\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename gmp.info
@settitle GNU MP 2.1
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

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

@c smallbook

@iftex
@finalout
@end iftex

@c Note: the edition number is listed in *three* places; please update
@c all three.  Also, update the month and year where appropriate.

@c ==> Update edition number for settitle and subtitle, and in the
@c ==> following paragraph; update date, too.


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

Copyright (C) 1991, 1993, 1994, 1995, 1996, 1998 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.

@ignore
Permission is granted to process this file through TeX and print the
results, provided the printed document carries copying permission
notice identical to this one except for the removal of this paragraph
(this paragraph not being relevant to the printed manual).

@end ignore
Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions,
except that this permission notice may be stated in a translation approved
by the Foundation.
@end ifinfo

@setchapternewpage on
@titlepage
@c  use the new format for titles

@title GNU MP
@subtitle The GNU Multiple Precision Arithmetic Library
@subtitle Edition 2.1 SNAPSHOT
@subtitle November 1998

@author by Torbj@"orn Granlund, TMG Datakonsult
tege@@gnu.org

@c Include the Distribution inside the titlepage so
@c that headings are turned off.

@tex
\global\parindent=0pt
\global\parskip=8pt
\global\baselineskip=13pt
@end tex

@page
@vskip 0pt plus 1filll
Copyright @copyright{} 1991, 1993, 1994, 1995, 1996, 1998 Free Software Foundation, Inc.

@sp 2

Published by the Free Software Foundation @*
59 Temple Place - Suite 330 @*
Boston, MA 02111-1307, USA @*

Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions,
except that this permission notice may be stated in a translation approved
by the Foundation.
@end titlepage
@headings double

@ifinfo
@node Top, Copying, (dir), (dir)

@top GNU MP

This manual documents how to install and use the GNU multiple precision
arithmetic library, version 2.1.

@end ifinfo

@menu
* Copying::                   GMP Copying Conditions (LGPL).
* Introduction to MP::        Brief introduction to GNU MP.
* Installing MP::             How to configure and compile the MP library.
* MP Basics::                 What every MP user should now.
* Reporting Bugs::            How to usefully report bugs.
* Integer Functions::         Functions for arithmetic on signed integers.
* Rational Number Functions:: Functions for arithmetic on rational numbers.
* Floating-point Functions::  Functions for arithmetic on floats.
* Low-level Functions::       Fast functions for natural numbers.
* BSD Compatible Functions::  All functions found in BSD MP.
* Custom Allocation::         How to customize the internal allocation.

* Contributors::
* References::
* Concept Index::
* Function Index::
@end menu

@node Copying, Introduction to MP, Top, Top
@comment  node-name, next, previous,  up
@unnumbered GNU MP Copying Conditions
@cindex Copying conditions
@cindex Conditions for copying GNU MP

This library is @dfn{free}; this means that everyone is free to use it and
free to redistribute it on a free basis.  The library is not in the public
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.@refill

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.@refill

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.@refill

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.@refill

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

@node Introduction to MP, Installing MP, Copying, Top
@comment  node-name,  next,  previous,  up
@chapter 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.

For up-to-date information on MP,
@samp{http://www.matematik.su.se/~tege/gmp/}, is a good place to look.


@section How to use this Manual

Everyone should read @ref{MP Basics}.  If you need to install the library
yourself, you need to read @ref{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.


@node Installing MP, MP Basics, Introduction to MP, Top
@comment  node-name,  next,  previous,  up
@chapter Installing MP
@cindex Installation

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 @samp{make} program, plus some other
standard Unix utility programs.

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

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

@enumerate
@item
In most cases, @samp{./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 @samp{make} programs have the necessary features (VPATH) to support
this.  In particular, SunOS and Slowaris @samp{make} have bugs that makes them
unable to build from a separate object directory.  Use GNU @samp{make}
instead.

In addition to the standard cpu-vendor-os tuples, MP recognizes sparc8 and
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 @code{gcc} in your @code{PATH}, it will be used by default.  To
override this, pass @samp{-with-gcc=no} to @file{configure}.

@item
@samp{make}

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

@item
@samp{make check}

This will make sure MP was built correctly.  If you get error messages, please
report this to @samp{bug-gmp@@gnu.org}.  (@xref{Reporting Bugs}, for
information on what to include in useful bug reports.)

@item
@samp{make install}

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

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

There are some other useful make targets:

@itemize @bullet
@item
@samp{doc}

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

@item
@samp{ps}

Create a Postscript version of the manual, in @file{gmp.ps}.

@item
@samp{html}

Create a HTML version of the manual, in @file{gmp.html}.

@item
@samp{clean}

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

@item
@samp{distclean}

Delete all files not included in the distribution.

@item
@samp{uninstall}

Delete all files copied by @samp{make install}.
@end itemize


@section Known Build Problems

(You might find more up-to-date information at
@samp{http://www.matematik.su.se/~tege/gmp/}.)

GCC 2.7.2 (or 2.6.3) for the POWER and PowerPC can not be used to compile MP,
due to a bug in GCC.  If you want to use GCC for these machines, get GCC
2.7.2.1 (or later), or apply the patch below to GCC and recompile GCC.

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

The system compiler on NeXT is a massacred and old GCC, even if the compiler
calls itself @file{cc}.  This compiler cannot be used to build MP.  You need
to get a real GCC, and install that before you compile MP.  (NeXT have fixed
this in release 3.3 of their system.)

The system C compiler under SunOS 4 has a bug that makes it miscompile
mpq/get_d.c.  Use GCC instead if plan to use the function @code{mpq_get_d}.

Please report other problems to @samp{bug-gmp@@gnu.org}.
@xref{Reporting Bugs}.


Patch to apply to GCC 2.6.3 and 2.7.2:

@example
*** 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 ""
@end example

@node MP Basics, Reporting Bugs, Installing MP, Top
@comment  node-name,  next,  previous,  up
@chapter MP Basics

@menu
* Random Number Functions::
@end menu

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


@section Nomenclature and Types

@cindex Integer
@tindex @code{mpz_t}
@noindent
In this manual, @dfn{integer} usually means a multiple precision integer, as
defined by the MP library.  The C data type for such integers is @code{mpz_t}.
Here are some examples of how to declare such integers:

@example
mpz_t sum;

struct foo @{ mpz_t x, y; @};

mpz_t vec[20];
@end example

@cindex Rational number
@tindex @code{mpq_t}
@noindent
@dfn{Rational number} means a multiple precision fraction.  The C data type
for these fractions is @code{mpq_t}.  For example:

@example
mpq_t quotient;
@end example

@cindex Floating-point number
@tindex @code{mpf_t}
@noindent
@dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision
mantissa with an limited precision exponent.  The C data type for such objects
is @code{mpf_t}.

@cindex Limb
@tindex @code{mp_limb_t}
@noindent
A @dfn{limb} means the part of a multi-precision number that fits in a single
word.  (We chose this word because a limb of the human body is analogous to a
digit, only larger, and containing several digits.)  Normally a limb contains
32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.


@section Function Classes

There are six classes of functions in the MP library:

@enumerate
@item
Functions for signed integer arithmetic, with names beginning with
@code{mpz_}.  The associated type is @code{mpz_t}.  There are about 100
functions in this class.

@item
Functions for rational number arithmetic, with names beginning with
@code{mpq_}.  The associated type is @code{mpq_t}.  There are about 20
functions in this class, but the functions in the previous class can be used
for performing arithmetic on the numerator and denominator separately.

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

@item
Functions compatible with Berkeley MP, such as @code{itom}, @code{madd}, and
@code{mult}.  The associated type is @code{MINT}.

@item
Fast low-level functions that operate on natural numbers.  These are used by
the functions in the preceding groups, and you can also call them directly
from very time-critical user programs.  These functions' names begin with
@code{mpn_}.  There are about 30 (hard-to-use) functions in this class.

The associated type is array of @code{mp_limb_t}.

@item
Miscellaneous functions.  Functions for setting up custom allocation.
@end enumerate


@section 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,
@code{mpz_mul}, can be used like this: @code{mpz_mul (x, x, x)}.  This
computes the square of @var{x} and puts the result back in @var{x}.

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

Once you have initialized an MP variable, you don't need to worry about space
allocation for that variable.  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.


@section MP and reentrancy

The MP code is reentrant and thred-safe, with some exceptions:

@itemize @bullet
@item
The function @code{mpf_set_default_prec} saves the selected precision in
a global variable.

@item
The function @code{mp_set_memory_functions} uses several global
variables for storing the selected memory allocation functions.

@item
The memory allocation functions, @code{malloc} and friends, that GMP
uses unless directed to uses custom allocation, may not be reentrant.

@item
The random number functions (@code{mpz_random}, etc) use an underlying
random number generator from the C library, typically @code{mrand48} or
@code{random}.  These routines are not reentrant, since they rely on
global state.
@end itemize

If the memory allocation functions (@code{malloc} and friends, or
whatever functions set by a call to @code{mp_set_memory_functions}),
are not reentrant, MP will not be reentrant either.


@section Useful Macros and Constants

@deftypevr {Global Constant} {const int} mp_bits_per_limb
The number of bits per limb.
@end deftypevr

@defmac __GNU_MP_VERSION
@defmacx __GNU_MP_VERSION_MINOR
@defmacx __GNU_MP_VERSION_PATCHLEVEL
The major and minor MP version, and patch level, respectively, as integers.
For GMP i.j, these numbers will be i, j, and 0, respectively.
For GMP i.j.k, these numbers will be i, j, and k, respectively.
@end defmac


@section Compatibility with Version 1.x

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

@enumerate
@item Integer division functions round the result differently.  The obsolete
functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv},
@code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the
quotient towards
@ifinfo
@minus{}infinity).
@end ifinfo
@iftex
@tex
$-\infty$
@end tex
@end iftex
There are a lot of new functions for integer division, giving the user better
control over the rounding.

@item The function @code{mpz_mod} now compute the true @strong{mod} function.

@item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use
@strong{mod} for reduction.

@item 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
@code{mpq_canonicalize}.  This change was made for efficiency.

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

@item Several @code{mpn} functions have changed.  But they were intentionally
undocumented in previous releases.

@item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui}
are now implementated as macros, and thereby sometimes evaluate their
arguments multiple times.

@item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1
for 0^0.  (In version 1, they yielded 0.)

@end enumerate


@section Compatibility with Version 2.0.x

This version of MP is upward compatible with version 2.0, 2.0.1, and 2.0.2.


@section Getting the Latest Version of MP

The latest version of the MP library is available by at
@samp{ftp://ftp.gnu.org/pub/gnu}.  The file name will be @file{gmp-I.J.tar.gz}
or @file{gmp-I.J.K.tar.gz}.  Many sites around the world mirror
@samp{ftp.gnu.org}; please use a mirror site near you.

@node Random Number Functions,,,MP Basics
@comment  node-name,  next,  previous,  up
@section 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 seed
parameter that supplants global state, and generate good random numbers.

The seed parameter is of the type @code{mpz_t}.  It must be initialized
just like any MP variable, but also assigned a good starting seed.  The
size of the seed determines the maximum cycle of the generated random
numbers, and thus to some extent the "randomness" of the generated
numbers.  If good uniform random numbers are required, the seed variable
needs to be set to a value considerably greater than the largest
required random number.

The algorithm for assigning seed is critical if the generated random
numbers are to be used for important applications, such as generating
cryptographic keys.  One should also avoid generating multiple random
numbers from the same seed for such applications, since a sequence of
random numbers generated from the same seed are not truly independent.

@node Reporting Bugs, Integer Functions, MP Basics, Top
@comment  node-name,  next,  previous,  up
@chapter Reporting Bugs
@cindex 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 @samp{what `which cc`}, or,
if you're using gcc, @samp{gcc -v}.  Also, include the output from @samp{uname
-a}.

If your bug report is good, we will do our best to help you to get a corrected
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: @samp{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.


@node Integer Functions, Rational Number Functions, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter Integer Functions
@cindex Integer functions

This chapter describes the MP functions for performing integer arithmetic.
These functions start with the prefix @code{mpz_}.

MP integers are stored in objects of type @code{mpz_t}.

@menu
* Initializing Integers::
* Assigning Integers::
* Simultaneous Integer Init & Assign::
* Converting Integers::
* Integer Arithmetic::
* Comparison Functions::
* Integer Logic and Bit Fiddling::
* I/O of Integers::
* Miscellaneous Integer Functions::
@end menu

@node Initializing Integers, Assigning Integers, , Integer Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions

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

@deftypefun void mpz_init (mpz_t @var{integer})
Initialize @var{integer} with limb space and set the initial numeric value to
0.  Each variable should normally only be initialized once, or at least cleared
out (using @code{mpz_clear}) between each initialization.
@end deftypefun

Here is an example of using @code{mpz_init}:

@example
@{
  mpz_t integ;
  mpz_init (integ);
  @dots{}
  mpz_add (integ, @dots{});
  @dots{}
  mpz_sub (integ, @dots{});

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

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

@deftypefun void mpz_clear (mpz_t @var{integer})
Free the limb space occupied by @var{integer}.  Make sure to call this
function for all @code{mpz_t} variables when you are done with them.
@end deftypefun

@deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
Change the limb space allocation to @var{new_alloc} limbs.  This function is
not normally called from user code, but it can be used to give memory back to
the heap, or to increase the space of a variable to avoid repeated automatic
re-allocation.
@end deftypefun

@deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, mp_size_t @var{fixed_num_bits})
Allocate @strong{fixed} limb space for all @var{array_size} integers in
@var{integer_array}.  The fixed allocation for each integer in the array is
enough to store @var{fixed_num_bits}.  If the fixed space will be insufficient
for storing the result of a subsequent calculation, the result is
unpredictable.

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 @code{mpz_clear}!
@end deftypefun


@node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@subsection Assignment Functions
@cindex Integer assignment functions

These functions assign new values to already initialized integers
(@pxref{Initializing Integers}).

@deftypefun void mpz_set (mpz_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpz_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_set_d (mpz_t @var{rop}, double @var{op})
@deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op})
@deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op})
Set the value of @var{rop} from @var{op}.
@end deftypefun

@deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
Set the value of @var{rop} from @var{str}, a '\0'-terminated C string in base
@var{base}.  White space is allowed in the string, and is simply ignored.  The
base may vary from 2 to 36.  If @var{base} is 0, the actual base is determined
from the leading characters: if the first two characters are `0x' or `0X',
hexadecimal is assumed, otherwise if the first character is `0', octal is
assumed, otherwise decimal is assumed.

This function returns 0 if the entire string up to the '\0' is a valid
number in base @var{base}.  Otherwise it returns @minus{}1.
@end deftypefun

@deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
Swap the values @var{rop1} and @var{rop2} efficiently.
@end deftypefun


@node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@subsection Combined Initialization and Assignment Functions
@cindex 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 @code{mpz_init_set@dots{}}

Here is an example of using one:

@example
@{
  mpz_t pie;
  mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
  @dots{}
  mpz_sub (pie, @dots{});
  @dots{}
  mpz_clear (pie);
@}
@end example

@noindent
Once the integer has been initialized by any of the @code{mpz_init_set@dots{}}
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!

@deftypefun void mpz_init_set (mpz_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpz_init_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_init_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_init_set_d (mpz_t @var{rop}, double @var{op})
Initialize @var{rop} with limb space and set the initial numeric value from
@var{op}.
@end deftypefun

@deftypefun int mpz_init_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
Initialize @var{rop} and set its value like @code{mpz_set_str} (see its
documentation above for details).

If the string is a correct base @var{base} number, the function returns 0;
if an error occurs it returns @minus{}1.  @var{rop} is initialized even if
an error occurs.  (I.e., you have to call @code{mpz_clear} for it.)
@end deftypefun


@node Converting Integers,  Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Integer conversion functions
@cindex Conversion functions

This section describes functions for converting MP integers to standard C
types.  Functions for converting @emph{to} MP integers are described in
@ref{Assigning Integers} and @ref{I/O of Integers}.

@deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
Return limb #@var{n} from @var{op}.  This function allows for very efficient
decomposition of a number in its limbs.

The function @code{mpz_size} can be used to determine the useful range for
@var{n}.
@end deftypefun

@deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})
Return the least significant part from @var{op}.  This function combined with
@* @code{mpz_tdiv_q_2exp(@dots{}, @var{op}, CHAR_BIT*sizeof(unsigned long
int))} can be used to decompose an integer into unsigned longs.
@end deftypefun

@deftypefun {signed long int} mpz_get_si (mpz_t @var{op})
If @var{op} fits into a @code{signed long int} return the value of @var{op}.
Otherwise return the least significant part of @var{op}, with the same sign
as @var{op}.

If @var{op} is too large to fit in a @code{signed long int}, the returned
result is probably not very useful.  @c To find out if the value will fit, use
@c the function @code{mpz_fits_si}.
@end deftypefun

@deftypefun double mpz_get_d (mpz_t @var{op})
Convert @var{op} to a double.
@end deftypefun

@deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base may vary
from 2 to 36.

If @var{str} is NULL, space for the result string is allocated using the
default allocation function.

If @var{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
@var{str}, use @code{mpz_sizeinbase (@var{op}, @var{base}) + 2}.  The two
extra bytes are for a possible minus sign, and for the terminating null
character.

A pointer to the result string is returned.  This pointer will will either
equal @var{str}, or if that is NULL, will point to the allocated storage.
@end deftypefun


@need 2000
@node Integer Arithmetic, Comparison Functions, Converting Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Integer arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} + @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} + @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} times @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Add @var{op1} times @var{op2} to @var{rop}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{rop} + @var{op1} \times @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} times 2 raised to @var{op2}.  This operation can
also be defined as a left shift, @var{op2} steps.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times 2^{op2}$.  This operation can also be
defined as a left shift, @var{op2} steps.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpz_abs (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@subsection Division and mod functions

Division is undefined if the divisor is zero, and passing a zero divisor to the
divide or modulo functions, as well passing a zero mod argument to the
@code{mpz_powm} and @code{mpz_powm_ui} functions, will make these functions
intentionally divide by zero.  This 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:
@itemize @bullet
@item
Functions that truncate the quotient towards 0.  The names of these functions
start with @code{mpz_tdiv}.  The @samp{t} in the name is short for
@samp{truncate}.
@item
Functions that round the quotient towards
@ifinfo
@minus{}infinity).
@end ifinfo
@iftex
@tex
$-\infty$
@end tex
@end iftex
The names of these routines start with @code{mpz_fdiv}.  The @samp{f} in the
name is short for @samp{floor}.
@item
Functions that round the quotient towards
@ifinfo
+infinity.
@end ifinfo
@iftex
@tex
$+\infty$
@end tex
@end iftex
The names of these routines start with @code{mpz_cdiv}.  The @samp{c} in the
name is short for @samp{ceil}.
@end itemize

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

@deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
Set @var{q} to [@var{n}/@var{d}], truncated towards 0.  The function
@code{mpz_tdiv_q_ui} returns the absolute value of remainder.

The function @code{mpz_tdiv_q_ui} returns the absolute value of the remainder.
@end deftypefun

@deftypefun void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{r} to (@var{n} - [@var{n}/@var{d}] * @var{d}), where the quotient is
truncated towards 0.  Unless @var{r} becomes zero, it will get the same sign as
@var{n}.
@end ifinfo
@iftex
@tex
Set @var{r} to $(@var{n} - [@var{n}/@var{d}] \times @var{d})$, where the
quotient is truncated towards 0.  Unless @var{r} becomes zero, it will get the
same sign as @var{n}.
@end tex
@end iftex

The function @code{mpz_tdiv_r_ui} returns the absolute value of the remainder.
@end deftypefun

@deftypefun void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to [@var{n}/@var{d}], truncated towards 0.  Set @var{r} to (@var{n}
- [@var{n}/@var{d}] * @var{d}).  Unless @var{r} becomes zero, it will get the
same sign as @var{n}.  If @var{q} and @var{r} are the same variable, the
results are undefined.
@end ifinfo
@iftex
@tex
Set @var{q} to [@var{n}/@var{d}], truncated towards 0.  Set @var{r} to $(@var{n}
- [@var{n}/@var{d}] \times @var{d})$.  Unless @var{r} becomes zero, it will get the
same sign as @var{n}.  If @var{q} and @var{r} are the same variable, the
results are undefined.
@end tex
@end iftex

The function @code{mpz_tdiv_qr_ui} returns the absolute value of the remainder.
@end deftypefun

@deftypefun {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
Like @code{mpz_tdiv_r_ui}, but the remainder is not stored anywhere; its
absolute value is just returned.
@end deftypefun

@deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n}/@var{d}, rounded towards @minus{}infinity.
@end ifinfo
@iftex
@tex
Set @var{q} to $\lfloor@var{n}/@var{d}\rfloor$.
@end tex
@end iftex

The function @code{mpz_fdiv_q_ui} returns the remainder.
@end deftypefun

@deftypefun void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{r} to (@var{n} - @var{n}/@var{d} * @var{d}), where the quotient is
rounded towards @minus{}infinity.  Unless @var{r} becomes zero, it will get the
same sign as @var{d}.
@end ifinfo
@iftex
@tex
Set @var{r} to $(@var{n} - \lfloor@var{n}/@var{d}\rfloor \times @var{d})$.
Unless @var{r} becomes zero, it will get the same sign as @var{d}.
@end tex
@end iftex

The function @code{mpz_fdiv_r_ui} returns the remainder.
@end deftypefun

@deftypefun void mpz_fdiv_qr (mpz_t @var{r}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n}/@var{d}, rounded towards @minus{}infinity.  Set @var{r}
to (@var{n} - @var{n}/@var{d} * @var{d}).  Unless @var{r} becomes zero, it
will get the same sign as @var{d}.  If @var{q} and @var{r} are the same
variable, the results are undefined.
@end ifinfo
@iftex
@tex
Set @var{q} to $\lfloor@var{n}/@var{d}\rfloor$.  Set @var{r} to $(@var{n} -
\lfloor@var{n}/@var{d}\rfloor \times @var{d})$.  Unless @var{r} becomes zero,
it will get the same sign as @var{d}.  If @var{q} and @var{r} are the same
variable, the results are undefined.
@end tex
@end iftex

The function @code{mpz_fdiv_qr_ui} returns the remainder.
@end deftypefun

@deftypefun {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
Like @code{mpz_fdiv_r_ui}, but the remainder is not stored anywhere; it is just
returned.
@end deftypefun

@deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n}/@var{d}, rounded towards +infinity.
@end ifinfo
@iftex
@tex
Set @var{q} to $\lceil@var{n}/@var{d}\rceil$.
@end tex
@end iftex

The function @code{mpz_cdiv_q_ui} returns the negated remainder.
@end deftypefun

@deftypefun void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{r} to (@var{n} - @var{n}/@var{d} * @var{d}), where the quotient is
rounded towards +infinity.  Unless @var{r} becomes zero, it will get the
opposite sign as @var{d}.
@end ifinfo
@iftex
@tex
Set @var{r} to $(@var{n} - \lceil@var{n}/@var{d}\rceil \times @var{d})$.  Unless
@var{r} becomes zero, it will get the opposite sign as @var{d}.
@end tex
@end iftex

The function @code{mpz_cdiv_r_ui} returns the negated remainder.
@end deftypefun

@deftypefun void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n}/@var{d}, rounded towards +infinity.  Set @var{r}
to (@var{n} - @var{n}/@var{d} * @var{d}).  Unless @var{r} becomes zero, it
will get the opposite sign as @var{d}.  If @var{q} and @var{r} are the same
variable, the results are undefined.
@end ifinfo
@iftex
@tex
Set @var{q} to $\lceil@var{n}/@var{d}\rceil$.  Set @var{r} to $(@var{n} -
\lceil@var{n}/@var{d}\rceil \times @var{d})$.  Unless @var{r} becomes zero, it will
get the opposite sign as @var{d}.  If @var{q} and @var{r} are the same
variable, the results are undefined.
@end tex
@end iftex

The function @code{mpz_cdiv_qr_ui} returns the negated remainder.
@end deftypefun

@deftypefun {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, unsigned long int @var{d})
Like @code{mpz_tdiv_r_ui}, but the remainder is not stored anywhere; its
negated value is just returned.
@end deftypefun

@deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
Set @var{r} to @var{n} @code{mod} @var{d}.  The sign of the divisor is ignored;
the result is always non-negative.

The function @code{mpz_mod_ui} returns the remainder.
@end deftypefun

@deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
Set @var{q} to @var{n}/@var{d}.  This function produces correct results only
when it is known in advance that @var{d} divides @var{n}.

Since mpz_divexact is much faster than any of the other routines that produce
the quotient (@pxref{References} Jebelean), it is the best choice for instances
in which exact division is known to occur, such as reducing a rational to
lowest terms.
@end deftypefun

@deftypefun void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n} divided by 2 raised to @var{d}.  The quotient is truncated
towards 0.
@end ifinfo
@iftex
@tex
Set @var{q} to $@var{n}/2^{d}$.  The quotient is truncated towards 0.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Divide @var{n} by (2 raised to @var{d}) and put the remainder in @var{r}.
Unless it is zero, @var{r} will have the same sign as @var{n}.
@end ifinfo
@iftex
@tex
Divide @var{n} by $2^{d}$ and put the remainder in @var{r}.  Unless it is zero,
@var{r} will have the same sign as @var{n}.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Set @var{q} to @var{n} divided by 2 raised to @var{d}, rounded towards
@minus{}infinity.
@end ifinfo
@iftex
@tex
Set @var{q} to $\lfloor@var{n}/2^{d}\rfloor$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, unsigned long int @var{d})
@ifinfo
Divide @var{n} by (2 raised to @var{d}) and put the remainder in @var{r}.  The
sign of @var{r} will always be positive.
@end ifinfo
@iftex
@tex
Divide @var{n} by $2^{d}$ and put the remainder in @var{r}.  The sign of
@var{r} will always be positive.
@end tex
@end iftex

This operation can also be defined as masking of the @var{d} least significant
bits.
@end deftypefun

@subsection Exponentiation Functions

@deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
@deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod})
Set @var{rop} to (@var{base} raised to @var{exp}) @code{mod} @var{mod}.  If
@var{exp} is negative, the result is undefined.
@end deftypefun

@deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp})
@deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp})
Set @var{rop} to @var{base} raised to @var{exp}.
@ifinfo
The case of 0^0 yields 1.
@end ifinfo
@iftex
@tex
The case of $0^0$ yields 1.
@end tex
@end iftex
@end deftypefun

@subsection Root Functions

@deftypefun void mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n})
@ifinfo
Set @var{rop} to the truncated integer part of the @var{n}th root of @var{op}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $\lfloor\root n \of {op}\rfloor$, the truncated integer part
of the @var{n}th root of @var{op}.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})
@ifinfo
Set @var{rop} to the truncated integer part of the square root of @var{op}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $\lfloor\sqrt{@var{op}}\rfloor$, the truncated integer part of
the square root of @var{op}.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
@ifinfo
Set @var{rop1} to the truncated integer part of the square root of @var{op},
like @code{mpz_sqrt}.  Set @var{rop2} to
@var{op}@minus{}@var{rop1}*@var{rop1},
@end ifinfo
@iftex
@tex
Set @var{rop1} to $\lfloor\sqrt{@var{op}}\rfloor$, like @code{mpz_sqrt}.
Set @var{rop2} to $(@var{op} - @var{rop1}^2)$,
@end tex
@end iftex
(i.e., zero if @var{op} is a perfect square).

If @var{rop1} and @var{rop2} are the same variable, the results are
undefined.
@end deftypefun

@deftypefun int mpz_perfect_power_p (mpz_t @var{op})
@ifinfo
Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
@var{a} and @var{b}, with @var{b} > 1, such that @var{op} equals a raised to
b.  Return zero otherwise.
@end ifinfo
@iftex
@tex
Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
$a$ and $b$, with $b>1$, such that $@var{op}=a^b$.  Return zero otherwise.
@end tex
@end iftex
@end deftypefun

@deftypefun int mpz_perfect_square_p (mpz_t @var{op})
Return non-zero if @var{op} is a perfect square, i.e., if the square root of
@var{op} is an integer.  Return zero otherwise.
@end deftypefun


@subsection Number Theoretic Functions

@deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
If this function returns 0, @var{n} is definitely not prime.  If it
returns 1, then @var{n} is `probably' prime.  If it returns 2, then
@var{n} is surely prime.  Reasonable values of reps vary from 5 to 10; a
higher value lowers the probability for a non-prime to pass as a
`probable' prime.

The function uses Miller-Rabin's probabilistic test.
@end deftypefun

@deftypefun int mpz_prime_p (mpz_t @var{n})
Return non-zero if @var{n} is prime and zero if @var{n} is a non-prime.
This function is far slower than @code{mpz_probab_prime_p}, but then it
never returns non-zero for composite numbers.

(For practical purposes, using @code{mpz_probab_prime_p} is adequate.
The likelyhood of a programming error or hardware malfunction is orders
of magnitudes greater than the likelyhood for a composite to pass as a
prime, if the @var{reps} argument is in the suggested range.)
@end deftypefun

@deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.
The result is always positive even if either of or both input operands
are negative.
@end deftypefun

@deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Compute the greatest common divisor of @var{op1} and @var{op2}.  If
@var{rop} is not NULL, store the result there.

If the result is small enough to fit in an @code{unsigned long int}, it is
returned.  If the result does not fit, 0 is returned, and the result is equal
to the argument @var{op1}.  Note that the result will always fit if @var{op2}
is non-zero.
@end deftypefun

@deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})
Compute @var{g}, @var{s}, and @var{t}, such that @var{a}@var{s} +
@var{b}@var{t} = @var{g} = @code{gcd}(@var{a}, @var{b}).  If @var{t} is
NULL, that argument is not computed.
@end deftypefun

@deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
@end deftypefun

@deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Compute the inverse of @var{op1} modulo @var{op2} and put the result in
@var{rop}.  Return non-zero if an inverse exist, zero otherwise.  When the
function returns zero, @var{rop} is undefined.
@end deftypefun

@deftypefun int mpz_jacobi (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx int mpz_legendre (mpz_t @var{op1}, mpz_t @var{op2})
Compute the Jacobi and Legendre symbols, respectively.
@end deftypefun

@deftypefun {unsigned long int} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f})
Remove all occurences of the factor @var{f} from @var{op} and store the
result in @var{rop}.  Return the multiplicity of @var{f} in @var{op}.
@end deftypefun

@deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{op})
Set @var{rop} to @var{op}!, the factorial of @var{op}.
@end deftypefun

@deftypefun void mpz_bin_ui (mpz_t @var{rop}, mpz_t @var{n}, unsigned long int @var{k})
@deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, unsigned long int @var{n}, unsigned long int @var{k})
@ifinfo
Compute the binomial coefficient @var{n} over @var{k} and store the result in
@end ifinfo
@iftex
@tex
Compute the binomial coefficient $\left({n}\atop{k}\right)$ and store the result in
@end tex
@end iftex
@var{rop}.
@end deftypefun

@deftypefun void mpz_fib_ui (mpz_t @var{rop}, unsigned long int @var{n})
Compute the @var{n}th Fibonacci number and store the result in @var{rop}.
@end deftypefun


@node Comparison Functions, Integer Logic and Bit Fiddling, Integer Arithmetic, Integer Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions

@deftypefun int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
@ifinfo
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >
@var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}
< @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefn Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})
@deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2})
@ifinfo
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >
@var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}
< @var{op2}$.
@end tex
@end iftex

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


@deftypefun int mpz_cmp_abs (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx int mpz_cmp_abs_ui (mpz_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
value if @var{op1} > @var{op2}, zero if @var{op1} = @var{op2}, and a negative
value if @var{op1} < @var{op2}.
@end ifinfo
@iftex
@tex
Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
value if $|@var{op1}| > |@var{op2}|$, zero if $|@var{op1}| = |@var{op2}|$, and a
negative value if $|@var{op1}| < |@var{op2}|$.
@end tex
@end iftex
@end deftypefun

@deftypefn Macro int mpz_sgn (mpz_t @var{op})
@ifinfo
Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
@end ifinfo
@iftex
@tex
Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
@end tex
@end iftex

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

@node Integer Logic and Bit Fiddling, I/O of Integers, Comparison Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Logical and Bit Manipulation Functions
@cindex Logical functions
@cindex Bit manipulation functions

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

@deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} logical-and @var{op2}.
@end deftypefun

@deftypefun void mpz_ior (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} inclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} exclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the one's complement of @var{op}.
@end deftypefun

@deftypefun {unsigned long int} mpz_popcount (mpz_t @var{op})
For non-negative numbers, return the population count of @var{op}.  For
negative numbers, return the largest possible value (@var{MAX_ULONG}).
@end deftypefun

@deftypefun {unsigned long int} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
If @var{op1} and @var{op2} are both non-negative, return the hamming distance
between the two operands.  Otherwise, return the largest possible value
(@var{MAX_ULONG}).

It is possible to extend this function to return a useful value when the
operands are both negative, but the current implementation returns
@var{MAX_ULONG} in this case.  @strong{Do not depend on this behavior, since
it will change in a future release.}
@end deftypefun

@deftypefun {unsigned long int} mpz_scan0 (mpz_t @var{op}, unsigned long int @var{starting_bit})
Scan @var{op}, starting with bit @var{starting_bit}, towards more significant
bits, until the first clear bit is found.  Return the index of the found bit.
@end deftypefun

@deftypefun {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})
Scan @var{op}, starting with bit @var{starting_bit}, towards more significant
bits, until the first set bit is found.  Return the index of the found bit.
@end deftypefun

@deftypefun void mpz_setbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
Set bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun void mpz_clrbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
Clear bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun int mpz_tstbit (mpz_t @var{op}, unsigned long int @var{bit_index})
Check bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
@end deftypefun

@node I/O of Integers, Miscellaneous Integer Functions, Integer Logic and Bit Fiddling, Integer Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Integer input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream.  Passing a NULL pointer for a @var{stream} argument to any of
these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

@deftypefun size_t mpz_out_str (FILE *@var{stream}, int @var{base}, mpz_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base may vary from 2 to 36.

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

@deftypefun size_t mpz_inp_str (mpz_t @var{rop}, FILE *@var{stream}, int @var{base})
Input a possibly white-space preceded string in base @var{base} from stdio
stream @var{stream}, and put the read integer in @var{rop}.  The base may vary
from 2 to 36.  If @var{base} is 0, the actual base is determined from the
leading characters: if the first two characters are `0x' or `0X', hexadecimal
is assumed, otherwise if the first character is `0', octal is assumed,
otherwise decimal is assumed.

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

@deftypefun size_t mpz_out_raw (FILE *@var{stream}, mpz_t @var{op})
Output @var{op} on stdio stream @var{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 @code{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 @code{mpz_inp_raw} from GMP 1, because
of changes necessary for compatibility between 32-bit and 64-bit machines.
@end deftypefun

@deftypefun size_t mpz_inp_raw (mpz_t @var{rop}, FILE *@var{stream})
Input from stdio stream @var{stream} in the format written by
@code{mpz_out_raw}, and put the result in @var{rop}.  Return the number of
bytes read, or if an error occurred, return 0.

This routine can read the output from @code{mpz_out_raw} also from GMP 1, in
spite of changes necessary for compatibility between 32-bit and 64-bit
machines.
@end deftypefun


@need 2000
@node Miscellaneous Integer Functions,, I/O of Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous integer functions

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

@deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{max_size} limbs.  The generated
random number doesn't satisfy any particular requirements of randomness.
Negative random numbers are generated when @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_urandomb} or
@code{mpz_urandomm} instead.
@end deftypefun

@deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{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 @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_rrandomb} instead.
@end deftypefun

@deftypefun void mpz_urandomb (mpz_t @var{rop}, unsigned long int @var{n}, mpz_t @var{seed})
Generate a uniform random integer in the range
@ifinfo
0 to 2^@var{n} @minus{} 1,
@end ifinfo
@iftex
@tex
0 to $2^n-1$,
@end tex
@end iftex
inclusive.

The variable @var{seed} is used as a seed for generating the random
number.  It is updated with a new seed.
@end deftypefun

@deftypefun void mpz_urandomm (mpz_t @var{rop}, mpz_t @var{n}, mpz_t @var{seed})
Generate a uniform random integer in the range
@ifinfo
0 to @var{n}
@end ifinfo
@iftex
@tex
0 to $n-1$
@end tex
@end iftex

The variable @var{seed} is used as a seed for generating the random number.
It is updated with a new seed.
@end deftypefun

@deftypefun void mpz_rrandomb (mpz_t @var{rop}, unsigned long int @var{n}, mpz_t @var{seed})
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
@ifinfo
0 to 2^@var{n} @minus{} 1,
@end ifinfo
@iftex
@tex
0 to $2^n-1$,
@end tex
@end iftex
inclusive.

The variable @var{seed} is used as a seed for generating the random
number.  It is updated with a new seed.  Unlike other random number
functions, the @var{seed} variable should probably be much smaller than
the generated numbers, since the generated random numbers are of such
special nature.  In fact, it should probably be kept under
@ifinfo
2^100.
@end ifinfo
@iftex
@tex
$2^{100}$.
@end tex
@end iftex
@end deftypefun

@deftypefun size_t mpz_size (mpz_t @var{op})
Return the size of @var{op} measured in number of limbs.  If @var{op} is zero,
the returned value will be zero.
@c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)
@end deftypefun

@deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base})
Return the size of @var{op} measured in number of digits in base @var{base}.
The base may vary from 2 to 36.  The returned value will be exact or 1 too
big.  If @var{base} is a power of 2, the returned value will always be exact.

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


@node Rational Number Functions, Floating-point Functions, Integer Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Rational Number Functions
@cindex Rational number functions

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

Rational numbers are stored in objects of type @code{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.  @strong{Note that
this is an incompatible change from version 1 of the library.}

@deftypefun void mpq_canonicalize (mpq_t @var{op})
Remove any factors that are common to the numerator and denominator of
@var{op}, and make the denominator positive.
@end deftypefun

@menu
* Initializing Rationals::
* Assigning Rationals::
* Simultaneous Integer Init & Assign::
* Comparing Rationals::
* Applying Integer Functions::
* Miscellaneous Rational Functions::
@end menu

@node Initializing Rationals, Assigning Rationals, Rational Number Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions

@deftypefun void mpq_init (mpq_t @var{dest_rational})
Initialize @var{dest_rational} and set it to 0/1.  Each variable should
normally only be initialized once, or at least cleared out (using the function
@code{mpq_clear}) between each initialization.
@end deftypefun

@deftypefun void mpq_clear (mpq_t @var{rational_number})
Free the space occupied by @var{rational_number}.  Make sure to call this
function for all @code{mpq_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpq_set (mpq_t @var{rop}, mpq_t @var{op})
@deftypefunx void mpq_set_z (mpq_t @var{rop}, mpz_t @var{op})
Assign @var{rop} from @var{op}.
@end deftypefun

@deftypefun void mpq_set_ui (mpq_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2})
@deftypefunx void mpq_set_si (mpq_t @var{rop}, signed long int @var{op1}, unsigned long int @var{op2})
Set the value of @var{rop} to @var{op1}/@var{op2}.  Note that if @var{op1} and
@var{op2} have common factors, @var{rop} has to be passed to
@code{mpq_canonicalize} before any operations are performed on @var{rop}.
@end deftypefun

@node Assigning Rationals, Comparing Rationals, Initializing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions

@deftypefun void mpq_add (mpq_t @var{sum}, mpq_t @var{addend1}, mpq_t @var{addend2})
Set @var{sum} to @var{addend1} + @var{addend2}.
@end deftypefun

@deftypefun void mpq_sub (mpq_t @var{difference}, mpq_t @var{minuend}, mpq_t @var{subtrahend})
Set @var{difference} to @var{minuend} @minus{} @var{subtrahend}.
@end deftypefun

@deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})
@ifinfo
Set @var{product} to @var{multiplier} times @var{multiplicand}.
@end ifinfo
@iftex
@tex
Set @var{product} to $@var{multiplier} \times @var{multiplicand}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor})
Set @var{quotient} to @var{dividend}/@var{divisor}.
@end deftypefun

@deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand})
Set @var{negated_operand} to @minus{}@var{operand}.
@end deftypefun

@deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number})
Set @var{inverted_number} to 1/@var{number}.  If the new denominator is
zero, this routine will divide by zero.
@end deftypefun

@node Comparing Rationals, Applying Integer Functions, Assigning Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions

@deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})
@ifinfo
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >
@var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}
< @var{op2}$.
@end tex
@end iftex

To determine if two rationals are equal, @code{mpq_equal} is faster than
@code{mpq_cmp}.
@end deftypefun

@deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
@ifinfo
Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
@var{op1} > @var{num2}/@var{den2}, zero if @var{op1} = @var{num2}/@var{den2},
and a negative value if @var{op1} < @var{num2}/@var{den2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
$@var{op1} > @var{num2}/@var{den2}$, zero if $@var{op1} =
@var{num2}/@var{den2}$, and a negative value if $@var{op1} <
@var{num2}/@var{den2}$.
@end tex
@end iftex

This routine allows that @var{num2} and @var{den2} have common factors.

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

@deftypefn Macro int mpq_sgn (mpq_t @var{op})
@ifinfo
Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
@end ifinfo
@iftex
@tex
Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
@end tex
@end iftex

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

@deftypefun int mpq_equal (mpq_t @var{op1}, mpq_t @var{op2})
Return non-zero if @var{op1} and @var{op2} are equal, zero if they are
non-equal.  Although @code{mpq_cmp} can be used for the same purpose, this
function is much faster.
@end deftypefun

@node Applying Integer Functions, Miscellaneous Rational Functions, Comparing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Applying Integer Functions to Rationals

The set of @code{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 @code{mpz} function on the numerator or denominator of a rational
number.  If these macros are used to assign to the rational number,
@code{mpq_canonicalize} normally need to be called afterwards.

@deftypefn Macro mpz_t mpq_numref (mpq_t @var{op})
@deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})
Return a reference to the numerator and denominator of @var{op}, respectively.
The @code{mpz} functions can be used on the result of these macros.
@end deftypefn

@need 2000
@node Miscellaneous Rational Functions, , Applying Integer Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions

@deftypefun double mpq_get_d (mpq_t @var{op})
Convert @var{op} to a double.
@end deftypefun

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

@deftypefun void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
Copy @var{numerator} to the numerator of @var{rational}.  When this risks to
make the numerator and denominator of @var{rational} have common factors, you
have to pass @var{rational} to @code{mpq_canonicalize} before any operations
are performed on @var{rational}.

This function is equivalent to
@code{mpz_set (mpq_numref (@var{rational}), @var{numerator})}.
@end deftypefun

@deftypefun void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
Copy @var{denominator} to the denominator of @var{rational}.  When this risks
to make the numerator and denominator of @var{rational} have common factors,
or if the denominator might be negative, you have to pass @var{rational} to
@code{mpq_canonicalize} before any operations are performed on @var{rational}.

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

This function is equivalent to
@code{mpz_set (mpq_denref (@var{rational}), @var{denominators})}.
@end deftypefun

@deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
Copy the numerator of @var{rational} to the integer @var{numerator}, to
prepare for integer operations on the numerator.

This function is equivalent to
@code{mpz_set (@var{numerator}, mpq_numref (@var{rational}))}.
@end deftypefun

@deftypefun void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
Copy the denominator of @var{rational} to the integer @var{denominator}, to
prepare for integer operations on the denominator.

This function is equivalent to
@code{mpz_set (@var{denominator}, mpq_denref (@var{rational}))}.
@end deftypefun


@node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions

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

The floating-point functions expect arguments of type @code{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
@code{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.

@cindex User-defined precision
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 @emph{not} intended as a smooth extension
to the IEEE P754 arithmetic.  Specifically, the results obtained on one
computer often differs from the results obtained on a computer with a
different word size.

@menu
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::
@end menu

@node Initializing Floats, Assigning Floats, , Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions

@deftypefun void mpf_set_default_prec (unsigned long int @var{prec})
Set the default precision to be @strong{at least} @var{prec} bits.  All
subsequent calls to @code{mpf_init} will use this precision, but previously
initialized variables are unaffected.
@end deftypefun

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

@deftypefun void mpf_init (mpf_t @var{x})
Initialize @var{x} to 0.  Normally, a variable should be initialized once only
or at least be cleared, using @code{mpf_clear}, between initializations.  The
precision of @var{x} is undefined unless a default precision has already been
established by a call to @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun void mpf_init2 (mpf_t @var{x}, unsigned long int @var{prec})
Initialize @var{x} to 0 and set its precision to be @strong{at least}
@var{prec} bits.  Normally, a variable should be initialized once only or at
least be cleared, using @code{mpf_clear}, between initializations.
@end deftypefun

@deftypefun void mpf_clear (mpf_t @var{x})
Free the space occupied by @var{x}.  Make sure to call this function for all
@code{mpf_t} variables when you are done with them.
@end deftypefun

@need 2000
Here is an example on how to initialize floating-point variables:
@example
@{
  mpf_t x, y;
  mpf_init (x);			/* use default precision */
  mpf_init2 (y, 256);		/* precision @emph{at least} 256 bits */
  @dots{}
  /* Unless the program is about to exit, do ... */
  mpf_clear (x);
  mpf_clear (y);
@}
@end example

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.

@deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.
Since changing the precision involves calls to @code{realloc}, this routine
should not be called in a tight loop.
@end deftypefun

@deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})
Return the precision actually used for assignments of @var{op}.
@end deftypefun

@deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  This
is a low-level function that does not change the allocation.  The @var{prec}
argument must not be larger that the precision previously returned by
@code{mpf_get_prec}.  It is crucial that the precision of @var{rop} is
ultimately reset to exactly the value returned by @code{mpf_get_prec} before
any calls to @code{mpf_set_prec_raw}.
@end deftypefun


@node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@subsection Assignment Functions
@cindex Float assignment functions

These functions assign new values to already initialized floats
(@pxref{Initializing Floats}).

@deftypefun void mpf_set (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_set_d (mpf_t @var{rop}, double @var{op})
@deftypefunx void mpf_set_z (mpf_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpf_set_q (mpf_t @var{rop}, mpq_t @var{op})
Set the value of @var{rop} from @var{op}.
@end deftypefun

@deftypefun int mpf_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
Set the value of @var{rop} from the string in @var{str}.  The string is of the
form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.
@samp{M} is the mantissa and @samp{N} is the exponent.  The mantissa is always
in the specified base.  The exponent is either in the specified base or, if
@var{base} is negative, in decimal.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

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

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 @var{base}.  Otherwise it returns @minus{}1.
@end deftypefun


@node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@subsection Combined Initialization and Assignment Functions
@cindex 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 @code{mpf_init_set@dots{}}

Once the float has been initialized by any of the @code{mpf_init_set@dots{}}
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!

@deftypefun void mpf_init_set (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_init_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_init_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_init_set_d (mpf_t @var{rop}, double @var{op})
Initialize @var{rop} and set its value from @var{op}.

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun int mpf_init_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
Initialize @var{rop} and set its value from the string in @var{str}.  See
@code{mpf_set_str} above for details on the assignment operation.

Note that @var{rop} is initialized even if an error occurs.  (I.e., you have to
call @code{mpf_clear} for it.)

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun


@node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Conversion functions

@deftypefun double mpf_get_d (mpf_t @var{op})
Convert @var{op} to a double.
@end deftypefun

@deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base may vary
from 2 to 36.  Generate at most @var{n_digits} significant digits, or if
@var{n_digits} is 0, the maximum number of digits accurately representable by
@var{op}.

If @var{str} is NULL, space for the mantissa is allocated using the default
allocation function.

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

The exponent is written through the pointer @var{expptr}.

If @var{n_digits} is 0, the maximum number of digits meaningfully achievable
from the precision of @var{op} will be generated.  Note that the space
requirements for @var{str} in this case will be impossible for the user to
predetermine.  Therefore, you need to pass NULL for the string argument
whenever @var{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 @var{expptr}.

A pointer to the result string is returned.  This pointer will will either
equal @var{str}, or if that is NULL, will point to the allocated storage.
@end deftypefun


@node Float Arithmetic, Float Comparison, Converting Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Float arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} + @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} + @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_ui_sub (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_sub_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} times @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times @var{op2}$.
@end tex
@end iftex
@end deftypefun

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.

@deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @var{op1}/@var{op2}.
@end deftypefun

@deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op})
@ifinfo
Set @var{rop} to the square root of @var{op}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $\sqrt{@var{op}}$.
@end tex
@end iftex
@end deftypefun

@c @deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@c Set @var{rop} to @var{op1} raised to @var{op2}.
@c @end deftypefun

@deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpf_abs (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} times 2 raised to @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times 2^{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@ifinfo
Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1}/2^{op2}$.
@end tex
@end iftex
@end deftypefun

@node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Float comparisons functions
@cindex Comparison functions

@deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, unsigned long int @var{op2})
@deftypefunx int mpf_cmp_si (mpf_t @var{op1}, signed long int @var{op2})
@ifinfo
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >
@var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}
< @var{op2}$.
@end tex
@end iftex
@end deftypefun

@deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, unsigned long int op3)
Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
equal, zero otherwise.  I.e., test of @var{op1} and @var{op2} are
approximately equal.
@end deftypefun

@deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
Compute the relative difference between @var{op1} and @var{op2} and store the
result in @var{rop}.
@end deftypefun

@deftypefn Macro int mpf_sgn (mpf_t @var{op})
@ifinfo
Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
@end ifinfo
@iftex
@tex
Return $+1$ if $@var{op} > 0$, 0 if $@var{op} = 0$, and $-1$ if $@var{op} < 0$.
@end tex
@end iftex

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

@node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Float input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream.  Passing a NULL pointer for a @var{stream} argument to any of
these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

@deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in
base @var{base}.  The base may vary from 2 to 36.  Print at most
@var{n_digits} significant digits, or if @var{n_digits} is 0, the maximum
number of digits accurately representable by @var{op}.

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

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

@deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
Input a string in base @var{base} from stdio stream @var{stream}, and put the
read float in @var{rop}.  The string is of the form @samp{M@@N} or, if the
base is 10 or less, alternatively @samp{MeN}.  @samp{M} is the mantissa and
@samp{N} is the exponent.  The mantissa is always in the specified base.  The
exponent is either in the specified base or, if @var{base} is negative, in
decimal.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

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

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

@c @deftypefun void mpf_out_raw (FILE *@var{stream}, mpf_t @var{float})
@c Output @var{float} on stdio stream @var{stream}, in raw binary
@c format.  The float is written in a portable format, with 4 bytes of
@c size information, and that many bytes of limbs.  Both the size and the
@c limbs are written in decreasing significance order.
@c @end deftypefun

@c @deftypefun void mpf_inp_raw (mpf_t @var{float}, FILE *@var{stream})
@c Input from stdio stream @var{stream} in the format written by
@c @code{mpf_out_raw}, and put the result in @var{float}.
@c @end deftypefun


@node Miscellaneous Float Functions, , I/O of Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous float functions

@deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{max_exp})
Generate a random float of at most @var{max_size} limbs, with long strings of
zeros and ones in the binary representation.  The exponent of the number is in
the interval @minus{}@var{exp} to @var{exp}.  This function is useful for
testing functions and algorithms, since this kind of random numbers have
proven to be more likely to trigger corner-case bugs.  Negative random numbers
are generated when @var{max_size} is negative.
@end deftypefun

@c @deftypefun size_t mpf_size (mpf_t @var{op})
@c Return the size of @var{op} measured in number of limbs.  If @var{op} is
@c zero, the returned value will be zero.  (@xref{Nomenclature}, for an
@c explanation of the concept @dfn{limb}.)
@c
@c @strong{This function is obsolete.  It will disappear from future MP
@c releases.}
@c @end deftypefun

@node Low-level Functions, BSD Compatible Functions, Floating-point Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Low-level Functions
@cindex 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 @code{mpn_}.

@c 1. Some of these function clobber input operands.
@c

The @code{mpn} functions are designed to be as fast as possible, @strong{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 @code{mpn} functions is the base for the implementation of the @code{mpz_},
@code{mpf_}, and @code{mpq_} functions.

This example adds the number beginning at @var{s1p} and the number
beginning at @var{s2p} and writes the sum at @var{destp}.  All areas
have @var{size} limbs.

@example
cy = mpn_add_n (destp, s1p, s2p, size)
@end example

@noindent
In the notation used here, a source operand is identified by the pointer to
the least significant limb, and the limb count in braces.  For example,
@{s1p, s1size@}.

@deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{size})
Add @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@}, and
write the @var{size} least significant limbs of the result to @var{rp}.
Return carry, either 0 or 1.

This is the lowest-level function for addition.  It is the preferred function
for addition, since it is written in assembly for most targets.  For addition
of a variable to itself (i.e., @var{s1p} equals @var{s2p}, use
@code{mpn_lshift} with a count of 1 for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
Add @{@var{s1p}, @var{size}@} and @var{s2limb}, and write the
@var{size} least significant limbs of the result to @var{rp}.  Return
carry, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
Add @{@var{s1p}, @var{s1size}@} and @{@var{s2p},
@var{s2size}@}, and write the @var{s1size} least significant limbs of
the result to @var{rp}.  Return carry, either 0 or 1.

This function requires that @var{s1size} is greater than or equal to
@var{s2size}.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{size})
Subtract @{@var{s2p}, @var{s2size}@} from @{@var{s1p},
@var{size}@}, and write the @var{size} least significant limbs of the result
to @var{rp}.  Return borrow, either 0 or 1.

This is the lowest-level function for subtraction.  It is the preferred
function for subtraction, since it is written in assembly for most targets.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
Subtract @var{s2limb} from @{@var{s1p}, @var{size}@}, and write the
@var{size} least significant limbs of the result to @var{rp}.  Return
borrow, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
Subtract @{@var{s2p}, @var{s2size}@} from @{@var{s1p},
@var{s1size}@}, and write the @var{s1size} least significant limbs of
the result to @var{rp}.  Return borrow, either 0 or 1.

This function requires that @var{s1size} is greater than or equal to
@var{s2size}.
@end deftypefun

@deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{size})
Multiply @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@},
and write the @strong{entire} result to @var{rp}.

The destination has to have space for 2@var{size} limbs, even if the
significant result might be one limb smaller.
@end deftypefun

@deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and write the
@var{size} least significant limbs of the product to @var{rp}.  Return
the most significant limb of the product.

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 @var{s2limb} is a power of 2; use
@code{mpn_lshift} with a count equal to the logarithm of @var{s2limb}
instead, for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and add the
@var{size} least significant limbs of the product to @{@var{rp},
@var{size}@} and write the result to @var{rp}.  Return
the most significant limb of the product, plus carry-out from the addition.

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.
@end deftypefun

@deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{size}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{size}@} and @var{s2limb}, and subtract the
@var{size} least significant limbs of the product from @{@var{rp},
@var{size}@} and write the result to @var{rp}.  Return the most
significant limb of the product, minus borrow-out from the subtraction.

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.
@end deftypefun

@deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
Multiply @{@var{s1p}, @var{s1size}@} and @{@var{s2p},
@var{s2size}@}, and write the result to @var{rp}.  Return the most
significant limb of the result.

The destination has to have space for @var{s1size} + @var{s2size}
limbs, even if the result might be one limb smaller.

This function requires that @var{s1size} is greater than or equal to
@var{s2size}.  The destination must be distinct from either input operands.
@end deftypefun

@deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{xsize}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2size}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3size})
Divide @{@var{rs2p}, @var{rs2size}@} by @{@var{s3p}, @var{s3size}@}, and write
the quotient at @var{r1p}, with the exception of the most significant limb,
which is returned.  The remainder replaces the dividend at @var{rs2p}; it will
be @var{s3size} limbs long (i.e., as many limbs as the divisor).

In addition to an integer quotient, @var{xsize} fraction limbs are developed,
and stored after the integral limbs.  For most usages, @var{xsize} will be
zero.

It is required that @var{rs2size} is greater than or equal to @var{s3size}.
It is required that the most significant bit of the divisor is set.

If the quotient is not needed, pass @var{rs2p} + @var{s3size} as @var{r1p}.
Aside from that special case, no overlap between arguments is permitted.

Return the most significant limb of the quotient, either 0 or 1.

The area at @var{r1p} needs to be @var{rs2size} @minus{} @var{s3size} +
@var{xsize} limbs large.
@end deftypefun

@deftypefun mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{xsize}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size}, mp_limb_t @var{s3limb})
Divide @{@var{s2p}, @var{s2size}@} by @var{s3limb}, and write the quotient
at @var{r1p}.  Return the remainder.

In addition to an integer quotient, @var{xsize} fraction limbs are developed,
and stored after the integral limbs.  For most usages, @var{xsize} will be
zero.

The areas at @var{r1p} and @var{s2p} have to be identical or completely
separate, not partially overlapping.
@end deftypefun

@deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2size}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3size})
@strong{This interface is obsolete.  It will disappear from future releases.
Use @code{mpn_divrem} in its stead.}
@end deftypefun

@deftypefun mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size}, mp_limb_t @var{s3limb})
@strong{This interface is obsolete.  It will disappear from future releases.
Use @code{mpn_divrem_1} in its stead.}
@end deftypefun

@deftypefun mp_limb_t mpn_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb})
Divide @{@var{s1p}, @var{s1size}@} by @var{s2limb}, and return the remainder.
@end deftypefun

@deftypefun mp_limb_t mpn_preinv_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb}, mp_limb_t @var{s3limb})
@strong{This interface is obsolete.  It will disappear from future releases.
Use @code{mpn_mod_1} in its stead.}
@end deftypefun

@deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2size}, unsigned long int @var{d})
The function puts the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of
@var{q} =
@{@var{s1p}, @var{s1size}@}/@{@var{s2p}, @var{s2size}@}
mod 2^@var{d}
at @var{rp},
and returns the high @var{d} mod @var{BITS_PER_MP_LIMB} bits of @var{q}.

@{@var{s1p}, @var{s1size}@} - @var{q} * @{@var{s2p}, @var{s2size}@}
mod 2^(@var{s1size}*@var{BITS_PER_MP_LIMB})
is placed at @var{s1p}.
Since the low [@var{d}/@var{BITS_PER_MP_LIMB}] limbs of
this difference are zero, it is possible to overwrite the low limbs at
@var{s1p} with this difference,
provided @var{rp} <= @var{s1p}.

This function requires that @var{s1size} * @var{BITS_PER_MP_LIMB} >= @var{D},
and that @{@var{s2p}, @var{s2size}@} is odd.

@strong{This interface is preliminary.  It might change incompatibly in
future revisions.}
@end deftypefun

@deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})
Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the left, and
write the @var{src_size} least significant limbs of the result to
@var{rp}.  @var{count} might be in the range 1 to n @minus{} 1, on an
n-bit machine. The bits shifted out to the left are returned.

Overlapping of the destination space and the source space is allowed in this
function, provided @var{rp} >= @var{src_ptr}.

This function is written in assembly for most targets.
@end deftypefun

@deftypefun mp_limp_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{src_ptr}, mp_size_t @var{src_size}, unsigned long int @var{count})
Shift @{@var{src_ptr}, @var{src_size}@} @var{count} bits to the right, and
write the @var{src_size} most significant limbs of the result to
@var{rp}.  @var{count} might be in the range 1 to n @minus{} 1, on an
n-bit machine.  The bits shifted out to the right are returned.

Overlapping of the destination space and the source space is allowed in this
function, provided @var{rp} <= @var{src_ptr}.

This function is written in assembly for most targets.
@end deftypefun

@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{size})
Compare @{@var{s1p}, @var{size}@} and @{@var{s2p}, @var{size}@} and
return a positive value if s1 > src2, 0 of they are equal, and a negative
value if s1 < src2.
@end deftypefun

@deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
Puts at @var{rp} the greatest common divisor of @{@var{s1p},
@var{s1size}@} and @{@var{s2p}, @var{s2size}@}; both source
operands are destroyed by the operation.  The size in limbs of the greatest
common divisor is returned.

@{@var{s1p}, @var{s1size}@} must have at least as many bits as
@{@var{s2p}, @var{s2size}@}, and @{@var{s2p}, @var{s2size}@} must be odd.

@strong{This interface is preliminary.  It might change incompatibly in
future revisions.}
@end deftypefun

@deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t @var{s2limb})
Return the greatest common divisor of @{@var{s1p}, @var{s1size}@}
and @var{s2limb}, where @var{s2limb} (as well as @var{s1size})
must be different from 0.
@end deftypefun

@deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, mp_size_t *@var{r2size}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size}, mp_limb_t *@var{s2p}, mp_size_t @var{s2size})
Compute the greatest common divisor of @{@var{s1p}, @var{s1size}@} and
@{@var{s2p}, @var{s2size}@}, and store it at @var{r1p}.  Write the the
first cofactor at @var{r2p}.  Both source operands are clobbered.

@{@var{s1p}, @var{s1size}@} must be greater or equal to @{@var{s2p},
@var{s2size}@}.  Neither operand may equal 0.

The size and sign of the first cofactor are written at *@var{r2size}; the
actual size of the cofactor is the absolute value of *@var{r2size}; the sign
of the cofactor is negative iff *@var{r2size} is negative.

The size in limbs of the greatest common divisor is returned.
@end deftypefun

@deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{size})
Compute the square root of @{@var{sp}, @var{size}@} and put the result at
@var{r1p}.  Write the remainder at @var{r2p}, unless @var{r2p} is NULL.

Return the size of the remainder, whether @var{r2p} was NULL or non-NULL.
Iff the operand was a perfect square, the return value will be 0.

The areas at @var{r1p} and @var{sp} have to be distinct.  The areas at
@var{r2p} and @var{sp} have to be identical or completely separate, not
partially overlapping.

@ifinfo
The area at @var{r1p} needs to have space for ceil(@var{size}/2) limbs.
@end ifinfo
@iftex
@tex
The area at @var{r1p} needs to have space for $\lceil@var{size}/2\rceil$ limbs.
@end tex
@end iftex
The area at @var{r2p} needs to be @var{size} limbs large.

@strong{This interface is preliminary.  It might change incompatibly in
future revisions.}
@end deftypefun

@deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1size})
Convert @{@var{s1p}, @var{s1size}@} to a raw unsigned char array in base
@var{base}.  The string is not in ASCII; to convert it to printable format,
add the ASCII codes for @samp{0} or @samp{A}, depending on the base and
range.  There may be leading zeros in the string.

The area at @var{s1p} is clobbered.

Return the number of characters in @var{str}.

The area at @var{str} has to have space for the largest possible number
represented by a @var{s1size} long limb array, plus one extra character.
@end deftypefun

@deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{r1p}, const char *@var{str}, size_t @var{strsize}, int @var{base})
Convert the raw unsigned char array at @var{str} of length @var{strsize} to
a limb array @{@var{s1p}, @var{s1size}@}.  The base of @var{str} is
@var{base}.

Return the number of limbs stored in @var{r1p}.
@end deftypefun

@deftypefun {unsigned long int} mpn_scan0 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next clear bit.

It is required that there be a clear bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.

@strong{This interface is preliminary.  It might change incompatibly in
future revisions.}
@end deftypefun

@deftypefun {unsigned long int} mpn_scan1 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next set bit.

It is required that there be a set bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.

@strong{This interface is preliminary.  It might change incompatibly in
future revisions.}
@end deftypefun

@deftypefun void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1size})
Generate a random number of length @var{r1size} with long strings of zeros
and ones in the binary representation, and store it at @var{r1p}.

The generated random numbers are intended for testing the correctness of the
implementation of the @code{mpn} routines.
@end deftypefun

@deftypefun {unsigned long int} mpn_popcount (const mp_limb_t *@var{s1p}, unsigned long int @var{size})
Count the number of set bits in @{@var{s1p}, @var{size}@}.
@end deftypefun

@deftypefun {unsigned long int} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, unsigned long int @var{size})
Compute the hamming distance between @{@var{s1p}, @var{size}@} and
@{@var{s2p}, @var{size}@}.
@end deftypefun

@deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{size})
Return non-zero iff @{@var{s1p}, @var{size}@} is a perfect square.
@end deftypefun


@node BSD Compatible Functions, Custom Allocation, Low-level Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Berkeley MP Compatible Functions
@cindex BSD 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
@code{MINT} objects is more error prone, and the @code{pow} function collides
with @code{pow} in @file{libm.a}.

@cindex @file{mp.h}
Include the header @file{mp.h} to get the definition of the necessary types
and functions.  If you are on a BSD derived system, make sure to include GNU
@file{mp.h} if you are going to link the GNU @file{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 @file{mp.h}.

@deftypefun {MINT *} itom (signed short int @var{initial_value})
Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
Initialize the integer to @var{initial_value}.  Return a pointer to the
@code{MINT} object.
@end deftypefun

@deftypefun {MINT *} xtom (char *@var{initial_value})
Allocate an integer consisting of a @code{MINT} object and dynamic limb space.
Initialize the integer from @var{initial_value}, a hexadecimal, '\0'-terminate
C string.  Return a pointer to the @code{MINT} object.
@end deftypefun

@deftypefun void move (MINT *@var{src}, MINT *@var{dest})
Set @var{dest} to @var{src} by copying.  Both variables must be previously
initialized.
@end deftypefun

@deftypefun void madd (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination})
Add @var{src_1} and @var{src_2} and put the sum in @var{destination}.
@end deftypefun

@deftypefun void msub (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination})
Subtract @var{src_2} from @var{src_1} and put the difference in
@var{destination}.
@end deftypefun

@deftypefun void mult (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination})
Multiply @var{src_1} and @var{src_2} and put the product in
@var{destination}.
@end deftypefun

@deftypefun void mdiv (MINT *@var{dividend}, MINT *@var{divisor}, MINT *@var{quotient}, MINT *@var{remainder})
@deftypefunx void sdiv (MINT *@var{dividend}, signed short int @var{divisor}, MINT *@var{quotient}, signed short int *@var{remainder})
Set @var{quotient} to @var{dividend}/@var{divisor}, and @var{remainder} to
@var{dividend} mod @var{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.
@end deftypefun

@deftypefun void msqrt (MINT *@var{operand}, MINT *@var{root}, MINT *@var{remainder})
@ifinfo
Set @var{root} to the truncated integer part of the square root of
@var{operand}.  Set @var{remainder} to
@var{operand}@minus{}@var{root}*@var{root},
@end ifinfo
@iftex
@tex
Set @var{root} to $\lfloor\sqrt{@var{operand}}\rfloor$, like
@code{mpz_sqrt}.  Set @var{remainder} to $(operand - root^2)$,
@end tex
@end iftex
(i.e., zero if @var{operand} is a perfect square).

If @var{root} and @var{remainder} are the same variable, the results are
undefined.
@end deftypefun

@deftypefun void pow (MINT *@var{base}, MINT *@var{exp}, MINT *@var{mod}, MINT *@var{dest})
Set @var{dest} to (@var{base} raised to @var{exp}) modulo @var{mod}.
@end deftypefun

@deftypefun void rpow (MINT *@var{base}, signed short int @var{exp}, MINT *@var{dest})
Set @var{dest} to @var{base} raised to @var{exp}.
@end deftypefun

@deftypefun void gcd (MINT *@var{operand1}, MINT *@var{operand2}, MINT *@var{res})
Set @var{res} to the greatest common divisor of @var{operand1} and
@var{operand2}.
@end deftypefun

@deftypefun int mcmp (MINT *@var{operand1}, MINT *@var{operand2})
Compare @var{operand1} and @var{operand2}.  Return a positive value if
@var{operand1} > @var{operand2}, zero if @var{operand1} =
@var{operand2}, and a negative value if @var{operand1} < @var{operand2}.
@end deftypefun

@deftypefun void min (MINT *@var{dest})
Input a decimal string from @code{stdin}, and put the read integer in
@var{dest}.  SPC and TAB are allowed in the number string, and are ignored.
@end deftypefun

@deftypefun void mout (MINT *@var{src})
Output @var{src} to @code{stdout}, as a decimal string.  Also output a newline.
@end deftypefun

@deftypefun {char *} mtox (MINT *@var{operand})
Convert @var{operand} to a hexadecimal string, and return a pointer to the
string.  The returned string is allocated using the default memory allocation
function, @code{malloc} by default.
@end deftypefun

@deftypefun void mfree (MINT *@var{operand})
De-allocate, the space used by @var{operand}.  @strong{This function should
only be passed a value returned by @code{itom} or @code{xtom}.}
@end deftypefun

@node Custom Allocation, Contributors, BSD Compatible Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Custom Allocation

By default, the MP functions use @code{malloc}, @code{realloc}, and
@code{free} for memory allocation.  If @code{malloc} or @code{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.

@deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t))
Replace the current allocation functions from the arguments.  If an argument
is NULL, the corresponding default function is retained.

@strong{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.}
@end deftypefun

The functions you supply should fit the following declarations:

@deftypefun {void *} allocate_function (size_t @var{alloc_size})
This function should return a pointer to newly allocated space with at least
@var{alloc_size} storage units.
@end deftypefun

@deftypefun {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})
This function should return a pointer to newly allocated space of at least
@var{new_size} storage units, after copying at least the first @var{old_size}
storage units from @var{ptr}.  It should also de-allocate the space at
@var{ptr}.

You can assume that the space at @var{ptr} was formerly returned from
@code{allocate_function} or @code{reallocate_function}, for a request for
@var{old_size} storage units.
@end deftypefun

@deftypefun void deallocate_function (void *@var{ptr}, size_t @var{size})
De-allocate the space pointed to by @var{ptr}.

You can assume that the space at @var{ptr} was formerly returned from
@code{allocate_function} or @code{reallocate_function}, for a request for
@var{size} storage units.
@end deftypefun

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


@node Contributors, References, Custom Allocation, Top
@comment  node-name,  next,  previous,  up
@unnumbered Contributors

Torbjorn Granlund authored the original GMP library and keeps maintaining
it.  But several other individuals and organizations have contributed to GMP
in various ways.

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
@code{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 @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and
@code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2.

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

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

Bennet Yee contributed the functions @code{mpz_jacobi} and @code{mpz_legendre}.

Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
@file{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 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 a template Karatsuba multiplication function for GMP
2.1 on which the current implementation is based.

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

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

@node References, , Contributors, Top
@comment  node-name,  next,  previous,  up
@unnumbered References

@itemize @bullet

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

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

@item
Richard M. Stallman, "Using and Porting GCC", Free Software Foundation,
1995.

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

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

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

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

@node Concept Index, , , Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

@node Function Index, , , Top
@comment  node-name,  next,  previous,  up
@unnumbered Function and Type Index
@printindex fn


@contents
@bye