     START-INFO-DIR-ENTRY
     * gmp: (gmp.info).               GNU Multiple Precision Arithmetic Library.
     END-INFO-DIR-ENTRY

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

Copyright (C) 1991, 1993, 1994, 1995, 1996 Free Software Foundation, Inc.

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 2.0.2.

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 Library 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.  MP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum.

The speed of MP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance).

There is carefully optimized assembly code for these CPUs: DEC Alpha, Amd 29000, HPPA 1.0 and 1.1, Intel Pentium and generic x86, Intel i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, and DEC VAX.  Some optimizations also for ARM, Clipper, IBM ROMP (RT), and Pyramid AP/XP.

This version of MP is released under a more liberal license than previous versions.  It is now permitted to link MP to non-free programs, as long as MP source code is provided when distributing the non-free program.

How to use this Manual
======================

Everyone should read MP Basics.  If you need to install the library yourself, you need to read Installing MP, too.

The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.

Installing MP
*************

To build MP, you first have to configure it for your CPU and operating system. You need a C compiler, preferably GCC, but any reasonable compiler should work.  And you need a standard Unix make program, plus some other standard Unix utility programs.

(If you're on an MS-DOS machine, your can build MP using make.bat.  It requires that djgpp is installed.  It does not require configuration, nor is make needed; make.bat both configures and builds the library.)

Here are the steps needed to install the library on Unix systems:

 1. In most cases, ./configure --target=cpu-vendor-os, should work both for native and cross-compilation.  If you get error messages, your machine might not be supported.

     If you want to compile in a separate object directory, cd to that directory, and prefix the configure command with the path to the MP source directory. Not all make programs have the necessary features to support this.  In particular, SunOS and Slowaris make have bugs that makes them unable to build from a separate object directory.  Use GNU make instead.

     In addition to the standard cpu-vendor-os tuples, MP recognizes sparc8 and supersparc as valid CPU names.  Specifying these CPU names for relevant systems will improve performance significantly.

     In general, if you want a library that runs as fast as possible, you should make sure you configure MP for the exact CPU type your system uses.

     If you have gcc in your PATH, it will be used by default.  To override this, pass -with-gcc=no to configure.

 2. make

     This will compile MP, and create a library archive file libgmp.a in the working directory.

 3. make check

     This will make sure MP was built correctly.  If you get error messages, please report this to bug-gmp@prep.ai.mit.edu.  (see Reporting Bugs, for information on what to include in useful bug reports.)

 4. make install

     This will copy the file gmp.h and libgmp.a, as well as the info files, to /usr/local (or if you passed the --prefix option to configure, to the directory given as argument to --prefix).

If you wish to build and install the BSD MP compatible functions, use make libmp.a and make install-bsdmp.

There are some other useful make targets:

  * doc

     Create a DVI version of the manual, in gmp.dvi and a set of info files, in gmp.info, gmp.info-1, gmp.info-2, etc.

  * ps

     Create a Postscript version of the manual, in gmp.ps.

  * html

     Create a HTML version of the manual, in gmp.html.

  * clean

     Delete all object files and archive files, but not the configuration files.

  * distclean

     Delete all files not included in the distribution.

  * uninstall

     Delete all files copied by make install.

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

GCC 2.7.2 (as well as 2.6.3) for the RS/6000 and PowerPC can not be used to compile MP, due to a bug in GCC.  If you want to use GCC for these machines, you need to apply the patch below to GCC, or use a later version of the compiler.

If you are on a Sequent Symmetry, use the GNU assembler instead of the system's assembler, since the latter has serious bugs.

The system compiler on NeXT is a massacred and old gcc, even if the compiler calls itself cc.  This compiler cannot be used to build MP.  You need to get a real gcc, and install that before you compile MP.  (NeXT might have fixed this in newer releases of their system.)

The system C compiler under SunOS 4 has a bug that makes it miscompile mpq/get_d.c.  This will make make check fail.

Please report other problems to bug-gmp@prep.ai.mit.edu. See Reporting Bugs.

Patch to apply to GCC 2.6.3 and 2.7.2:

     *** config/rs6000/rs6000.md	Sun Feb 11 08:22:11 1996
     --- config/rs6000/rs6000.md.new	Sun Feb 18 03:33:37 1996
     ***************
     *** 920,926 ****
          (set (match_operand:SI 0 "gpc_reg_operand" "=r")
       	(not:SI (match_dup 1)))]
         ""
     !   "nor. %0,%2,%1"
         [(set_attr "type" "compare")])

       (define_insn ""
     --- 920,926 ----
          (set (match_operand:SI 0 "gpc_reg_operand" "=r")
       	(not:SI (match_dup 1)))]
         ""
     !   "nor. %0,%1,%1"
         [(set_attr "type" "compare")])

       (define_insn ""

MP Basics
*********

All declarations needed to use MP are collected in the include file gmp.h.  It is designed to work with both C and C++ compilers.

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

In this manual, integer usually means a multiple precision integer, as defined by the MP library.  The C data type for such integers is 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 an 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 MP 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 MP, 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.

MP Variable Conventions
=======================

As a general rule, all MP functions expect output arguments before input arguments.  This notation is based on an analogy with the assignment operator. (The BSD MP compatibility functions disobey this rule, having the output argument(s) last.)

MP allows you to use the same variable for both input and output in the same expression.  For example, the main function for integer multiplication, mpz_mul, can be used like this: mpz_mul (x, x, x).  This computes the square of X and puts the result back in X.

Before you can assign to an MP 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 to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited.

You don't need to be concerned about allocating additional space for MP variables.  All functions in MP automatically allocate additional space when a variable does not already have enough space.  They do not, however, reduce the space when a smaller number is stored in the object.  Most of the time, this policy is best, since it avoids frequent re-allocation.

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
     The major and minor MP version, respectively, as integers.

Compatibility with Version 1.x
==============================

This version of MP is upward compatible with previous versions of MP, with a few exceptions.

 1. Integer division functions round the result differently.  The old functions (mpz_div, mpz_divmod, mpz_mdiv, mpz_mdivmod, etc) now all use floor rounding (i.e., they round the quotient to -infinity).  There are a lot of new functions for integer division, giving the user better control over the rounding.

 2. The function mpz_mod now compute the true mod function.

 3. The functions mpz_powm and mpz_powm_ui now use mod for reduction.

 4. The assignment functions for rational numbers do no longer canonicalize their results.  In the case a non-canonical result could arise from an assignment, the user need to insert an explicit call to mpq_canonicalize.  This change was made for efficiency.

 5. Output generated by mpz_out_raw in this release cannot be read by mpz_inp_raw in previous releases.  This change was made for making the file format truly portable between machines with different word sizes.

 6. Several mpn functions have changed.  But they were intentionally undocumented in previous releases.

 7. The functions mpz_cmp_ui, mpz_cmp_si, and mpq_cmp_ui are now implementated as macros, and thereby sometimes evaluate their arguments multiple times.

 8. The functions mpz_pow_ui and mpz_ui_pow_ui now yield 1 for 0^0.  (In version 1, they yielded 0.)

Getting the Latest Version of MP
================================

The latest version of the MP library is available by anonymous ftp from from prep.ai.mit.edu.  The file name is /pub/gnu/gmp-M.N.tar.gz.  Many sites around the world mirror prep; please use a mirror site near you.

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

If you think you have found a bug in the MP library, please investigate it and report it.  We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find.

There are a few things you should think about when you put your bug report together.

You have to send us a test case that makes it possible for us to reproduce the bug.  Include instructions on how to run the test case.

You also have to explain what is wrong; if you get a crash, or if the results printed are incorrect and in that case, in what way.

It is not uncommon that an observed problem is actually due to a bug in the compiler used when building MP; the MP code tends to explore interesting corners in compilers.  Therefore, please include compiler version information in your bug report.  This can be extracted using what `which cc`, or, if you're using gcc, gcc -v.  Also, include the output from uname -a.

If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we won't do anything about it (aside of chiding you to send better bug reports).

Send your bug report to: bug-gmp@prep.ai.mit.edu.

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 MP functions for performing integer arithmetic. These functions start with the prefix mpz_.

Arbitrary precision integers are stored in objects of type mpz_t.

Initialization and Assignment 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.

Combined Initialization and Assignment Functions
------------------------------------------------

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

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 arbitrary precision integers to standard C types.  Functions for converting to arbitrary precision integers are described in Assigning Integers and I/O of Integers.

 -- Function: unsigned long int mpz_get_ui (mpz_t OP)
     Return the least significant part from OP.  This function combined with
      mpz_tdiv_q_2exp(..., OP, CHAR_BIT*sizeof(unsigned
     long int)) can be used to extract the limbs of an integer.

 -- 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.

 -- 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, and a pointer to the string is returned.

     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.

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_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1 times OP2.

 -- 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.

 -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
     Set ROP to OP!, the factorial 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 gives the user the possibility to 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 ROP, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_tdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to [OP1/OP2].  The quotient is truncated towards 0.

 -- Function: void mpz_tdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_tdiv_r_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to (OP1 - [OP1/OP2] * OP2). Unless the remainder is zero, it has the same sign as the dividend.

 -- Function: void mpz_tdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_tdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by OP2 and put the quotient in ROP1 and the remainder in ROP2.  The quotient is rounded towards 0.  Unless the remainder is zero, it has the same sign as the dividend.

     If ROP1 and ROP2 are the same variable, the results are undefined.

 -- Function: void mpz_fdiv_q (mpz_t ROP1, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_fdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1/OP2.  The quotient is rounded towards -infinity.

 -- Function: void mpz_fdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by OP2 and put the remainder in ROP.  Unless the remainder is zero, it has the same sign as the divisor.

     For mpz_fdiv_r_ui the remainder is small enough to fit in an unsigned long int, and is therefore returned.

 -- Function: void mpz_fdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t OP2)
 -- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by OP2 and put the quotient in ROP1 and the remainder in ROP2.  The quotient is rounded towards -infinity. Unless the remainder is zero, it has the same sign as the divisor.

     For mpz_fdiv_qr_ui the remainder is small enough to fit in an unsigned long int, and is therefore returned.

     If ROP1 and ROP2 are the same variable, the results are undefined.

 -- Function: unsigned long int mpz_fdiv_ui (mpz_t OP1, unsigned long int OP2)
     This function is similar to mpz_fdiv_r_ui, but the remainder is only returned; it is not stored anywhere.

 -- Function: void mpz_cdiv_q (mpz_t ROP1, mpz_t OP1, mpz_t OP2)
 -- Function: void mpz_cdiv_q_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1/OP2.  The quotient is rounded towards +infinity.

 -- Function: void mpz_cdiv_r (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by OP2 and put the remainder in ROP.  Unless the remainder is zero, it has the opposite sign as the divisor.

     For mpz_cdiv_r_ui the negated remainder is small enough to fit in an unsigned long int, and it is therefore returned.

 -- Function: void mpz_cdiv_qr (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, mpz_t OP2)
 -- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t ROP1, mpz_t ROP2, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by OP2 and put the quotient in ROP1 and the remainder in ROP2.  The quotient is rounded towards +infinity.  Unless the remainder is zero, it has the opposite sign as the divisor.

     For mpz_cdiv_qr_ui the negated remainder is small enough to fit in an unsigned long int, and it is therefore returned.

     If ROP1 and ROP2 are the same variable, the results are undefined.

 -- Function: unsigned long int mpz_cdiv_ui (mpz_t OP1, unsigned long int OP2)
     Return the negated remainder, similar to mpz_cdiv_r_ui.  (The difference is that this function doesn't store the remainder anywhere.)

 -- Function: void mpz_mod (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 -- Function: unsigned long int mpz_mod_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1 mod OP2.  The sign of the divisor is ignored, and the result is always non-negative.

     For mpz_mod_ui the remainder is small enough to fit in an unsigned long int, and is therefore returned.

 -- Function: void mpz_divexact (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to OP1/OP2.  This function produces correct results only when it is known in advance that OP2 divides OP1.

     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 ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1 divided by 2 raised to OP2.  The quotient is rounded towards 0.

 -- Function: void mpz_tdiv_r_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by (2 raised to OP2) and put the remainder in ROP.  Unless it is zero, ROP will have the same sign as OP1.

 -- Function: void mpz_fdiv_q_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Set ROP to OP1 divided by 2 raised to OP2.  The quotient is rounded towards -infinity.

 -- Function: void mpz_fdiv_r_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Divide OP1 by (2 raised to OP2) and put the remainder in ROP.  The sign of ROP will always be positive.

     This operation can also be defined as masking of the OP2 least significant bits.

Exponentialization 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.

Square Root Functions
---------------------

 -- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
     Set ROP to the truncated integer part of the square root of OP.

 -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
     Set ROP1 to the truncated integer part of the square root of OP, like mpz_sqrt.  Set ROP2 to OP-ROP1*ROP1, (i.e., zero if OP is a perfect square).

     If ROP1 and ROP2 are the same variable, the results are undefined.

 -- Function: int mpz_perfect_square_p (mpz_t OP)
     Return non-zero if OP is a perfect square, i.e., if the square root of OP is an integer.  Return zero otherwise.

Number Theoretic Functions
--------------------------

 -- Function: int mpz_probab_prime_p (mpz_t OP, int REPS)
     If this function returns 0, OP is definitely not prime.  If it returns 1, then OP is `probably' prime.  The probability of a false positive is (1/4)**REPS. A reasonable value of reps is 25.

     An implementation of the probabilistic primality test found in Seminumerical Algorithms (see References Knuth).

 -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to the greatest common divisor of OP1 and OP2.

 -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2)
     Compute the greatest common divisor of OP1 and OP2.  If ROP is not NULL, store the result there.

     If the result is small enough to fit in an unsigned long int, it is returned.  If the result does not fit, 0 is returned, and the result is equal to the argument OP1.  Note that the result will always fit if OP2 is non-zero.

 -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t B)
     Compute G, S, and T, such that AS + BT = G = gcd (A, B).  If T is NULL, that argument is not computed.

 -- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Compute the inverse of OP1 modulo OP2 and put the result in ROP.  Return non-zero if an inverse exist, zero otherwise.  When the function returns zero, do not assume anything about the value in ROP.

 -- Function: int mpz_jacobi (mpz_t OP1, mpz_t OP2)
 -- Function: int mpz_legendre (mpz_t OP1, mpz_t OP2)
     Compute the Jacobi and Legendre symbols, respectively.

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

 -- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
     Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2.

 -- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
 -- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
     Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2.

     These functions are actually implemented as macros.  They evaluate their arguments multiple times.

 -- Macro: int mpz_sgn (mpz_t OP)
     Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.

     This function is actually implemented as a macro.  It evaluates its arguments multiple times.

Logical and Bit Manipulation Functions
======================================

These functions behave as if two's complement arithmetic were used (although sign-magnitude is used by the actual implementation).

 -- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to OP1 logical-and OP2.

 -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
     Set ROP to OP1 inclusive-or OP2.

 -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
     Set ROP to the one's complement of OP.

 -- Function: unsigned long int mpz_popcount (mpz_t OP)
     For non-negative numbers, return the population count of OP.  For negative numbers, return the largest possible value (MAX_ULONG).

 -- Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
     If OP1 and OP2 are both non-negative, return the hamming distance between the two operands.  Otherwise, return the largest possible value (MAX_ULONG).

     It is possible to extend this function to return a useful value when the operands are both negative, but the current implementation returns MAX_ULONG in this case.  Do not depend on this behavior, since it will change in future versions of the library.

 -- Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int STARTING_BIT)
     Scan OP, starting with bit STARTING_BIT, towards more significant bits, until the first clear bit is found.  Return the index of the found bit.

 -- Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int STARTING_BIT)
     Scan OP, starting with bit STARTING_BIT, towards more significant bits, until the first set bit is found.  Return the index of the found bit.

 -- Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
     Set bit BIT_INDEX in OP1.

 -- Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
     Clear bit BIT_INDEX in OP1.

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.

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

 -- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
     Generate a random integer of at most MAX_SIZE limbs.  The generated random number doesn't satisfy any particular requirements of randomness. Negative random numbers are generated when MAX_SIZE is negative.

 -- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
     Generate a random integer of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation.  Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs.  Negative random numbers are generated when MAX_SIZE is negative.

 -- Function: size_t mpz_size (mpz_t OP)
     Return the size of OP measured in number of limbs.  If OP is zero, the returned value will be zero.

     This function is obsolete.  It will disappear from future MP releases.

 -- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
     Return the size of OP measured in number of digits in base BASE. The base may vary from 2 to 36.  The returned value will be exact or 1 too big.  If BASE is a power of 2, the returned value will always be exact.

     This function is useful in order to allocate the right amount of space before converting OP to a string.  The right amount of allocation is normally two more than the value returned by mpz_sizeinbase (one extra for a minus sign and one for the terminating '\0').

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

This chapter describes the MP functions for performing arithmetic on rational numbers.  These functions start with the prefix mpq_.

Rational numbers are stored in objects of type mpq_t.

All rational arithmetic functions assume operands have a canonical form, and canonicalize their result.  The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1.

Pure assignment functions do not canonicalize the assigned variable.  It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable.  Note that this is an incompatible change from version 1 of the library.

 -- Function: void mpq_canonicalize (mpq_t OP)
     Remove any factors that are common to the numerator and denominator of OP, and make the denominator positive.

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.

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 no functions for either input or output.  But there are two macros that allow us to apply any mpz function on the numerator or denominator of a rational number.  If these macros are used to assign to the rational number, mpq_canonicalize normally need to be called afterwards.

 -- Macro: mpz_t mpq_numref (mpq_t OP)
 -- Macro: mpz_t mpq_denref (mpq_t OP)
     Return a reference to the numerator and denominator of OP, respectively. The mpz functions can be used on the result of these macros.

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

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

These functions assign between either the numerator or denominator of a rational, and an integer.  Instead of using these functions, it is preferable to use the more general mechanisms mpq_numref and mpq_denref, together with mpz_set.

 -- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
     Copy NUMERATOR to the numerator of RATIONAL.  When this risks to make the numerator and denominator of RATIONAL have common factors, you have to pass RATIONAL to mpq_canonicalize before any operations are performed on RATIONAL.

     This function is equivalent to mpz_set (mpq_numref (RATIONAL), NUMERATOR).

 -- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
     Copy DENOMINATOR to the denominator of RATIONAL.  When this risks to make the numerator and denominator of RATIONAL have common factors, or if the denominator might be negative, you have to pass RATIONAL to mpq_canonicalize before any operations are performed on RATIONAL.

     In version 1 of the library, negative denominators were handled by copying the sign to the numerator.  That is no longer done.

     This function is equivalent to mpz_set (mpq_denref (RATIONAL), DENOMINATORS).

 -- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
     Copy the numerator of RATIONAL to the integer NUMERATOR, to prepare for integer operations on the numerator.

     This function is equivalent to mpz_set (NUMERATOR, mpq_numref (RATIONAL)).

 -- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
     Copy the denominator of RATIONAL to the integer DENOMINATOR, to prepare for integer operations on the denominator.

     This function is equivalent to mpz_set (DENOMINATOR, mpq_denref (RATIONAL)).

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

This is a description of the preliminary interface for floating-point arithmetic in GNU MP 2.

The floating-point functions expect arguments of type mpf_t.

The MP floating-point functions have an interface that is similar to the MP 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 MP 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, MP will not calculate with superfluous digits.  For example, if two low-precision numbers of nearly equal magnitude are added, the precision of the result will be limited to what is required to represent the result accurately.

The MP floating-point functions are 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 and Assignment 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.

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 function returns 0 if the entire string up to the '\0' is a valid number in base BASE.  Otherwise it returns -1.

Combined Initialization and Assignment Functions
------------------------------------------------

For convenience, MP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there.  These functions' names have the form 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, and a pointer to the string is returned.

     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.

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 gives the user the possibility to 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_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_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 MP functions, used to implement the high-level MP 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.

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

This example adds the number beginning at SRC1_PTR and the number beginning at SRC2_PTR and writes the sum at DEST_PTR.  All areas have SIZE limbs.

     cy = mpn_add_n (dest_ptr, src1_ptr, src2_ptr, 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, {s1_ptr, s1_size}.

 -- Function: mp_limb_t mpn_add_n (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)
     Add {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE}, and write the SIZE least significant limbs of the result to DEST_PTR. 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., SRC1_PTR equals SRC2_PTR, use mpn_lshift with a count of 1 for optimal speed.

 -- Function: mp_limb_t mpn_add_1 (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)
     Add {SRC1_PTR, SIZE} and SRC2_LIMB, and write the SIZE least significant limbs of the result to DEST_PTR.  Return carry, either 0 or 1.

 -- Function: mp_limb_t mpn_add (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR, mp_size_t SRC2_SIZE)
     Add {SRC1_PTR, SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}, and write the SRC1_SIZE least significant limbs of the result to DEST_PTR.  Return carry, either 0 or 1.

     This function requires that SRC1_SIZE is greater than or equal to SRC2_SIZE.

 -- Function: mp_limb_t mpn_sub_n (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)
     Subtract {SRC2_PTR, SRC2_SIZE} from {SRC1_PTR, SIZE}, and write the SIZE least significant limbs of the result to DEST_PTR.  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 * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)
     Subtract SRC2_LIMB from {SRC1_PTR, SIZE}, and write the SIZE least significant limbs of the result to DEST_PTR.  Return borrow, either 0 or 1.

 -- Function: mp_limb_t mpn_sub (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR, mp_size_t SRC2_SIZE)
     Subtract {SRC2_PTR, SRC2_SIZE} from {SRC1_PTR, SRC1_SIZE}, and write the SRC1_SIZE least significant limbs of the result to DEST_PTR.  Return borrow, either 0 or 1.

     This function requires that SRC1_SIZE is greater than or equal to SRC2_SIZE.

 -- Function: void mpn_mul_n (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)
     Multiply {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE}, and write the entire result to DEST_PTR.

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

 -- Function: mp_limb_t mpn_mul_1 (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)
     Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and write the SIZE least significant limbs of the product to DEST_PTR.  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 MP.  It is written in assembly for most targets.

     Don't call this function if SRC2_LIMB is a power of 2; use mpn_lshift with a count equal to the logarithm of SRC2_LIMB instead, for optimal speed.

 -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)
     Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and add the SIZE least significant limbs of the product to {DEST_PTR, SIZE} and write the result to DEST_PTR DEST_PTR.  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 MP.  It is written in assembly for most targets.

 -- Function: mp_limb_t mpn_submul_1 (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)
     Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and subtract the SIZE least significant limbs of the product from {DEST_PTR, SIZE} and write the result to DEST_PTR.  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 MP.  It is written in assembly for most targets.

 -- Function: mp_limb_t mpn_mul (mp_limb_t * DEST_PTR, const mp_limb_t * SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR, mp_size_t SRC2_SIZE)
     Multiply {SRC1_PTR, SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}, and write the result to DEST_PTR.  Return the most significant limb of the result.

     The destination has to have space for SRC1_SIZE + SRC1_SIZE limbs, even if the result might be one limb smaller.

     This function requires that SRC1_SIZE is greater than or equal to SRC2_SIZE.  The destination must be distinct from either input operands.

 -- Function: mp_size_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)
     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.

     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)
     Divide {S2P, S2SIZE} by S3LIMB, and write the quotient at R1P.  Return the remainder.

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

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

 -- Function: mp_size_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.

 -- Function: mp_limb_t mpn_divmod_1 (mp_limb_t * R1P, mp_limb_t * S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)
     This interface is obsolete.  It will disappear from future releases. Use mpn_divrem_1 in its stead.

 -- 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.

 -- 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 * DEST_PTR, 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 DEST_PTR, 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 DEST_PTR <= 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 * DEST_PTR, 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 DEST_PTR.  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 DEST_PTR >= SRC_PTR.

     This function is written in assembly for most targets.

 -- Function: mp_limp_t mpn_rshift (mp_limb_t * DEST_PTR, 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 DEST_PTR.  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 DEST_PTR <= SRC_PTR.

     This function is written in assembly for most targets.

 -- Function: int mpn_cmp (const mp_limb_t * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)
     Compare {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE} and return a positive value if src1 > src2, 0 of they are equal, and a negative value if src1 < src2.

 -- Function: mp_size_t mpn_gcd (mp_limb_t * DEST_PTR, mp_limb_t * SRC1_PTR, mp_size_t SRC1_SIZE, mp_limb_t * SRC2_PTR, mp_size_t SRC2_SIZE)
     Puts at DEST_PTR the greatest common divisor of {SRC1_PTR, SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}; both source operands are destroyed by the operation.  The size in limbs of the greatest common divisor is returned.

     {SRC1_PTR, SRC1_SIZE} must be odd, and {SRC2_PTR, SRC2_SIZE} must have at least as many bits as {SRC1_PTR, SRC1_SIZE}.

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

 -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t * SRC1_PTR, mp_size_t SRC1_SIZE, mp_limb_t SRC2_LIMB)
     Return the greatest common divisor of {SRC1_PTR, SRC1_SIZE} and SRC2_LIMB, where SRC2_LIMB (as well as SRC1_SIZE) must be different from 0.

 -- Function: mp_size_t mpn_gcdext (mp_limb_t * R1P, mp_limb_t * R2P, mp_limb_t * S1P, mp_size_t S1SIZE, mp_limb_t * S2P, mp_size_t S2SIZE)
     Puts at R1P the greatest common divisor of {S1P, S1SIZE} and {S2P, S2SIZE}.  The first cofactor is written at R2P.  Both source operands are destroyed by the operation.  The size in limbs of the greatest common divisor is returned.

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

 -- 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.

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

 -- 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.

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

 -- 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.

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

 -- Function: void mpn_random2 (mp_limb_t * R1P, mp_size_t R1SIZE)
     Generate a random number of length R1SIZE with long strings of zeros and ones in the binary representation, and store it at R1P.

     The generated random numbers are intended for testing the correctness of the implementation 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.

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 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 you 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, the MP functions use malloc, realloc, and free for memory allocation.  If malloc or realloc fails, the MP library terminates execution after printing a fatal error message to standard error.

For some applications, you may wish to allocate memory in other ways, or you may not want to have a fatal error when there is no more memory available.  To accomplish this, you can specify alternative memory allocation functions.

 -- 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.

     Make sure to call this function in such a way that there are no active MP objects that were allocated using the previously active allocation function!  Usually, that means that you have to call this function before any other MP 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
************

I would like to thank Gunnar Sjoedin and Hans Riesel for their help with mathematical problems, Richard Stallman for his help with design issues and for revising the first version of this manual, Brian Beuning and Doug Lea for their testing of early versions of the library.

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 MP 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 TMG Datakonsult, Sodermannagatan 5, 116 23 STOCKHOLM, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA.

References
**********

  * Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.

  * 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, 1995.

  * 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.

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

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