     [file version.texi not included]

This file documents GNU MP, a library for arbitrary-precision arithmetic.

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 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.

GNU MP
******

This manual documents how to install and use the GNU multiple precision arithmetic library, version VERSION.

GNU MP Copying Conditions
*************************

This library is 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 domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do.  What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.

Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.

To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights.  For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have.  You must make sure that they, too, receive or can get the source code.  And you must tell them their rights.

Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library.  If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.

The precise conditions of the license for the GNU MP library are found in the Lesser General Public License that accompany the source code.

Introduction to GNU MP
**********************

GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers.  It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types.

Many applications use just a few hundred bits of precision; but some 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 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: 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.

There is a mailing list for GMP users.  To join it, send a mail to gmp-request@swox.com 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 GMP Basics.  If you need to install the library yourself, you need to read 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.

Installing GMP
**************

GMP has an autoconf/automake/libtool based configuration system.  On a Unix-like system a basic build can be done with

     ./configure
     make

Some self-tests can be run with

     make check

And you can install (under /usr/local by default) with

     make install

If you experience problems, please report them to bug-gmp@gnu.org. (see Reporting Bugs, for information on what to include in useful bug reports.)

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 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 configure.in for which mpn subdirectories get used by each.

       * Alpha: alpha, alphaev5, alphaev6

       * Hitachi: sh, sh2

       * HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0w

       * MIPS: mips, mips3,

       * Motorola: m68000, m68k, m88k, m88110

       * POWER: power1, power2, power2sc, powerpc, powerpc64

       * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9, ultrasparc, sparc64

       * 80x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro, pentium2, pentium3, k6, k62, k63, athlon

       * Other: a29k, arm, clipper, i960, ns32k, pyramid, vax, z8k

     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.

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.

     For some configurations specific compiler flags are set based on the target CPU and compiler, see CFLAGS in the generated Makefiles.  The usual CFLAGS="-whatever" can be passed to ./configure to use something different or to set good flags for systems GMP doesn't otherwise know.

     Note that 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.

--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.

     When not using alloca, it's actually the allocation function selected with mp_set_memory_functions that's used, this being malloc by default.  See Custom Allocation.

     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.

--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.

--enable-mpbsd

     The Berkeley MP compatibility library (libmp.a) and header file (mp.h) are built and installed only if --enable-mpbsd is used. See BSD Compatible Functions.

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.

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.

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.

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.

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.

Known Build Problems
====================

You might find more up-to-date information at http://www.swox.com/gmp/.

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.

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.)

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).

Sequent Symmetry

     Use the GNU assembler instead of the system assembler, since the latter has serious bugs.

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.

     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(<GSYM_PREFIX>, <_>).

     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.

GMP Basics
**********

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.

Nomenclature and Types
======================

In this manual, integer usually means a multiple precision integer, as 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;

     struct foo { mpz_t x, y; };

     mpz_t vec[20];

Rational number means a multiple precision fraction.  The C data type for these fractions is mpq_t.  For example:

     mpq_t quotient;

Floating point number or Float for short, is an arbitrary precision 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 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 mp_limb_t.

Function Classes
================

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 functions in this class.

 2. Functions for rational number arithmetic, with names beginning with mpq_.  The associated type is 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.

 3. Functions for floating-point arithmetic, with names beginning with mpf_.  The associated type is mpf_t.  There are about 50 functions is this class.

 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 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 mpn_.  There are about 30 (hard-to-use) functions in this class.

     The associated type is array of mp_limb_t.

 6. Miscellaneous functions.  Functions for setting up custom allocation and functions for generating random numbers.

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.)

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

     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 of variable.  See the 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.

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.

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.

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.)

  * If alloca is not available, or GMP is configured with --disable-alloca, the library is not reentrant, due to the current implementation of stack-alloc.c.  In the generated config.h, USE_STACK_ALLOC set to 1 will mean not reentrant.

Useful Macros and Constants
===========================

 -- Global Constant: const int mp_bits_per_limb
     The number of bits per limb.

 -- Macro: __GNU_MP_VERSION
 -- Macro: __GNU_MP_VERSION_MINOR
 -- 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 older versions
=================================

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.

  * mpn_gcd had its source arguments swapped as of GMP 3.0 for consistency with other mpn functions.

  * 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.

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.

Getting the Latest Version of GMP
=================================

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.

Reporting Bugs
**************

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 http://www.swox.com/gmp/ for patches for this release.

Please include the following in any report,

  * The GMP version number, and if pre-packaged or patched then say so.

  * 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 straces.

  * 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; the GMP code tends to explore interesting corners in compilers.

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 (except maybe ask you to send a better report).

Send your report to: bug-gmp@gnu.org.

If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.

Integer Functions
*****************

This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix mpz_.

GMP integers are stored in objects of type mpz_t.

Initialization Functions
========================

The functions for integer arithmetic assume that all integer objects are initialized.  You do that by calling the function mpz_init.

 -- Function: void mpz_init (mpz_t INTEGER)
     Initialize 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 mpz_clear) between each initialization.

Here is an example of using mpz_init:

     {
       mpz_t integ;
       mpz_init (integ);
       ...
       mpz_add (integ, ...);
       ...
       mpz_sub (integ, ...);

       /* Unless the program is about to exit, do ... */
       mpz_clear (integ);
     }

As you can see, you can store new values any number of times, once an object is initialized.

 -- Function: void mpz_clear (mpz_t INTEGER)
     Free the limb space occupied by INTEGER.  Make sure to call this function for all mpz_t variables when you are done with them.

 -- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
     Change the limb space allocation to 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.

 -- Function: void mpz_array_init (mpz_t INTEGER_ARRAY[], size_t ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
     Allocate fixed limb space for all ARRAY_SIZE integers in INTEGER_ARRAY.  The fixed allocation for each integer in the array is enough to store FIXED_NUM_BITS.  If the fixed space will be insufficient for storing the result of a subsequent calculation, the result is unpredictable.

     This function is useful for decreasing the working set for some algorithms that use large integer arrays.

     There is no way to de-allocate the storage allocated by this function. Don't call mpz_clear!

Assignment Functions
====================

These functions assign new values to already initialized integers (see Initializing Integers).

 -- Function: void mpz_set (mpz_t ROP, mpz_t OP)
 -- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
 -- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
 -- Function: void mpz_set_d (mpz_t ROP, double OP)
 -- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
 -- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
     Set the value of ROP from OP.

 -- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
     Set the value of ROP from STR, a '\0'-terminated C string in base BASE.  White space is allowed in the string, and is simply ignored.  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.

     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.

Combined Initialization and Assignment Functions
================================================

For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there.  These functions' names have the form mpz_init_set...

Here is an example of using one:

     {
       mpz_t pie;
       mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
       ...
       mpz_sub (pie, ...);
       ...
       mpz_clear (pie);
     }

Once the integer has been initialized by any of the mpz_init_set... functions, it can be used as the source or destination operand for the ordinary integer functions.  Don't use an initialize-and-set function on a variable already initialized!

 -- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
 -- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
 -- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
 -- Function: void mpz_init_set_d (mpz_t ROP, double OP)
     Initialize ROP with limb space and set the initial numeric value from OP.

 -- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
     Initialize ROP and set its value like mpz_set_str (see its documentation above for details).

     If the string is a correct base BASE number, the function returns 0; if an error occurs it returns -1.  ROP is initialized even if an error occurs.  (I.e., you have to call mpz_clear for it.)

Conversion Functions
====================

This section describes functions for converting GMP integers to standard C types.  Functions for converting to GMP integers are described in Assigning Integers and 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 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. Otherwise return the least significant part of OP, with the same sign as OP.

     If OP is too large to fit in a signed long int, the returned 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.

 -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
     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.

     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.

Arithmetic Functions
====================

 -- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1 + OP2.

 -- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     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 defined as a left shift, OP2 steps.

 -- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
     Set ROP to -OP.

 -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
     Set ROP to the absolute value of OP.

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 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 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 these routines start with mpz_cdiv.  The c in the name is short for ceil.

For each rounding mode, there are a couple of variants.  Here q means that the quotient is computed, while r means that the remainder is computed.  Functions that compute both the quotient and remainder have qr in the name.

 -- 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.

     The function mpz_tdiv_q_ui returns the absolute value of the true remainder.

 -- 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.

     The function mpz_tdiv_r_ui returns the absolute value of the remainder.

 -- 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.

     The function mpz_tdiv_qr_ui returns the absolute value of the remainder.

 -- 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_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.

     The function mpz_fdiv_q_ui returns the remainder.

 -- 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.

     The function mpz_fdiv_r_ui returns the remainder.

 -- 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.

     The function mpz_fdiv_qr_ui returns the remainder.

 -- 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_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.

     The function mpz_cdiv_q_ui returns the negated remainder.

 -- 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.

     The function mpz_cdiv_r_ui returns the negated remainder.

 -- 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.

     The function mpz_cdiv_qr_ui returns the negated remainder.

 -- 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.

 -- 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.

     The function mpz_mod_ui returns the remainder.

 -- 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.

     Since mpz_divexact is much faster than any of the other routines that produce the quotient (see 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_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.

 -- 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.

 -- 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_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.

Exponentiation Functions
========================

 -- 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.

Root Extraction Functions
=========================

 -- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
     Set ROP to the truncated integer part of the Nth root of OP. Return non-zero if the computation was exact, i.e., if OP is ROP to the Nth power.

 -- 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_power_p (mpz_t OP)
     Return non-zero if OP is a perfect power, i.e., if there exist integers A and B, with B > 1, such that OP equals a raised to b.  Return zero otherwise.

 -- 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 N, int REPS)
     If this function returns 0, N is definitely not prime.  If it returns 1, then N is `probably' prime.  If it returns 2, then 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.

     The function uses Miller-Rabin's probabilistic test.

 -- Function: int mpz_nextprime (mpz_t ROP, mpz_t OP)
     Set ROP to the next prime greater than 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.

 -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to the greatest common divisor of OP1 and OP2. The result is always positive even if either of or both input operands are negative.

 -- 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: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to the least common multiple of OP1 and OP2.

 -- 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 exists, zero otherwise.  When the function returns zero, ROP is undefined.

 -- 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.  OP2 should be odd and must be positive.

 -- Function: int mpz_si_kronecker (long A, mpz_t B);
 -- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B);
 -- Function: int mpz_kronecker_si (mpz_t A, long B);
 -- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B);
     Calculate the value of the Kronecker/Jacobi symbol (A/B), with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even. All values of A and B give a well-defined result.  See Henri Cohen, section 1.4.2, for more information (see References).  See also the example program demos/qcn.c which uses mpz_kronecker_ui.

 -- Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
     Remove all occurrences of the factor F from OP and store the result in ROP.  Return the multiplicity of F in OP.

 -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
     Set ROP to OP!, the factorial of OP.

 -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
 -- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N, unsigned long int K)
     Compute the binomial coefficient N over K and store the result in ROP.  Negative values of N are supported by mpz_bin_ui, using the identity bin(-n,k) = (-1)^k * bin(n+k-1,k) (see Knuth volume 1 section 1.2.6 part G).

 -- Function: void mpz_fib_ui (mpz_t ROP, unsigned long int N)
     Compute the Nth Fibonacci number and store the result in ROP.

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.

 -- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
 -- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
     Compare the absolute values of OP1 and OP2.  Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2.

 -- 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.

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_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to OP1 exclusive-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 a future release.

 -- 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 ROP.

 -- Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
     Clear bit BIT_INDEX in ROP.

 -- Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
     Check bit BIT_INDEX in OP and return 0 or 1 accordingly.

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.

Random Number Functions
=======================

The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified.  Please see the Random Number Functions for more information on how to use and not to use random number functions.

 -- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
     unsigned long int N) Generate a uniformly distributed random integer in the range 0 to 2^N - 1, inclusive.

     The variable STATE must be initialized by calling one of the gmp_randinit functions (Random State Initialization) before invoking this function.

 -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
     mpz_t N) Generate a uniform random integer in the range 0 to N - 1, inclusive.

     The variable STATE must be initialized by calling one of the gmp_randinit functions (Random State Initialization) before invoking this function.

 -- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE, unsigned long int N)
     Generate a random integer with long strings of zeros and ones in the binary representation.  Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs.  The random number will be in the range 0 to 2^N - 1, inclusive.

     The variable STATE must be initialized by calling one of the gmp_randinit functions (Random State Initialization) before invoking this function.

 -- 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.

     This function is obsolete.  Use mpz_urandomb or mpz_urandomm instead.

 -- 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.

     This function is obsolete.  Use mpz_rrandomb instead.

Miscellaneous Functions
=======================

 -- Function: int mpz_fits_ulong_p (mpz_t OP)
 -- Function: int mpz_fits_slong_p (mpz_t OP)
 -- Function: int mpz_fits_uint_p (mpz_t OP)
 -- Function: int mpz_fits_sint_p (mpz_t OP)
 -- Function: int mpz_fits_ushort_p (mpz_t OP)
 -- Function: int mpz_fits_sshort_p (mpz_t OP)
     Return non-zero iff the value of OP fits in an unsigned long int, signed long int, unsigned int, signed int, unsigned short int, or signed short int, respectively.  Otherwise, return zero.

 -- Macro: int mpz_odd_p (mpz_t OP)
 -- Macro: int mpz_even_p (mpz_t OP)
     Determine whether OP is odd or even, respectively.  Return non-zero if yes, zero if no.  These macros evaluate their arguments more than once.

 -- 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.

 -- 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').

Rational Number Functions
*************************

This chapter describes the GMP 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.

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.

 -- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
     Swap the values ROP1 and ROP2 efficiently.

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.

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.

Applying Integer Functions to Rationals
=======================================

The set of 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 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.

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 mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
     Output OP on stdio stream STREAM, as a string of digits in base BASE.  The base may vary from 2 to 36.  Output is in the form num/den or if the denominator is 1 then just num.

     Return the number of bytes written, or if an error occurred, return 0.

Miscellaneous Functions
=======================

 -- Function: double mpq_get_d (mpq_t OP)
     Convert OP to a double.

 -- Function: void mpq_set_d (mpq_t ROP, double D)
     Set ROP to the value of d, without rounding.

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)).

Floating-point Functions
************************

This chapter describes the GMP functions for performing floating point arithmetic.  These functions start with the prefix mpf_.

GMP floating point numbers are stored in objects of type mpf_t.

The GMP floating-point functions have an interface that is similar to the GMP integer functions.  The function prefix for floating-point operations is mpf_.

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.

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.

The GMP floating-point functions are 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.

Initialization Functions
========================

 -- Function: void mpf_set_default_prec (unsigned long int PREC)
     Set the default precision to be at least PREC bits.  All subsequent calls to mpf_init will use this precision, but previously initialized variables are unaffected.

An mpf_t object must be initialized before storing the first value in it.  The functions mpf_init and mpf_init2 are used for that purpose.

 -- Function: void mpf_init (mpf_t X)
     Initialize X to 0.  Normally, a variable should be initialized once only or at least be cleared, using mpf_clear, between initializations.  The precision of X is undefined unless a default precision has already been established by a call to mpf_set_default_prec.

 -- Function: void mpf_init2 (mpf_t X, unsigned long int PREC)
     Initialize X to 0 and set its precision to be at least PREC bits.  Normally, a variable should be initialized once only or at least be cleared, using mpf_clear, between initializations.

 -- Function: void mpf_clear (mpf_t X)
     Free the space occupied by X.  Make sure to call this function for all mpf_t variables when you are done with them.

Here is an example on how to initialize floating-point variables:

     {
       mpf_t x, y;
       mpf_init (x);			/* use default precision */
       mpf_init2 (y, 256);		/* precision at least 256 bits */
       ...
       /* Unless the program is about to exit, do ... */
       mpf_clear (x);
       mpf_clear (y);
     }

The following three functions are useful for changing the precision during a calculation.  A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers.

 -- Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)
     Set the precision of ROP to be at least PREC bits. Since changing the precision involves calls to realloc, this routine should not be called in a tight loop.

 -- Function: unsigned long int mpf_get_prec (mpf_t OP)
     Return the precision actually used for assignments of OP.

 -- Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)
     Set the precision of ROP to be at least PREC bits.  This is a low-level function that does not change the allocation.  The PREC argument must not be larger that the precision previously returned by mpf_get_prec.  It is crucial that the precision of ROP is ultimately reset to exactly the value returned by mpf_get_prec before the first call to mpf_set_prec_raw.

Assignment Functions
====================

These functions assign new values to already initialized floats (see Initializing Floats).

 -- Function: void mpf_set (mpf_t ROP, mpf_t OP)
 -- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
 -- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
 -- Function: void mpf_set_d (mpf_t ROP, double OP)
 -- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
 -- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
     Set the value of ROP from OP.

 -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
     Set the value of ROP from the string in STR.  The string is of the form M@N or, if the base is 10 or less, alternatively MeN. M is the mantissa and N is the exponent.  The mantissa is always in the specified base.  The exponent is either in the specified base or, if BASE is negative, in decimal.

     The argument BASE may be in the ranges 2 to 36, or -36 to -2.  Negative values are used to specify that the exponent is in decimal.

     Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if BASE is 0.  This is so that numbers like 0.23 are not interpreted as octal.

     White space is allowed in the string, and is simply ignored.  [This is not really true; white-space is ignored in the beginning of the string and within the mantissa, but not in other places, such as after a minus sign or in the exponent.  We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense.  Please tell us your opinion about this change.  Do you really want it to accept "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 BASE.  Otherwise it returns -1.

 -- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
     Swap the values ROP1 and ROP2 efficiently.

Combined Initialization and Assignment Functions
================================================

For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there.  These functions' names have the form mpf_init_set...

Once the float has been initialized by any of the mpf_init_set... functions, it can be used as the source or destination operand for the ordinary float functions.  Don't use an initialize-and-set function on a variable already initialized!

 -- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
 -- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
 -- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
 -- Function: void mpf_init_set_d (mpf_t ROP, double OP)
     Initialize ROP and set its value from OP.

     The precision of ROP will be taken from the active default precision, as set by mpf_set_default_prec.

 -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
     Initialize ROP and set its value from the string in STR.  See mpf_set_str above for details on the assignment operation.

     Note that ROP is initialized even if an error occurs.  (I.e., you have to call mpf_clear for it.)

     The precision of ROP will be taken from the active default precision, as set by mpf_set_default_prec.

Conversion Functions
====================

 -- Function: double mpf_get_d (mpf_t OP)
     Convert OP to a double.

 -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpf_t OP)
     Convert OP to a string of digits in base BASE.  The base may vary from 2 to 36.  Generate at most N_DIGITS significant digits, or if N_DIGITS is 0, the maximum number of digits accurately representable by OP.

     If STR is NULL, space for the mantissa is allocated using the default allocation function.

     If STR is not NULL, it should point to a block of storage enough large for the mantissa, i.e., N_DIGITS + 2.  The two extra bytes are for a possible minus sign, and for the terminating null character.

     The exponent is written through the pointer EXPPTR.

     If N_DIGITS is 0, the maximum number of digits meaningfully achievable from the precision of OP will be generated.  Note that the space requirements for STR in this case will be impossible for the user to predetermine.  Therefore, you need to pass NULL for the string argument whenever N_DIGITS is 0.

     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 EXPPTR.

     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.

Arithmetic Functions
====================

 -- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 -- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 + OP2.

 -- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t OP2)
 -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 - OP2.

 -- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 -- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 times OP2.

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.

 -- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t OP2)
 -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1/OP2.

 -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
 -- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
     Set ROP to the square root of OP.

 -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 raised to the power OP2.

 -- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
     Set ROP to -OP.

 -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
     Set ROP to the absolute value of OP.

 -- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 times 2 raised to OP2.

 -- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2)
     Set ROP to OP1 divided by 2 raised to OP2.

Comparison Functions
====================

 -- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
 -- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
 -- Function: int mpf_cmp_si (mpf_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.

 -- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)
     Return non-zero if the first OP3 bits of OP1 and OP2 are equal, zero otherwise.  I.e., test of OP1 and OP2 are approximately equal.

 -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
     Compute the relative difference between OP1 and OP2 and store the result in ROP.

 -- Macro: int mpf_sgn (mpf_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.

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 mpf_out_str (FILE *STREAM, int BASE, size_t N_DIGITS, mpf_t OP)
     Output OP on stdio stream STREAM, as a string of digits in base BASE.  The base may vary from 2 to 36.  Print at most N_DIGITS significant digits, or if N_DIGITS is 0, the maximum number of digits accurately representable by OP.

     In addition to the significant digits, a leading 0. and a trailing exponent, in the form eNNN, are printed.  If BASE is greater than 10, @ will be used instead of e as exponent delimiter.

     Return the number of bytes written, or if an error occurred, return 0.

 -- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
     Input a string in base BASE from stdio stream STREAM, and put the read float in ROP.  The string is of the form M@N or, if the base is 10 or less, alternatively MeN.  M is the mantissa and N is the exponent.  The mantissa is always in the specified base.  The exponent is either in the specified base or, if BASE is negative, in decimal.

     The argument BASE may be in the ranges 2 to 36, or -36 to -2.  Negative values are used to specify that the exponent is in decimal.

     Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if BASE is 0.  This is so that numbers like 0.23 are not interpreted as octal.

     Return the number of bytes read, or if an error occurred, return 0.

Miscellaneous Functions
=======================

 -- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
 -- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
 -- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
     Set ROP to OP rounded to an integer.  mpf_ceil rounds to the next higher integer, mpf_floor to the next lower, and mpf_trunc to the integer towards zero.

 -- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE, unsigned long int NBITS)
     Generate a uniformly distributed random float in ROP, such that 0 <= ROP < 1, with NBITS significant bits in the mantissa.

     The variable STATE must be initialized by calling one of the gmp_randinit functions (Random State Initialization) before invoking this function.

 -- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t MAX_EXP)
     Generate a random float of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation.  The exponent of the number is in the interval -EXP to 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 MAX_SIZE is negative.

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.

These functions start with the prefix mpn_.

The mpn functions are designed to be as fast as possible, not to provide a coherent calling interface.  The different functions have somewhat similar interfaces, but there are variations that make them hard to use.  These functions do as little as possible apart from the real multiple precision computation, so that no time is spent on things that not all callers need.

A source operand is specified by a pointer to the least significant limb and a limb count.  A destination operand is specified by just a pointer.  It is the 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 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.

The mpn functions are the base for the implementation of the mpz_, mpf_, and mpq_ functions.

This example adds the number beginning at S1P and the number beginning at S2P and writes the sum at DESTP.  All areas have SIZE limbs.

     cy = mpn_add_n (destp, s1p, s2p, size)

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}.

 -- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t SIZE)
     Add {S1P, SIZE} and {S2P, SIZE}, and write the SIZE least significant limbs of the result to 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., S1P equals S2P, use mpn_lshift with a count of 1 for optimal speed.

 -- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
     Add {S1P, SIZE} and S2LIMB, and write the SIZE least significant limbs of the result to RP.  Return carry, either 0 or 1.

 -- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
     Add {S1P, S1SIZE} and {S2P, S2SIZE}, and write the S1SIZE least significant limbs of the result to RP.  Return carry, either 0 or 1.

     This function requires that S1SIZE is greater than or equal to S2SIZE.

 -- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t SIZE)
     Subtract {S2P, S2SIZE} from {S1P, SIZE}, and write the SIZE least significant limbs of the result to 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: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
     Subtract S2LIMB from {S1P, SIZE}, and write the SIZE least significant limbs of the result to RP.  Return borrow, either 0 or 1.

 -- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
     Subtract {S2P, S2SIZE} from {S1P, S1SIZE}, and write the S1SIZE least significant limbs of the result to RP.  Return borrow, either 0 or 1.

     This function requires that S1SIZE is greater than or equal to S2SIZE.

 -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t SIZE)
     Multiply {S1P, SIZE} and {S2P, SIZE}, and write the entire result to RP.

     The destination has to have space for 2*SIZE limbs, even if the significant result might be one limb smaller.

 -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
     Multiply {S1P, SIZE} and S2LIMB, and write the SIZE least significant limbs of the product to RP.  Return the most significant limb of the product.

     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.

     Don't call this function if S2LIMB is a power of 2; use mpn_lshift with a count equal to the logarithm of S2LIMB instead, for optimal speed.

 -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
     Multiply {S1P, SIZE} and S2LIMB, and add the SIZE least significant limbs of the product to {RP, SIZE} and write the result to 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.

 -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)
     Multiply {S1P, SIZE} and S2LIMB, and subtract the SIZE least significant limbs of the product from {RP, SIZE} and write the result to 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.

 -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)
     Multiply {S1P, S1SIZE} and {S2P, S2SIZE}, and write the result to RP.  Return the most significant limb of the result.

     The destination has to have space for S1SIZE + S2SIZE limbs, even if the result might be one limb smaller.

     This function requires that S1SIZE is greater than or equal to S2SIZE.  The destination must be distinct from either input operands.

 -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP, mp_size_t DN)
     Divide {NP, NN} by {DP, DN}.  Write the quotient at QP and the remainder at RP.

     The quotient written at QP will be NN - DN + 1 limbs. The remainder written at RP will be DN limbs.

     It is required that NN is greater than or equal to DN.  The QXN operand must be zero.

     The quotient is rounded towards 0.

     No overlap between arguments is permitted.

 -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t XSIZE, mp_limb_t *RS2P, mp_size_t RS2SIZE, const mp_limb_t *S3P, mp_size_t S3SIZE)
     [This function is obsolete.  Please call mpn_tdiv_qr instead for best performance.]

     Divide {RS2P, RS2SIZE} by {S3P, S3SIZE}, and write the quotient at R1P, with the exception of the most significant limb, which is returned.  The remainder replaces the dividend at RS2P; it will be S3SIZE limbs long (i.e., as many limbs as the divisor).

     In addition to an integer quotient, XSIZE fraction limbs are developed, and stored after the integral limbs.  For most usages, XSIZE will be zero.

     It is required that RS2SIZE is greater than or equal to S3SIZE. It is required that the most significant bit of the divisor is set.

     If the quotient is not needed, pass RS2P + S3SIZE as 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 R1P needs to be RS2SIZE - S3SIZE + XSIZE limbs large.

 -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t XSIZE, mp_limb_t *S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)
 -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)
     Divide {S2P, S2SIZE} by S3LIMB, and write the quotient at R1P.  Return the remainder.

     The integer quotient is written to {R1P+XSIZE, S2SIZE} and in addition XSIZE fraction limbs are developed and written to {R1P, XSIZE}.  Either or both S2SIZE and XSIZE can be zero.  For most usages, XSIZE will be zero.

     mpn_divmod_1 exists for upward source compatibility and is simply a macro calling mpn_divrem_1 with an XSIZE of 0.

     The areas at R1P and S2P have to be identical or completely separate, not partially overlapping.

 -- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P, mp_size_t RS2SIZE, const mp_limb_t *S3P, mp_size_t S3SIZE)
     This interface is obsolete.  It will disappear from future releases. Use mpn_divrem in its stead.

 -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP, mp_size_t SIZE)
 -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP, mp_size_t SIZE, mp_limb_t CARRY)
     Divide {SP, SIZE} by 3, expecting it to divide exactly, and writing the result to {RP, 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.

     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.  mpn_divexact_by3 is simply a macro calling mpn_divexact_by3c with a 0 carry parameter.

     These routines use a multiply-by-inverse and will be faster than mpn_divrem_1 on CPUs with fast multiplication but slow division.

     The source a, result q, size n, initial carry i, and return value c satisfy c*b^n + a-i = 3*q, where b is the size of a limb (2^32 or 2^64). c is always 0, 1 or 2, and the initial carry must also be 0, 1 or 2 (these are both borrows really).  When c=0, clearly q=(a-i)/3. When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b == 1 mod 3.

 -- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1SIZE, mp_limb_t S2LIMB)
     Divide {S1P, S1SIZE} by S2LIMB, and return the remainder. S1SIZE can be zero.

 -- Function: mp_limb_t mpn_preinv_mod_1 (mp_limb_t *S1P, mp_size_t S1SIZE, mp_limb_t S2LIMB, mp_limb_t S3LIMB)
     This interface is obsolete.  It will disappear from future releases. Use mpn_mod_1 in its stead.

 -- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P, mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE, unsigned long int D)
     The function puts the low [D/BITS_PER_MP_LIMB] limbs of Q = {S1P, S1SIZE}/{S2P, S2SIZE} mod 2^D at RP, and returns the high D mod BITS_PER_MP_LIMB bits of Q.

     {S1P, S1SIZE} - Q * {S2P, S2SIZE} mod 2^(S1SIZE*BITS_PER_MP_LIMB) is placed at S1P. Since the low [D/BITS_PER_MP_LIMB] limbs of this difference are zero, it is possible to overwrite the low limbs at S1P with this difference, provided RP <= S1P.

     This function requires that S1SIZE * BITS_PER_MP_LIMB >= D, and that {S2P, S2SIZE} is odd.

     This interface is preliminary.  It might change incompatibly in future revisions.

 -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)
     Shift {SRC_PTR, SRC_SIZE} COUNT bits to the left, and write the SRC_SIZE least significant limbs of the result to RP.  COUNT might be in the range 1 to n - 1, on an n-bit machine. The bits shifted out to the left are returned.

     Overlapping of the destination space and the source space is allowed in this function, provided RP >= SRC_PTR.

     This function is written in assembly for most targets.

 -- Function: mp_limp_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)
     Shift {SRC_PTR, SRC_SIZE} COUNT bits to the right, and write the SRC_SIZE most significant limbs of the result to RP.  COUNT might be in the range 1 to n - 1, on an n-bit machine.  The bits shifted out to the right are returned.

     Overlapping of the destination space and the source space is allowed in this function, provided RP <= SRC_PTR.

     This function is written in assembly for most targets.

 -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t SIZE)
     Compare {S1P, SIZE} and {S2P, SIZE} and return a positive value if s1 > src2, 0 of they are equal, and a negative value if s1 < src2.

 -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P, mp_size_t S1SIZE, mp_limb_t *S2P, mp_size_t S2SIZE)
     Puts at RP the greatest common divisor of {S1P, S1SIZE} and {S2P, S2SIZE}; both source operands are destroyed by the operation.  The size in limbs of the greatest common divisor is returned.

     {S1P, S1SIZE} must have at least as many bits as {S2P, S2SIZE}, and {S2P, S2SIZE} must be odd.

 -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1SIZE, mp_limb_t S2LIMB)
     Return the greatest common divisor of {S1P, S1SIZE} and S2LIMB, where S2LIMB (as well as S1SIZE) must be different from 0.

 -- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P, mp_size_t *R2SIZE, mp_limb_t *S1P, mp_size_t S1SIZE, mp_limb_t *S2P, mp_size_t S2SIZE)
     Compute the greatest common divisor of {S1P, S1SIZE} and {S2P, S2SIZE}.  Store the gcd at R1P and return its size in limbs.  Write the first cofactor at R2P and store its size in *R2SIZE.  If the cofactor is negative, *R2SIZE is negative and R2P is the absolute value of the cofactor.

     {S1P, S1SIZE} must be greater than or equal to {S2P, S2SIZE}.  Neither operand may equal 0.  Both source operands are destroyed, plus one limb past the end of each, ie. {S1P, S1SIZE+1} and {S2P, S2SIZE+1}.

 -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P, const mp_limb_t *SP, mp_size_t SIZE)
     Compute the square root of {SP, SIZE} and put the result at R1P.  Write the remainder at R2P, unless R2P is NULL.

     Return the size of the remainder, whether R2P was NULL or non-NULL. Iff the operand was a perfect square, the return value will be 0.

     The areas at R1P and SP have to be distinct.  The areas at R2P and SP have to be identical or completely separate, not partially overlapping.

     The area at R1P needs to have space for ceil(SIZE/2) limbs. The area at R2P needs to be SIZE limbs large.

 -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE, mp_limb_t *S1P, mp_size_t S1SIZE)
     Convert {S1P, S1SIZE} to a raw unsigned char array in base BASE.  The string is not in ASCII; to convert it to printable format, add the ASCII codes for 0 or A, depending on the base and range.  There may be leading zeros in the string.

     The area at S1P is clobbered.

     Return the number of characters in STR.

     The area at STR has to have space for the largest possible number represented by a S1SIZE long limb array, plus one extra character.

 -- Function: mp_size_t mpn_set_str (mp_limb_t *R1P, const char *STR, size_t STRSIZE, int BASE)
     Convert the raw unsigned char array at STR of length STRSIZE to a limb array {S1P, S1SIZE}.  The base of STR is BASE.

     Return the number of limbs stored in R1P.

 -- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P, unsigned long int BIT)
     Scan S1P from bit position BIT for the next clear bit.

     It is required that there be a clear bit within the area at S1P at or beyond bit position BIT, so that the function has something to return.

 -- Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P, unsigned long int BIT)
     Scan S1P from bit position BIT for the next set bit.

     It is required that there be a set bit within the area at S1P at or beyond bit position BIT, so that the function has something to return.

 -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1SIZE)
 -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1SIZE)
     Generate a random number of length R1SIZE and store it at R1P. The most significant limb is always non-zero.  mpn_random generates uniformly distributed limb data, mpn_random2 generates long strings of zeros and ones in the binary representation.

     mpn_random2 is intended for testing the correctness of the mpn routines.

 -- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P, unsigned long int SIZE)
     Count the number of set bits in {S1P, SIZE}.

 -- Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const mp_limb_t *S2P, unsigned long int SIZE)
     Compute the hamming distance between {S1P, SIZE} and {S2P, SIZE}.

 -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t SIZE)
     Return non-zero iff {S1P, SIZE} is a perfect square.

Random Number Functions
***********************

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.

The random state parameter is of the type gmp_randstate_t.  It must be initialized by a call to one of the gmp_randinit functions (Random State Initialization).  The initial seed is set using one of the gmp_randseed functions (Random State Initialization).

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.

The algorithm for assigning seed is critical if the generated random numbers are to be used for important applications, such as generating cryptographic keys.

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.

On some systems there is a special device, often called /dev/random, which provides a source of somewhat random numbers more usable as seed.

The functions actually generating random functions are documented under ``Miscellaneous Functions'' in their respective function class: Miscellaneous Integer Functions, Miscellaneous Float Functions.

Random State Initialization
===========================

See Random Number Functions for a discussion on how to choose the initial seed value passed to these functions.

 -- Function: void gmp_randinit (gmp_randstate_t STATE, gmp_randalg_t ALG, ...)
     Initialize random state variable STATE.

     ALG denotes what algorithm to use for random number generation. Use one of

       - GMP_RAND_ALG_LC --- Linear congruential.

          A fast generator defined by X = (aX + c) mod m.

          A third argument SIZE of type unsigned long int is required.  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 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.

          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 SIZE parameter.  The maximum SIZE supported by this algorithm is 128.  If you need bigger random numbers, use your own scheme and call one of the other gmp_randinit functions.

     If 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 SIZE argument (see GMP_RAND_ALG_LC).

     When you're done with a STATE variable, call gmp_randclear to deallocate any memory allocated by this function.

     gmp_randinit may set the following bits in GMP_ERRNO:

       * GMP_ERROR_UNSUPPORTED_ARGUMENT --- ALG is unsupported

       * GMP_ERROR_INVALID_ARGUMENT --- SIZE is too big

 -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A,
     unsigned long int C, unsigned long int M2EXP)

     Initialize random state variable STATE with given linear congruential scheme.

     Parameters A, C, and 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 M = 2^M2EXP.

     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 M2EXP / 2 (rounded upwards) bits when this function has been used for initializing the random state.

     When you're done with a STATE variable, call gmp_randclear to deallocate any memory allocated by this function.

 -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
 -- Function: void gmp_randseed_ui (gmp_randstate_t STATE, unsigned long int SEED)

     Set the initial seed value.

     Parameter SEED is the initial random seed.  The function gmp_randseed_ui takes the SEED as an unsigned long int rather than as an mpz_t.

 -- Function: void gmp_randclear (gmp_randstate_t STATE)
     Free all memory occupied by STATE.  Make sure to call this function for all gmp_randstate_t variables when you are done with them.

Berkeley MP Compatible Functions
********************************

These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U*ix systems.  The --enable-mpbsd option must be used when building GNU MP to make these available (see Installing GMP).

The original Berkeley MP library has a usage restriction: you cannot use the same variable as both source and destination in a single function call.  The compatible functions in GNU MP do not share this restriction---inputs and outputs may overlap.

It is not recommended that new programs are written using these functions. Apart from the incomplete set of functions, the interface for initializing MINT objects is more error prone, and the pow function collides with pow in libm.a.

Include the header 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 mp.h if you are going to link the GNU libmp.a to your program. This means that you probably need to give the -I<dir> option to the compiler, where <dir> is the directory where you have GNU mp.h.

 -- Function: MINT * itom (signed short int INITIAL_VALUE)
     Allocate an integer consisting of a MINT object and dynamic limb space. Initialize the integer to INITIAL_VALUE.  Return a pointer to the MINT object.

 -- Function: MINT * xtom (char *INITIAL_VALUE)
     Allocate an integer consisting of a MINT object and dynamic limb space. Initialize the integer from INITIAL_VALUE, a hexadecimal, '\0'-terminate C string.  Return a pointer to the MINT object.

 -- Function: void move (MINT *SRC, MINT *DEST)
     Set DEST to SRC by copying.  Both variables must be previously initialized.

 -- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
     Add SRC_1 and SRC_2 and put the sum in DESTINATION.

 -- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
     Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.

 -- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
     Multiply SRC_1 and SRC_2 and put the product in DESTINATION.

 -- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT, MINT *REMAINDER)
 -- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT *QUOTIENT, signed short int *REMAINDER)
     Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod DIVISOR.  The quotient is rounded towards zero; the remainder has the same sign as the dividend unless it is zero.

     Some implementations of these functions work differently---or not at all---for negative arguments.

 -- Function: void msqrt (MINT *OPERAND, MINT *ROOT, MINT *REMAINDER)
     Set ROOT to the truncated integer part of the square root of OPERAND.  Set REMAINDER to OPERAND-ROOT*ROOT, (i.e., zero if OPERAND is a perfect square).

     If ROOT and REMAINDER are the same variable, the results are undefined.

 -- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
     Set DEST to (BASE raised to EXP) modulo MOD.

 -- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
     Set DEST to BASE raised to EXP.

 -- Function: void gcd (MINT *OPERAND1, MINT *OPERAND2, MINT *RES)
     Set RES to the greatest common divisor of OPERAND1 and OPERAND2.

 -- Function: int mcmp (MINT *OPERAND1, MINT *OPERAND2)
     Compare OPERAND1 and OPERAND2.  Return a positive value if OPERAND1 > OPERAND2, zero if OPERAND1 = OPERAND2, and a negative value if OPERAND1 < OPERAND2.

 -- Function: void min (MINT *DEST)
     Input a decimal string from stdin, and put the read integer in DEST.  SPC and TAB are allowed in the number string, and are ignored.

 -- Function: void mout (MINT *SRC)
     Output SRC to stdout, as a decimal string.  Also output a newline.

 -- Function: char * mtox (MINT *OPERAND)
     Convert OPERAND to a hexadecimal string, and return a pointer to the string.  The returned string is allocated using the default memory allocation function, malloc by default.

 -- Function: void mfree (MINT *OPERAND)
     De-allocate, the space used by OPERAND.  This function should only be passed a value returned by itom or xtom.

Custom Allocation
*****************

By default, GMP uses malloc, realloc and free for memory allocation.  If malloc or realloc fails, GMP prints a message to the standard error output and terminates execution.

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.

This can be done in the Berkeley compatibility library as well as the main GMP library.

 -- Function: void mp_set_memory_functions (
 void *(*ALLOC_FUNC_PTR) (size_t),
 void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
 void (*FREE_FUNC_PTR) (void *, size_t))
     Replace the current allocation functions from the arguments.  If an argument is NULL, the corresponding default function is retained.

     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.

The functions you supply should fit the following declarations:

 -- Function: void * allocate_function (size_t ALLOC_SIZE)
     This function should return a pointer to newly allocated space with at least ALLOC_SIZE storage units.

 -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE, size_t NEW_SIZE)
     This function should return a pointer to newly allocated space of at least NEW_SIZE storage units, after copying at least the first OLD_SIZE storage units from PTR.  It should also de-allocate the space at PTR.

     You can assume that the space at PTR was formerly returned from allocate_function or reallocate_function, for a request for OLD_SIZE storage units.

 -- Function: void deallocate_function (void *PTR, size_t SIZE)
     De-allocate the space pointed to by PTR.

     You can assume that the space at PTR was formerly returned from allocate_function or reallocate_function, for a request for SIZE storage units.

(A storage unit is the unit in which the sizeof operator returns the size of an object, normally an 8 bit byte.)

Contributors
************

Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it.  Several other individuals and organizations have contributed to GMP in various ways.  Here is a list in chronological order:

Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library.

Richard Stallman contributed to the interface design and revised the first version of this manual.

Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions.

John Amanatides of York University in Canada contributed the function mpz_probab_prime_p.

Paul Zimmermann of Inria sparked the development of GMP 2, with his comparisons between bignum packages.

Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed mpz_gcd, mpz_divexact, mpn_gcd, and mpn_bdivmod, partially supported by CNPq (Brazil) grant 301314194-2.

Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases.

Joachim Hollman was involved in the design of the mpf interface, and in the mpz design revisions for version 2.

Bennet Yee contributed the functions mpz_jacobi and mpz_legendre.

Andreas Schwab contributed the files mpn/m68k/lshift.S and mpn/m68k/rshift.S.

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.

Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count.

Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3.  He also contributed the ARM assembly code.

Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development.  His mathematical expertise helped improve several algorithms.

Paul Zimmermann wrote the 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.

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.

Steve Root helped write the optimized alpha 21264 assembly code.

GNU MP 3.1 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn's work was partially funded by the IDA Center for Computing Sciences, USA.

(This list is chronological, not ordered after significance.  If you have contributed to GMP but are not listed above, please tell tege@swox.com about the omission!)

References
**********

  * Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1988.

  * John D. Lipson, "Elements of Algebra and Algebraic Computing", The Benjamin Cummings Publishing Company Inc, 1981.

  * Richard M. Stallman, "Using and Porting GCC", Free Software Foundation, 1999, available online http://www.gnu.org/software/gcc/onlinedocs/, and in the GCC package ftp://ftp.gnu.org/pub/gnu/gcc/.

  * Peter L. Montgomery, "Modular Multiplication Without Trial Division", in Mathematics of Computation, volume 44, number 170, April 1985.

  * Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant Integers using Multiplication", in Proceedings of the SIGPLAN PLDI'94 Conference, June 1994.  Available online,

     ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz (and .psl.gz too).

  * Tudor Jebelean, "An algorithm for exact division", Journal of Symbolic Computation, v. 15, 1993, pp. 169-180. Research report version available online

     ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz

  * Kenneth Weber, "The accelerated integer GCD algorithm", ACM Transactions on Mathematical Software, v. 21 (March), 1995, pp. 111-122.

  * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,

     http://www.mpi-sb.mpg.de/~ziegler/TechRep.ps.gz.

  * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of Applied Cryptography", http://cacr.math.uwaterloo.ca/hac/.

  * Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.  Errata available online
      http://www.math.u-bordeaux.fr/~cohen