version of MP -================================ - - The latest version of the MP library is available by anonymous ftp -from from `prep.ai.mit.edu'. The file name is -`/pub/gnu/gmp-M.N.tar.gz'. Many sites around the world mirror `prep'; -please use a mirror site near you. -  -File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: MP Basics, Up: Top +File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top Reporting Bugs ************** - If you think you have found a bug in the MP library, please + 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 to ask too much from you, to ask you to report the -bugs that you find. +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 +`http://www.swox.com/gmp/' for patches for this release. - There are a few things you should think about when you put your bug -report together. + Please include the following in any report, - You have to send us a test case that makes it possible for us to -reproduce the bug. Include instructions on how to run the test case. + * The GMP version number, and if pre-packaged or patched then say so. - You also have to explain what is wrong; if you get a crash, or if -the results printed are incorrect and in that case, in what way. + * A test program that makes it possible for us to reproduce the bug. + Include instructions on how to run the program. + * A description of what is wrong. If the results are incorrect, in + what way. If you get a crash, say so. + + * If you get a crash, include a stack backtrace from the debugger if + it's informative (`where' in `gdb', or `$C' in `adb'). + + * *Please do not send core dumps, executables or `strace's.* + + * The configuration options you used when building GMP, if any. + + * The name of the compiler and its version. For `gcc', get the + version with `gcc -v', otherwise perhaps `what `which cc`', or + similar. + + * The output from running `uname -a'. + + * The output from running `./config.guess'. + + * If the bug is related to `configure', then the contents of + `config.log'. + + * If the bug is related to an `asm' file not assembling, then the + contents of `config.m4'. + It is not uncommon that an observed problem is actually due to a bug -in the compiler used when building MP; the MP code tends to explore -interesting corners in compilers. Therefore, please include compiler -version information in your bug report. This can be extracted using -`what `which cc`', or, if you're using gcc, `gcc -v'. Also, include -the output from `uname -a'. +in the compiler; the GMP code tends to explore interesting corners in +compilers. - If your bug report is good, we will do our best to help you to get a + If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't -do anything about it (aside of chiding you to send better bug reports). +do anything about it (except maybe ask you to send a better report). - Send your bug report to: `bug-gmp@prep.ai.mit.edu'. + Send your report to: . If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note @@ -471,10 +855,10 @@ File: gmp.info, Node: Integer Functions, Next: Ratio Integer Functions ***************** - This chapter describes the MP functions for performing integer + This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix `mpz_'. - Arbitrary precision integers are stored in objects of type `mpz_t'. + GMP integers are stored in objects of type `mpz_t'. * Menu: @@ -483,16 +867,21 @@ arithmetic. These functions start with the prefix `mp * Simultaneous Integer Init & Assign:: * Converting Integers:: * Integer Arithmetic:: -* Comparison Functions:: +* Integer Division:: +* Integer Exponentiation:: +* Integer Roots:: +* Number Theoretic Functions:: +* Integer Comparisons:: * Integer Logic and Bit Fiddling:: * I/O of Integers:: +* Integer Random Numbers:: * Miscellaneous Integer Functions::  -File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Up: Integer Functions +File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions -Initialization and Assignment Functions -======================================= +Initialization Functions +======================== The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function `mpz_init'. @@ -548,10 +937,10 @@ object is initialized. File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions Assignment Functions --------------------- +==================== These functions assign new values to already initialized integers -(*note Initializing Integers::.). +(*note Initializing Integers::). - Function: void mpz_set (mpz_t ROP, mpz_t OP) - Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP) @@ -573,15 +962,26 @@ Assignment Functions This function returns 0 if the entire string up to the '\0' is a valid number in base BASE. Otherwise it returns -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?] + + - Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2) + Swap the values ROP1 and ROP2 efficiently. +  File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions Combined Initialization and Assignment Functions ------------------------------------------------- +================================================ - For convenience, MP provides a parallel series of initialize-and-set -functions which initialize the output and then store the value there. -These functions' names have the form `mpz_init_set...' + For convenience, GMP provides a parallel series of +initialize-and-set functions which initialize the output and then store +the value there. These functions' names have the form `mpz_init_set...' Here is an example of using one: @@ -620,16 +1020,22 @@ File: gmp.info, Node: Converting Integers, Next: Int Conversion Functions ==================== - This section describes functions for converting arbitrary precision -integers to standard C types. Functions for converting *to* arbitrary -precision integers are described in *Note Assigning Integers:: and -*Note I/O of Integers::. + This section describes functions for converting GMP integers to +standard C types. Functions for converting _to_ GMP integers are +described in *Note Assigning Integers:: and *Note I/O of Integers::. + - Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N) + Return limb #N from OP. This function allows for very efficient + decomposition of a number in its limbs. + + The function `mpz_size' can be used to determine the useful range + for N. + - Function: unsigned long int mpz_get_ui (mpz_t OP) Return the least significant part from OP. This function combined with `mpz_tdiv_q_2exp(..., OP, CHAR_BIT*sizeof(unsigned long int))' can - be used to extract the limbs of an integer. + be used to decompose an integer into unsigned longs. - Function: signed long int mpz_get_si (mpz_t OP) If OP fits into a `signed long int' return the value of OP. @@ -637,7 +1043,8 @@ precision integers are described in *Note Assigning In sign as OP. If OP is too large to fit in a `signed long int', the returned - result is probably not very useful. + result is probably not very useful. To find out if the value will + fit, use the function `mpz_fits_slong_p'. - Function: double mpz_get_d (mpz_t OP) Convert OP to a double. @@ -646,18 +1053,21 @@ precision integers are described in *Note Assigning In Convert OP to a string of digits in base BASE. The base may vary from 2 to 36. - If STR is NULL, space for the result string is allocated using the - default allocation function, and a pointer to the string is - returned. + If STR is `NULL', space for the result string is allocated using + the default allocation function. - If STR is not NULL, it should point to a block of storage enough + If STR is not `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 STR, use `mpz_sizeinbase (OP, BASE) + 2'. The two extra bytes are for a possible minus sign, and for the terminating null character. + A pointer to the result string is returned. This pointer will + will either equal STR, or if that is `NULL', will point to the + allocated storage. +  -File: gmp.info, Node: Integer Arithmetic, Next: Comparison Functions, Prev: Converting Integers, Up: Integer Functions +File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions Arithmetic Functions ==================== @@ -673,10 +1083,15 @@ Arithmetic Functions Set ROP to OP1 - OP2. - Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2) + - Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2) - Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. + - Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long + int OP2) + Add OP1 times OP2 to ROP. + - Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. This operation can also be @@ -688,26 +1103,26 @@ Arithmetic Functions - Function: void mpz_abs (mpz_t ROP, mpz_t OP) Set ROP to the absolute value of OP. - - Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP) - Set ROP to OP!, the factorial of OP. + +File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions -Division functions ------------------- +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 `mpz_powm' and `mpz_powm_ui' functions, will make these -functions intentionally divide by zero. This gives the user the -possibility to handle arithmetic exceptions in these functions in the -same manner as other arithmetic exceptions. +functions intentionally divide by zero. This lets the user handle +arithmetic exceptions in these functions in the same manner as other +arithmetic exceptions. There are three main groups of division functions: - * Functions that truncate the quotient towards 0. The names of these - functions start with `mpz_tdiv'. The `t' in the name is short for - `truncate'. + * Functions that truncate the quotient towards 0. The names of + these functions start with `mpz_tdiv'. The `t' in the name is + short for `truncate'. - * Functions that round the quotient towards -infinity. The names of - these routines start with `mpz_fdiv'. The `f' in the name is + * Functions that round the quotient towards -infinity). The names + of these routines start with `mpz_fdiv'. The `f' in the name is short for `floor'. * Functions that round the quotient towards +infinity. The names of @@ -719,565 +1134,135 @@ means that the quotient is computed, while `r' means t is computed. Functions that compute both the quotient and remainder have `qr' in the name. - - Function: void mpz_tdiv_q (mpz_t ROP, mpz_t OP1, mpz_t OP2) - - Function: void mpz_tdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to [OP1/OP2]. The quotient is truncated towards 0. + - Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N, + unsigned long int D) + Set Q to [N/D], truncated towards 0. - - Function: void mpz_tdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2) - - Function: void mpz_tdiv_r_ui (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to (OP1 - [OP1/OP2] * OP2). Unless the remainder is zero, - it has the same sign as the dividend. + The function `mpz_tdiv_q_ui' returns the absolute value of the true + remainder. - - Function: void mpz_tdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t - OP2) - - Function: void mpz_tdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, - unsigned long int OP2) - Divide OP1 by OP2 and put the quotient in ROP1 and the remainder - in ROP2. The quotient is rounded towards 0. Unless the remainder - is zero, it has the same sign as the dividend. + - Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N, + unsigned long int D) + Set R to (N - [N/D] * D), where the quotient is truncated towards + 0. Unless R becomes zero, it will get the same sign as N. - If ROP1 and ROP2 are the same variable, the results are undefined. + The function `mpz_tdiv_r_ui' returns the absolute value of the + remainder. - - Function: void mpz_fdiv_q (mpz_t ROP1, mpz_t OP1, mpz_t OP2) - - Function: void mpz_fdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to OP1/OP2. The quotient is rounded towards -infinity. + - Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t + N, unsigned long int D) + Set Q to [N/D], truncated towards 0. Set R to (N - [N/D] * D). + Unless R becomes zero, it will get the same sign as N. If Q and R + are the same variable, the results are undefined. - - Function: void mpz_fdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2) - - Function: unsigned long int mpz_fdiv_r_ui (mpz_t ROP, mpz_t OP1, - unsigned long int OP2) - Divide OP1 by OP2 and put the remainder in ROP. Unless the - remainder is zero, it has the same sign as the divisor. + The function `mpz_tdiv_qr_ui' returns the absolute value of the + remainder. - For `mpz_fdiv_r_ui' the remainder is small enough to fit in an - `unsigned long int', and is therefore returned. + - Function: unsigned long int mpz_tdiv_ui (mpz_t N, unsigned long int + D) + Like `mpz_tdiv_r_ui', but the remainder is not stored anywhere; its + absolute value is just returned. - - Function: void mpz_fdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t - OP2) - - Function: unsigned long int mpz_fdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, - mpz_t OP1, unsigned long int OP2) - Divide OP1 by OP2 and put the quotient in ROP1 and the remainder - in ROP2. The quotient is rounded towards -infinity. Unless the - remainder is zero, it has the same sign as the divisor. + - Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N, + unsigned long int D) + Set Q to N/D, rounded towards -infinity. - For `mpz_fdiv_qr_ui' the remainder is small enough to fit in an - `unsigned long int', and is therefore returned. + The function `mpz_fdiv_q_ui' returns the remainder. - If ROP1 and ROP2 are the same variable, the results are undefined. + - Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N, + unsigned long int D) + Set R to (N - N/D * D), where the quotient is rounded towards + -infinity. Unless R becomes zero, it will get the same sign as D. - - Function: unsigned long int mpz_fdiv_ui (mpz_t OP1, unsigned long - int OP2) - This function is similar to `mpz_fdiv_r_ui', but the remainder is - only returned; it is not stored anywhere. + The function `mpz_fdiv_r_ui' returns the remainder. - - Function: void mpz_cdiv_q (mpz_t ROP1, mpz_t OP1, mpz_t OP2) - - Function: void mpz_cdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to OP1/OP2. The quotient is rounded towards +infinity. + - Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t + N, unsigned long int D) + Set Q to N/D, rounded towards -infinity. Set R to (N - N/D * D). + Unless R becomes zero, it will get the same sign as D. If Q and R + are the same variable, the results are undefined. - - Function: void mpz_cdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2) - - Function: unsigned long int mpz_cdiv_r_ui (mpz_t ROP, mpz_t OP1, - unsigned long int OP2) - Divide OP1 by OP2 and put the remainder in ROP. Unless the - remainder is zero, it has the opposite sign as the divisor. + The function `mpz_fdiv_qr_ui' returns the remainder. - For `mpz_cdiv_r_ui' the negated remainder is small enough to fit - in an `unsigned long int', and it is therefore returned. + - Function: unsigned long int mpz_fdiv_ui (mpz_t N, unsigned long int + D) + Like `mpz_fdiv_r_ui', but the remainder is not stored anywhere; it + is just returned. - - Function: void mpz_cdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t - OP2) - - Function: unsigned long int mpz_cdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, - mpz_t OP1, unsigned long int OP2) - Divide OP1 by OP2 and put the quotient in ROP1 and the remainder - in ROP2. The quotient is rounded towards +infinity. Unless the - remainder is zero, it has the opposite sign as the divisor. + - Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N, + unsigned long int D) + Set Q to N/D, rounded towards +infinity. - For `mpz_cdiv_qr_ui' the negated remainder is small enough to fit - in an `unsigned long int', and it is therefore returned. + The function `mpz_cdiv_q_ui' returns the negated remainder. - If ROP1 and ROP2 are the same variable, the results are undefined. + - Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N, + unsigned long int D) + Set R to (N - N/D * D), where the quotient is rounded towards + +infinity. Unless R becomes zero, it will get the opposite sign + as D. - - Function: unsigned long int mpz_cdiv_ui (mpz_t OP1, unsigned long - int OP2) - Return the negated remainder, similar to `mpz_cdiv_r_ui'. (The - difference is that this function doesn't store the remainder - anywhere.) + The function `mpz_cdiv_r_ui' returns the negated remainder. - - Function: void mpz_mod (mpz_t ROP, mpz_t OP1, mpz_t OP2) - - Function: unsigned long int mpz_mod_ui (mpz_t ROP, mpz_t OP1, - unsigned long int OP2) - Set ROP to OP1 `mod' OP2. The sign of the divisor is ignored, and - the result is always non-negative. + - Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t + N, unsigned long int D) + Set Q to N/D, rounded towards +infinity. Set R to (N - N/D * D). + Unless R becomes zero, it will get the opposite sign as D. If Q + and R are the same variable, the results are undefined. - For `mpz_mod_ui' the remainder is small enough to fit in an - `unsigned long int', and is therefore returned. + The function `mpz_cdiv_qr_ui' returns the negated remainder. - - Function: void mpz_divexact (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Set ROP to OP1/OP2. This function produces correct results only - when it is known in advance that OP2 divides OP1. + - Function: unsigned long int mpz_cdiv_ui (mpz_t N, unsigned long int + D) + Like `mpz_tdiv_r_ui', but the remainder is not stored anywhere; its + negated value is just returned. - Since mpz_divexact is much faster than any of the other routines - that produce the quotient (*note 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. + - Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D) + - Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N, unsigned + long int D) + Set R to N `mod' D. The sign of the divisor is ignored; the + result is always non-negative. - - Function: void mpz_tdiv_q_2exp (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to OP1 divided by 2 raised to OP2. The quotient is - rounded towards 0. + The function `mpz_mod_ui' returns the remainder. - - Function: void mpz_tdiv_r_2exp (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Divide OP1 by (2 raised to OP2) and put the remainder in ROP. - Unless it is zero, ROP will have the same sign as OP1. + - Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D) + Set Q to N/D. This function produces correct results only when it + is known in advance that D divides N. - - Function: void mpz_fdiv_q_2exp (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Set ROP to OP1 divided by 2 raised to OP2. The quotient is - rounded towards -infinity. + Since mpz_divexact is much faster than any of the other routines + that produce the quotient (*note 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. - - Function: void mpz_fdiv_r_2exp (mpz_t ROP, mpz_t OP1, unsigned long - int OP2) - Divide OP1 by (2 raised to OP2) and put the remainder in ROP. The - sign of ROP will always be positive. + - Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int + D) + Set Q to N divided by 2 raised to D. The quotient is truncated + towards 0. - This operation can also be defined as masking of the OP2 least - significant bits. + - Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int + D) + Divide N by (2 raised to D), rounding the quotient towards 0, and + put the remainder in R. Unless it is zero, R will have the same + sign as N. -Exponentialization Functions ----------------------------- + - Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int + D) + Set Q to N divided by 2 raised to D, rounded towards -infinity. + This operation can also be defined as arithmetic right shift D bit + positions. - - Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t MOD) - - Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long int - EXP, mpz_t MOD) - Set ROP to (BASE raised to EXP) `mod' MOD. If EXP is negative, - the result is undefined. - - - Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int - EXP) - - Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE, - unsigned long int EXP) - Set ROP to BASE raised to EXP. The case of 0^0 yields 1. - -Square Root Functions ---------------------- - - - Function: void mpz_sqrt (mpz_t ROP, mpz_t OP) - Set ROP to the truncated integer part of the square root of OP. - - - Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP) - Set ROP1 to the truncated integer part of the square root of OP, - like `mpz_sqrt'. Set ROP2 to OP-ROP1*ROP1, (i.e., zero if OP is a - perfect square). - - If ROP1 and ROP2 are the same variable, the results are undefined. - - - Function: int mpz_perfect_square_p (mpz_t OP) - Return non-zero if OP is a perfect square, i.e., if the square - root of OP is an integer. Return zero otherwise. - -Number Theoretic Functions --------------------------- - - - Function: int mpz_probab_prime_p (mpz_t OP, int REPS) - If this function returns 0, OP is definitely not prime. If it - returns 1, then OP is `probably' prime. The probability of a - false positive is (1/4)**REPS. A reasonable value of reps is 25. - - An implementation of the probabilistic primality test found in - Seminumerical Algorithms (*note References::. Knuth). - - - Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Set ROP to the greatest common divisor of OP1 and OP2. - - - Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1, - unsigned long int OP2) - Compute the greatest common divisor of OP1 and OP2. If ROP is not - NULL, store the result there. - - If the result is small enough to fit in an `unsigned long int', it - is returned. If the result does not fit, 0 is returned, and the - result is equal to the argument OP1. Note that the result will - always fit if OP2 is non-zero. - - - Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t - B) - Compute G, S, and T, such that AS + BT = G = `gcd' (A, B). If T is - NULL, that argument is not computed. - - - Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Compute the inverse of OP1 modulo OP2 and put the result in ROP. - Return non-zero if an inverse exist, zero otherwise. When the - function returns zero, do not assume anything about the value in - ROP. - - - Function: int mpz_jacobi (mpz_t OP1, mpz_t OP2) - - Function: int mpz_legendre (mpz_t OP1, mpz_t OP2) - Compute the Jacobi and Legendre symbols, respectively. - - -File: gmp.info, Node: Comparison Functions, Next: Integer Logic and Bit Fiddling, Prev: Integer Arithmetic, Up: Integer Functions - -Comparison Functions -==================== - - - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2) - Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero - if OP1 = OP2, and a negative value if OP1 < OP2. - - - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2) - - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2) - Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero - if OP1 = OP2, and a negative value if OP1 < OP2. - - These functions are actually implemented as macros. They evaluate - their arguments multiple times. - - - Macro: int mpz_sgn (mpz_t OP) - Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. - - This function is actually implemented as a macro. It evaluates its - arguments multiple times. - - -File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Comparison Functions, Up: Integer Functions - -Logical and Bit Manipulation Functions -====================================== - - These functions behave as if two's complement arithmetic were used -(although sign-magnitude is used by the actual implementation). - - - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Set ROP to OP1 logical-and OP2. - - - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Set ROP to OP1 inclusive-or OP2. - - - Function: void mpz_com (mpz_t ROP, mpz_t OP) - Set ROP to the one's complement of OP. - - - Function: unsigned long int mpz_popcount (mpz_t OP) - For non-negative numbers, return the population count of OP. For - negative numbers, return the largest possible value (MAX_ULONG). - - - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2) - If OP1 and OP2 are both non-negative, return the hamming distance - between the two operands. Otherwise, return the largest possible - value (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 MAX_ULONG in this case. *Do not depend on - this behavior, since it will change in future versions of the - library.* - - - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int - STARTING_BIT) - Scan OP, starting with bit STARTING_BIT, towards more significant - bits, until the first clear bit is found. Return the index of the - found bit. - - - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int - STARTING_BIT) - Scan OP, starting with bit STARTING_BIT, towards more significant - bits, until the first set bit is found. Return the index of the - found bit. - - - Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX) - Set bit BIT_INDEX in OP1. - - - Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX) - Clear bit BIT_INDEX in OP1. - - -File: gmp.info, Node: I/O of Integers, Next: Miscellaneous Integer Functions, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions - -Input and Output Functions -========================== - - Functions that perform input from a stdio stream, and functions that -output to a stdio stream. Passing a NULL pointer for a STREAM argument -to any of these functions will make them read from `stdin' and write to -`stdout', respectively. - - When using any of these functions, it is a good idea to include -`stdio.h' before `gmp.h', since that will allow `gmp.h' to define -prototypes for these functions. - - - Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP) - Output OP on stdio stream STREAM, as a string of digits in base - BASE. The base may vary from 2 to 36. - - Return the number of bytes written, or if an error occurred, - return 0. - - - Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE) - Input a possibly white-space preceded string in base BASE from - stdio stream STREAM, and put the read integer in ROP. The base - may vary from 2 to 36. If 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 assumed, otherwise decimal is - assumed. - - Return the number of bytes read, or if an error occurred, return 0. - - - Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP) - Output OP on stdio stream STREAM, in raw binary format. The - integer is written in a portable format, with 4 bytes of size - information, and that many bytes of limbs. Both the size and the - limbs are written in decreasing significance order (i.e., in - big-endian). - - The output can be read with `mpz_inp_raw'. - - Return the number of bytes written, or if an error occurred, - return 0. - - The output of this can not be read by `mpz_inp_raw' from GMP 1, - because of changes necessary for compatibility between 32-bit and - 64-bit machines. - - - Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM) - Input from stdio stream STREAM in the format written by - `mpz_out_raw', and put the result in ROP. Return the number of - bytes read, or if an error occurred, return 0. - - This routine can read the output from `mpz_out_raw' also from GMP - 1, in spite of changes necessary for compatibility between 32-bit - and 64-bit machines. - - -File: gmp.info, Node: Miscellaneous Integer Functions, Prev: I/O of Integers, Up: Integer Functions - -Miscellaneous Functions -======================= - - - Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE) - Generate a random integer of at most MAX_SIZE limbs. The generated - random number doesn't satisfy any particular requirements of - randomness. Negative random numbers are generated when MAX_SIZE - is negative. - - - Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE) - Generate a random integer of at most MAX_SIZE limbs, with long - strings of zeros and ones in the binary representation. Useful - for testing functions and algorithms, since this kind of random - numbers have proven to be more likely to trigger corner-case bugs. - Negative random numbers are generated when MAX_SIZE is negative. - - - Function: size_t mpz_size (mpz_t OP) - Return the size of OP measured in number of limbs. If OP is zero, - the returned value will be zero. - - *This function is obsolete. It will disappear from future MP - releases.* - - - Function: size_t mpz_sizeinbase (mpz_t OP, int BASE) - Return the size of OP measured in number of digits in base BASE. - The base may vary from 2 to 36. The returned value will be exact - or 1 too big. If BASE is a power of 2, the returned value will - always be exact. - - This function is useful in order to allocate the right amount of - space before converting OP to a string. The right amount of - allocation is normally two more than the value returned by - `mpz_sizeinbase' (one extra for a minus sign and one for the - terminating '\0'). - - -File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top - -Rational Number Functions -************************* - - This chapter describes the MP functions for performing arithmetic on -rational numbers. These functions start with the prefix `mpq_'. - - Rational numbers are stored in objects of type `mpq_t'. - - All rational arithmetic functions assume operands have a canonical -form, and canonicalize their result. The canonical from means that the -denominator and the numerator have no common factors, and that the -denominator is positive. 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. *Note that this is an incompatible change from version 1 of -the library.* - - - Function: void mpq_canonicalize (mpq_t OP) - Remove any factors that are common to the numerator and - denominator of OP, and make the denominator positive. - -* Menu: - -* Initializing Rationals:: -* Assigning Rationals:: -* Simultaneous Integer Init & Assign:: -* Comparing Rationals:: -* Applying Integer Functions:: -* Miscellaneous Rational Functions:: - - -File: gmp.info, Node: Initializing Rationals, Next: Assigning Rationals, Prev: Rational Number Functions, Up: Rational Number Functions - -Initialization and Assignment Functions -======================================= - - - Function: void mpq_init (mpq_t DEST_RATIONAL) - Initialize DEST_RATIONAL and set it to 0/1. Each variable should - normally only be initialized once, or at least cleared out (using - the function `mpq_clear') between each initialization. - - - Function: void mpq_clear (mpq_t RATIONAL_NUMBER) - Free the space occupied by RATIONAL_NUMBER. Make sure to call this - function for all `mpq_t' variables when you are done with them. - - - Function: void mpq_set (mpq_t ROP, mpq_t OP) - - Function: void mpq_set_z (mpq_t ROP, mpz_t OP) - Assign ROP from OP. - - - Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1, - unsigned long int OP2) - - Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned - long int OP2) - Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have - common factors, ROP has to be passed to `mpq_canonicalize' before - any operations are performed on ROP. - - -File: gmp.info, Node: Assigning Rationals, Next: Comparing Rationals, Prev: Initializing Rationals, Up: Rational Number Functions - -Arithmetic Functions -==================== - - - Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2) - Set SUM to ADDEND1 + ADDEND2. - - - Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t - SUBTRAHEND) - Set DIFFERENCE to MINUEND - SUBTRAHEND. - - - Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t - MULTIPLICAND) - Set PRODUCT to MULTIPLIER times MULTIPLICAND. - - - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t - DIVISOR) - Set QUOTIENT to DIVIDEND/DIVISOR. - - - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND) - Set NEGATED_OPERAND to -OPERAND. - - - Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER) - Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero, - this routine will divide by zero. - - -File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Assigning Rationals, Up: Rational Number Functions - -Comparison Functions -==================== - - - Function: int mpq_cmp (mpq_t OP1, mpq_t OP2) - Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero - if OP1 = OP2, and a negative value if OP1 < OP2. - - To determine if two rationals are equal, `mpq_equal' is faster than - `mpq_cmp'. - - - Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned - long int DEN2) - Compare OP1 and NUM2/DEN2. Return a positive value if OP1 > - NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 < - NUM2/DEN2. - - This routine allows that NUM2 and DEN2 have common factors. - - This function is actually implemented as a macro. It evaluates its - arguments multiple times. - - - Macro: int mpq_sgn (mpq_t OP) - Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. - - This function is actually implemented as a macro. It evaluates its - arguments multiple times. - - - Function: int mpq_equal (mpq_t OP1, mpq_t OP2) - Return non-zero if OP1 and OP2 are equal, zero if they are - non-equal. Although `mpq_cmp' can be used for the same purpose, - this function is much faster. - - -File: gmp.info, Node: Applying Integer Functions, Next: Miscellaneous Rational Functions, Prev: Comparing Rationals, Up: Rational Number Functions - -Applying Integer Functions to Rationals -======================================= - - The set of `mpq' functions is quite small. In particular, there are -no functions for either input or output. But there are two macros that -allow us to apply any `mpz' function on the numerator or denominator of -a rational number. If these macros are used to assign to the rational -number, `mpq_canonicalize' normally need to be called afterwards. - - - Macro: mpz_t mpq_numref (mpq_t OP) - - Macro: mpz_t mpq_denref (mpq_t OP) - Return a reference to the numerator and denominator of OP, - respectively. The `mpz' functions can be used on the result of - these macros. - - -File: gmp.info, Node: Miscellaneous Rational Functions, Prev: Applying Integer Functions, Up: Rational Number Functions - -Miscellaneous Functions -======================= - - - Function: double mpq_get_d (mpq_t OP) - Convert OP to a double. - - 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 `mpq_numref' and -`mpq_denref', together with `mpz_set'. - - - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR) - Copy NUMERATOR to the numerator of RATIONAL. When this risks to - make the numerator and denominator of RATIONAL have common - factors, you have to pass RATIONAL to `mpq_canonicalize' before - any operations are performed on RATIONAL. - - This function is equivalent to `mpz_set (mpq_numref (RATIONAL), - NUMERATOR)'. - - - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR) - Copy DENOMINATOR to the denominator of RATIONAL. When this risks - to make the numerator and denominator of RATIONAL have common - factors, or if the denominator might be negative, you have to pass - RATIONAL to `mpq_canonicalize' before any operations are performed - on RATIONAL. - - *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 `mpz_set (mpq_denref (RATIONAL), - DENOMINATORS)'. - - - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL) - Copy the numerator of RATIONAL to the integer NUMERATOR, to - prepare for integer operations on the numerator. - - This function is equivalent to `mpz_set (NUMERATOR, mpq_numref - (RATIONAL))'. - - - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL) - Copy the denominator of RATIONAL to the integer DENOMINATOR, to - prepare for integer operations on the denominator. - - This function is equivalent to `mpz_set (DENOMINATOR, mpq_denref - (RATIONAL))'. + - Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int + D) + Divide N by (2 raised to D), rounding the quotient towards + -infinity, and put the remainder in R. The sign of R will always + be positive. This operation can also be defined as masking of the + D least significant bits.