path: root/gmp.texi

\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename gmp.info
@include version.texi
@settitle GNU MP @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

@c  Texinfo version 4 or up will be needed to process this into .info files.
@c
@c  The edition number is in three places and the month/year in one, all taken
@c  from version.texi.  version.texi is created when you configure with
@c  --enable-maintainer-mode, and is included in a distribution made with
@c  "make dist".
@c
@c  "cindex" entries have been made for function categories and programming
@c  topics.  Minutiae like particular systems and processors mentioned in
@c  various places have been left out so as not to bury important topics under
@c  a lot of junk.  "mpn" functions aren't in the concept index because a
@c  beginner looking for "GCD" or something is only going to be confused by
@c  pointers to low level routines.

@dircategory GNU libraries
@direntry
* gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
@end direntry

@c smallbook
@finalout
@setchapternewpage on

@ifnottex
@node Top, Copying, (dir), (dir)
@top GNU MP
This manual describes how to install and use the GNU multiple precision
arithmetic library, version @value{VERSION}.
@end ifnottex

@iftex
@titlepage
@c  use the new format for titles
@title GNU MP
@subtitle The GNU Multiple Precision Arithmetic Library
@subtitle Edition @value{EDITION}
@subtitle @value{UPDATED}

@author by Torbj@"orn Granlund, Swox AB
@email{tege@@swox.com}

@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

@c  Ensure copyright stuff gets into info and html output.
@end iftex

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001
Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.1 or any later
version published by the Free Software Foundation; with no Invariant Sections,
with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts
being "You have freedom to copy and modify this GNU Manual, like GNU
software".  A copy of the license is included in @ref{GNU Free Documentation
License}.

@iftex
@end titlepage
@headings double

@c  Don't bother with contents for "makeinfo --html", the menus seem adequate.
@contents
@end iftex

@menu
* Copying::                    GMP Copying Conditions (LGPL).
* Introduction to GMP::        Brief introduction to GNU MP.
* Installing GMP::             How to configure and compile the GMP library.
* GMP Basics::                 What every GMP user should know.
* Reporting Bugs::             How to usefully report bugs.
* Integer Functions::          Functions for arithmetic on signed integers.
* Rational Number Functions::  Functions for arithmetic on rational numbers.
* Floating-point Functions::   Functions for arithmetic on floats.
* Low-level Functions::        Fast functions for natural numbers.
* Random Number Functions::    Functions for generating random numbers.
* Formatted Output::           @code{printf} style output.
* Formatted Input::            @code{scanf} style input.
* C++ Class Interface::        Class wrappers around GMP types.
* BSD Compatible Functions::   All functions found in BSD MP.
* Custom Allocation::          How to customize the internal allocation.
* Language Bindings::          Using GMP from other languages.
* Algorithms::                 What happens behind the scenes.
* Internals::                  How values are represented behind the scenes.

* Contributors::	       Who brings your this library?
* References::                 Some useful papers and books to read.
* GNU Free Documentation License::
* Concept Index::
* Function Index::
@end menu


@c  @m{T,N} is $T$ in tex or @math{N} otherwise.  This is an easy way to give
@c  different forms for math in tex and info.  Commas in N or T don't work,
@c  but @C{} can be used instead.  \, works in info but not in tex.
@iftex
@macro m {T,N}
@tex$\T\$@end tex
@end macro
@end iftex
@ifnottex
@macro m {T,N}
@math{\N\}
@end macro
@end ifnottex

@macro C {}
,
@end macro

@c  @ma{E} is $E$ for tex or @math{E} otherwise.  This suits expressions which
@c  want $$ rather than @math{} in tex, for example @ma{N^2}.
@iftex
@macro ma {E}
@tex$\E\$@end tex
@end macro
@end iftex
@ifnottex
@macro ma {E}
@math{\E\}
@end macro
@end ifnottex

@c  @ms{V,N} is $V_N$ in tex or just vn otherwise.  This suits simple
@c  subscripts like @ms{x,0}.
@iftex
@macro ms {V,N}
@tex$\V\_{\N\}$@end tex
@end macro
@end iftex
@ifnottex
@macro ms {V,N}
\V\\N\
@end macro
@end ifnottex

@c  @nicode{S} is plain S in info, or @code{S} elsewhere.  This can be used
@c  when the quotes that @code{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.  Doesn't work as @nicode{'\\0'}
@c  though (gives two backslashes in tex).
@ifinfo
@macro nicode {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nicode {S}
@code{\S\}
@end macro
@end ifnotinfo

@c  @nisamp{S} is plain S in info, or @samp{S} elsewhere.  This can be used
@c  when the quotes that @samp{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.
@ifinfo
@macro nisamp {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nisamp {S}
@samp{\S\}
@end macro
@end ifnotinfo

@c  Usage: @GMPtimes{}
@c  Give either \times or the word "times".
@tex
\gdef\GMPtimes{\times}
@end tex
@ifnottex
@macro GMPtimes
times
@end macro
@end ifnottex

@c  Usage: @GMPmultiply{}
@c  Give * in info, or nothing in tex.
@tex
\gdef\GMPmultiply{}
@end tex
@ifnottex
@macro GMPmultiply
*
@end macro
@end ifnottex

@c  Usage: @GMPabs{x}
@c  Give either |x| in tex, or abs(x) in info or html.
@tex
\gdef\GMPabs#1{|#1|}
@end tex
@ifnottex
@macro GMPabs {X}
@abs{}(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPfloor{x}
@c  Give either \lfloor x\rfloor in tex, or floor(x) in info or html.
@tex
\gdef\GMPfloor#1{\lfloor #1\rfloor}
@end tex
@ifnottex
@macro GMPfloor {X}
floor(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPceil{x}
@c  Give either \lceil x\rceil in tex, or ceil(x) in info or html.
@tex
\gdef\GMPceil#1{\lceil #1 \rceil}
@end tex
@ifnottex
@macro GMPceil {X}
ceil(\X\)
@end macro
@end ifnottex

@c  Math operators already available in tex, made available in info too.
@c  For example @bmod{} can be used in both tex and info.
@ifnottex
@macro bmod
mod
@end macro
@macro gcd
gcd
@end macro
@macro ge
>=
@end macro
@macro le
<=
@end macro
@macro log
log
@end macro
@macro min
min
@end macro
@macro rightarrow
->
@end macro
@end ifnottex

@c  New math operators.
@c  @abs{} can be used in both tex and info, or just \abs in tex.
@tex
\gdef\abs{\mathop{\rm abs}}
@end tex
@ifnottex
@macro abs
abs
@end macro
@end ifnottex

@c  @cross{} is a \times symbol in tex, or an "x" in info.  In tex it works
@c  inside or outside $ $.
@tex
\gdef\cross{\ifmmode\times\else$\times$\fi}
@end tex
@ifnottex
@macro cross
x
@end macro
@end ifnottex

@c  @times{} made available as a "*" in info and html (already works in tex).
@ifnottex
@macro times
*
@end macro
@end ifnottex

@c  Usage: @W{text}
@c  Like @w{} but working in math mode too.
@tex
\gdef\W#1{\ifmmode{#1}\else\w{#1}\fi}
@end tex
@ifnottex
@macro W {S}
@w{\S\}
@end macro
@end ifnottex

@c  Usage: \GMPdisplay{text}
@c  Put the given text in an @display style indent, but without turning off
@c  paragraph reflow etc.
@tex
\gdef\GMPdisplay#1{%
\noindent
\advance\leftskip by \lispnarrowing
#1\par}
@end tex

@c  Usage: \GMPhat
@c  A new \hat that will work in math mode, unlike the texinfo redefined
@c  version.
@tex
\gdef\GMPhat{\mathaccent"705E}
@end tex

@c  Usage: \GMPraise{text}
@c  For use in a $ $ math expression as an alternative to "^".  This is good
@c  for @code{} in an exponent, since there seems to be no superscript font
@c  for that.
@tex
\gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}}
@end tex

@c  Usage: @texlinebreak{}
@c  A line break as per @*, but only in tex.
@iftex
@macro texlinebreak
@*
@end macro
@end iftex
@ifnottex
@macro texlinebreak
@end macro
@end ifnottex

@c  Usage: @maybepagebreak
@c  Allow tex to insert a page break, if it feels the urge.
@c  Normally blocks of @deftypefun/funx are kept together, which can lead to
@c  some poor page break positioning if it's a big block, like the sets of
@c  division functions etc.
@tex
\gdef\maybepagebreak{\penalty0}
@end tex
@ifnottex
@macro maybepagebreak
@end macro
@end ifnottex


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

This library is @dfn{free}; this means that everyone is free to use it and
free to redistribute it on a free basis.  The library is not in the public
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
Lesser General Public License version 2.1 that accompanies the source code,
see @file{COPYING.LIB}.  Certain demonstration programs are provided under the
terms of the plain General Public License version 2, see @file{COPYING}.


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

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

Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits.  GMP is designed to
give good performance for both, by choosing algorithms based on the sizes of
the operands, and by carefully keeping the overhead at a minimum.

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

There is carefully optimized assembly code for these CPUs:
@cindex CPUs supported
ARM,
DEC Alpha 21064, 21164, and 21264,
AMD 29000,
AMD K6, K6-2 and Athlon,
Hitachi SuperH and SH-2,
HPPA 1.0, 1.1 and 2.0,
Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86,
Intel IA-64, i960,
Motorola MC68000, MC68020, MC88100, and MC88110,
Motorola/IBM PowerPC 32 and 64,
National NS32000,
IBM POWER,
MIPS R3000, R4000,
SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC,
DEC VAX,
and
Zilog Z8000.
Some optimizations also for
Cray vector systems,
Clipper,
IBM ROMP (RT),
and
Pyramid AP/XP.

@cindex Mailing list
There is a mailing list for GMP users.  To join it, send a mail to
@email{gmp-request@@swox.com} with the word @samp{subscribe} in the message
@strong{body} (not in the subject line).

@cindex Home page
@cindex Web page
For up-to-date information on GMP, please see the GMP web pages at

@display
@uref{http://swox.com/gmp/}
@end display

@cindex Latest version of GMP
@cindex Anonymous FTP of latest version
@cindex FTP of latest version
The latest version of the library is available at

@display
@uref{ftp://ftp.gnu.org/gnu/gmp}
@end display

Many sites around the world mirror @samp{ftp.gnu.org}, please use a mirror
near you, see @uref{http://www.gnu.org/order/ftp.html} for a full list.


@section How to use this Manual
@cindex About this manual

Everyone should read @ref{GMP Basics}.  If you need to install the library
yourself, then read @ref{Installing GMP}.  If you have a system with multiple
ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used
on applications.

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


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

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

@example
./configure
make
@end example

@noindent
Some self-tests can be run with

@example
make check
@end example

@noindent
And you can install (under @file{/usr/local} by default) with

@example
make install
@end example

@noindent
If you experience problems, please report them to @email{bug-gmp@@gnu.org}.
See @ref{Reporting Bugs}, for information on what to include in useful bug
reports.

@menu
* Build Options::               
* ABI and ISA::                 
* Notes for Package Builds::    
* Notes for Particular Systems::  
* Known Build Problems::        
@end menu


@node Build Options, ABI and ISA, Installing GMP, Installing GMP
@section Build Options
@cindex Build options

All the usual autoconf configure options are available, run @samp{./configure
--help} for a summary.  The file @file{INSTALL.autoconf} has some generic
installation information too.

@table @asis
@item Non-Unix Systems

@samp{configure} requires various Unix-like tools.  On an MS-DOS system
Cygwin, DJGPP or MINGW can be used.  See

@display
@uref{http://www.cygnus.com/cygwin}
@uref{http://www.delorie.com/djgpp}
@uref{http://www.mingw.org}
@end display

The @file{macos} directory contains an unsupported port to MacOS 9 on Power
Macintosh.  Note that MacOS X ``Darwin'' can use the normal
@samp{./configure}.

It might be possible to build without the help of @samp{configure}, certainly
all the code is there, but unfortunately you'll be on your own.

@item Build Directory

To compile in a separate build directory, @command{cd} to that directory, and
prefix the configure command with the path to the GMP source directory.  For
example

@example
cd /my/build/dir
/my/sources/gmp-@value{VERSION}/configure
@end example

Not all @samp{make} programs have the necessary features (@code{VPATH}) to
support this.  In particular, SunOS and Slowaris @command{make} have bugs that
make them unable to build in a separate directory.  Use GNU @command{make}
instead.

@item @option{--disable-shared}, @option{--disable-static}

By default both shared and static libraries are built (where possible), but
one or other can be disabled.  Shared libraries result in smaller executables
and permit code sharing between separate running processes, but on some CPUs
are slightly slower, having a small cost on each function call.

@item Native Compilation, @option{--build=CPU-VENDOR-OS}

For normal native compilation, the system can be specified with
@samp{--build}.  By default @samp{./configure} uses the output from running
@samp{./config.guess}.  On some systems @samp{./config.guess} can determine
the exact CPU type, on others it will be necessary to give it explicitly.  For
example,

@example
./configure --build=ultrasparc-sun-solaris2.7
@end example

In all cases the @samp{OS} part is important, since it controls how libtool
generates shared libraries.  Running @samp{./config.guess} is the simplest way
to see what it should be, if you don't know already.

@item Cross Compilation, @option{--host=CPU-VENDOR-OS}

When cross-compiling, the system used for compiling is given by @samp{--build}
and the system where the library will run is given by @samp{--host}.  For
example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,

@example
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
@end example

Compiler tools are sought first with the host system type as a prefix.  For
example @command{m68k-mac-linux-gnu-ranlib} is checked for, then plain
@command{ranlib}.  This makes it possible for a set of cross-compiling tools
to co-exist with native tools.  The prefix is the argument to @samp{--host},
and this can be an alias, such as @samp{m68k-linux}.  But note that tools
don't have to be setup this way, it's enough to just have a @env{PATH} with a
suitable cross-compiling @command{cc} etc.

Compiling for a different CPU in the same family as the build system is a form
of cross-compilation, though very possibly this would merely be with special
options on a native compiler.  In any case @samp{./configure} avoids depending
on being able to run code on the build system, which is important when
creating binaries for a newer CPU since they very possibly won't run on the
build system.

Currently a warning is given unless an explicit @samp{--build} is used when
cross-compiling, because it may not be possible to correctly guess the build
system type if the @env{PATH} has only a cross-compiling @command{cc}.

Note that the @samp{--target} option is not appropriate for GMP.  It's for use
when building compiler tools, with @samp{--host} being where they will run,
and @samp{--target} what they'll produce code for.  Ordinary programs or
libraries like GMP are only interested in the @samp{--host} part, being where
they'll run.  (Some past versions of GMP used @samp{--target} incorrectly.)

@item CPU types

In general, if you want a library that runs as fast as possible, you should
configure GMP for the exact CPU type your system uses.  However, this may mean
the binaries won't run on older members of the family, and might run slower on
other members, older or newer.  The best idea is always to build GMP for the
exact machine type you intend to run it on.

The following CPUs have specific support.  See @file{configure.in} for details
of what code and compiler options they select.

@itemize @bullet

@c Keep this formatting, it's easy to read and it can be grepped to
@c automatically test that CPUs listed get through ./config.sub

@item
Alpha:
@nisamp{alpha},
@nisamp{alphaev5},
@nisamp{alphaev56},
@nisamp{alphapca56},
@nisamp{alphaev6},
@nisamp{alphaev67}

@item
Cray:
@nisamp{c90},
@nisamp{j90},
@nisamp{t90},
@nisamp{sv1}

@item
HPPA:
@nisamp{hppa1.0},
@nisamp{hppa1.1},
@nisamp{hppa2.0},
@nisamp{hppa2.0n},
@nisamp{hppa2.0w}

@item
MIPS:
@nisamp{mips},
@nisamp{mips3},
@nisamp{mips64}

@item
Motorola:
@nisamp{m68k},
@nisamp{m68000},
@nisamp{m68010},
@nisamp{m68020},
@nisamp{m68030},
@nisamp{m68040},
@nisamp{m68060},
@nisamp{m68302},
@nisamp{m68360},
@nisamp{m88k},
@nisamp{m88110}

@item
POWER: 
@nisamp{power},
@nisamp{power1},
@nisamp{power2},
@nisamp{power2sc},
@nisamp{powerpc},
@nisamp{powerpc64}

@item
SPARC:
@nisamp{sparc},
@nisamp{sparcv8},
@nisamp{microsparc},
@nisamp{supersparc},
@nisamp{sparcv9},
@nisamp{ultrasparc},
@nisamp{sparc64}

@item
80x86 family:
@nisamp{i386},
@nisamp{i486},
@nisamp{i586},
@nisamp{pentium},
@nisamp{pentiummmx},
@nisamp{pentiumpro},
@nisamp{pentium2},
@nisamp{pentium3},
@nisamp{pentium4},
@nisamp{k6},
@nisamp{k62},
@nisamp{k63},
@nisamp{athlon}

@item
Other:
@nisamp{a29k},
@nisamp{arm},
@nisamp{clipper},
@nisamp{i960},
@nisamp{ns32k},
@nisamp{pyramid},
@nisamp{sh},
@nisamp{sh2},
@nisamp{vax},
@nisamp{z8k}
@end itemize

CPUs not listed will use generic C code.

@item Generic C Build

If some of the assembly code causes problems, or if otherwise desired, the
generic C code can be selected with CPU @samp{none}.  For example,

@example
./configure --build=none-unknown-freebsd3.5
@end example

Note that this will run quite slowly, but it should be portable and should at
least make it possible to get something running if all else fails.

@item @option{ABI}

On some systems GMP supports multiple ABIs (application binary interfaces),
meaning data type sizes and calling conventions.  By default GMP chooses the
best ABI available, but a particular ABI can be selected.  For example

@example
./configure --build=mips64-sgi-irix6 ABI=n32
@end example

See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what
applications need to do.

@item @option{CC}, @option{CFLAGS}

By default the C compiler used is chosen from among some likely candidates,
with @command{gcc} normally preferred if it's present.  The usual
@samp{CC=whatever} can be passed to @samp{./configure} to choose something
different.

For some systems, default compiler flags are set based on the CPU and
compiler.  The usual @samp{CFLAGS="-whatever"} can be passed to
@samp{./configure} to use something different or to set good flags for systems
GMP doesn't otherwise know.

The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure},
and can be found in each generated @file{Makefile}.  This is the easiest way
to check the defaults when considering changing or adding something.

Note that when @samp{CC} and @samp{CFLAGS} are specified on a system
supporting multiple ABIs it's important to give an explicit
@samp{ABI=whatever}, since GMP can't determine the ABI just from the flags and
won't be able to select the correct assembler code.

If just @samp{CC} is selected then normal default @samp{CFLAGS} for that
compiler will be used (if GMP recognises it).  For example @samp{CC=gcc} can
be used to force the use of GCC, with default flags (and default ABI).

@item @option{CPPFLAGS}

Any flags like @samp{-D} defines or @samp{-I} includes required by the
preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}.
Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but
preprocessing uses just @samp{CPPFLAGS}.  This distinction is because most
preprocessors won't accept all the flags the compiler does.  Preprocessing is
done separately in some configure tests, and in the @samp{ansi2knr} support
for K&R compilers.

@item C++ Support, @option{--enable-cxx}
C++ support in GMP can be enabled with @samp{--enable-cxx}, in which case a
C++ compiler will be required.  As a convenience @samp{--enable-cxx=detect}
can be used to enable C++ support only if a compiler can be found.  The C++
support consists of a library @file{libgmpxx.la} and header file
@file{gmpxx.h}.

A separate @file{libgmpxx.la} has been adopted rather than having C++ objects
within @file{libgmp.la} in order to ensure dynamic linked C programs aren't
bloated by a dependency on the C++ standard library, and to avoid any chance
that the C++ compiler could be required when linking plain C programs.

@file{libgmpxx.la} will use certain internals from @file{libgmp.la} and can
only be expected to work with @file{libgmp.la} from the same GMP version.
Future changes to the relevant internals will be accompanied by renaming, so a
mismatch will cause unresolved symbols rather than perhaps mysterious
misbehaviour.

In general @file{libgmpxx.la} will be usable only with the C++ compiler that
built it, since name mangling and runtime support are usually incompatible
between different compilers.

@item @option{CXX}, @option{CXXFLAGS}
When C++ support is enabled, the C++ compiler and its flags can be set with
variables @samp{CXX} and @samp{CXXFLAGS} in the usual way.  The default for
@samp{CXX} is the first compiler that works from a list of likely candidates,
with @command{g++} normally preferred when available.  The default for
@samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then
for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers
@samp{-g} or nothing.  Trying @samp{CFLAGS} this way is convenient when using
@samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will
usually suit @samp{g++}.

It's important that the C and C++ compilers match, meaning their startup and
runtime support routines are compatible and that they generate code in the
same ABI (if there's a choice of ABIs on the system).  @samp{./configure}
isn't currently able to check these things very well itself, so for that
reason @samp{--disable-cxx} is the default, to avoid a build failure due to a
compiler mismatch.  Perhaps this will change in the future.

Incidentally, it's normally not good enough to set @samp{CXX} to the same as
@samp{CC}.  Although @command{gcc} for instance recognises @file{foo.cc} as
C++ code, only @command{g++} will invoke the linker the right way when
building an executable or shared library from object files.

@item Temporary Memory, @option{--enable-alloca=<choice>}
@cindex Stack overflow segfaults
@cindex @code{alloca}

GMP allocates temporary workspace using one of the following three methods,
which can be selected with for instance
@samp{--enable-alloca=malloc-reentrant}.

@itemize @bullet
@item
@samp{alloca} - C library or compiler builtin.
@item
@samp{malloc-reentrant} - the heap, in a re-entrant fashion.
@item
@samp{malloc-notreentrant} - the heap, with global variables.
@end itemize

For convenience, the following choices are also available.
@samp{--disable-alloca} is the same as @samp{--enable-alloca=no}.

@itemize @bullet
@item
@samp{yes} - a synonym for @samp{alloca}.
@item
@samp{no} - a synonym for @samp{malloc-reentrant}.
@item
@samp{reentrant} - @code{alloca} if available, otherwise
@samp{malloc-reentrant}.  This is the default.
@item
@samp{notreentrant} - @code{alloca} if available, otherwise
@samp{malloc-notreentrant}.
@end itemize

@code{alloca} is reentrant and fast, and is recommended, but when working with
large numbers it can overflow the available stack space, in which case one of
the two malloc methods will need to be used.  Alternately it might be possible
to increase available stack with @command{limit}, @command{ulimit} or
@code{setrlimit}, or under DJGPP with @command{stubedit} or
@code{@w{_stklen}}.  Note that depending on the system the only indication of
stack overflow might be a segmentation violation.

@samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe,
but @samp{malloc-notreentrant} is faster and should be used if reentrancy is
not required.

The two malloc methods in fact use the memory allocation functions selected by
@code{mp_set_memory_functions}, these being @code{malloc} and friends by
default.  @xref{Custom Allocation}.

An additional choice @samp{--enable-alloca=debug} is available, to help when
debugging memory related problems (@pxref{Debugging}).

@item FFT Multiplication, @option{--disable-fft}

By default multiplications are done using Karatsuba, 3-way Toom-Cook, and
Fermat FFT.  The FFT is only used on large to very large operands and can be
disabled to save code size if desired.

@item Berkeley MP, @option{--enable-mpbsd}

The Berkeley MP compatibility library (@file{libmp}) and header file
(@file{mp.h}) are built and installed only if @option{--enable-mpbsd} is used.
@xref{BSD Compatible Functions}.

@item MPFR, @option{--enable-mpfr}
@cindex MPFR

The optional MPFR functions are built and installed only if
@option{--enable-mpfr} is used.  These are in a separate library
@file{libmpfr.a} and are documented separately too (@pxref{Introduction to
MPFR,, Introduction to MPFR, mpfr, MPFR}).

@item Assertion Checking, @option{--enable-assert}

This option enables some consistency checking within the library.  This can be
of use while debugging, @pxref{Debugging}.

@item Execution Profiling, @option{--enable-profiling=prof/gprof}

Profiling support can be enabled either for @command{prof} or @command{gprof}.
This adds @samp{-p} or @samp{-pg} respectively to @samp{CFLAGS}, and for some
systems adds corresponding @code{mcount} calls to the assembler code.
@xref{Profiling}.

@item @option{MPN_PATH}

Various assembler versions of mpn subroutines are provided, and, for a given
CPU, a search is made though a path to choose a version of each.  For example
@samp{sparcv8} has path @samp{sparc32/v8 sparc32 generic}, which means it
looks first for v8 code, then plain sparc32, and finally falls back on generic
C.  Knowledgeable users with special requirements can specify a path with
@samp{MPN_PATH="dir list"}.  This will normally be unnecessary because all
sensible paths should be available under one or other CPU.

@item Demonstration Programs
@cindex Demonstration programs
@cindex Example programs

The @file{demos} subdirectory has some sample programs using GMP.  These
aren't built or installed, but there's a @file{Makefile} with rules for them.
For instance,

@example
make pexpr
./pexpr 68^975+10
@end example

@item Documentation

The document you're now reading is @file{gmp.texi}.  The usual automake
targets are available to make PostScript @file{gmp.ps} and/or DVI
@file{gmp.dvi}.

HTML can be produced with @samp{makeinfo --html}, see @ref{makeinfo
html,Generating HTML,Generating HTML,texinfo,Texinfo}.  Or alternately
@samp{texi2html}, see @ref{Top,Texinfo to HTML,About,texi2html,Texinfo To
HTML}.

PDF can be produced with @samp{texi2dvi --pdf} (@pxref{PDF
Output,PDF,,texinfo,Texinfo}) or with @samp{pdftex}.

Some supplementary notes can be found in the @file{doc} subdirectory.

@end table


@need 2000
@node ABI and ISA, Notes for Package Builds, Build Options, Installing GMP
@section ABI and ISA
@cindex ABI
@cindex Application Binary Interface
@cindex ISA
@cindex Instruction Set Architecture

ABI (Application Binary Interface) refers to the calling conventions between
functions, meaning what registers are used and what sizes the various C data
types are.  ISA (Instruction Set Architecture) refers to the instructions and
registers a CPU has available.

Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
latter for compatibility with older CPUs in the family.  GMP supports some
CPUs like this in both ABIs.  In fact within GMP @samp{ABI} means a
combination of chip ABI, plus how GMP chooses to use it.  For example in some
32-bit ABIs, GMP may support a limb as either a 32-bit @code{long} or a 64-bit
@code{long long}.

By default GMP chooses the best ABI available for a given system, and this
generally gives significantly greater speed.  But an ABI can be chosen
explicitly to make GMP compatible with other libraries, or particular
application requirements.  In all cases it's vital that all object code used
in a given program is compiled for the same ABI.

Usually a limb is implemented as a @code{long}.  When a @code{long long} limb
is used in a particular ABI, this is encoded in a generated @file{gmp.h}.
This is convenient for applications, but it does mean that @file{gmp.h} will
vary, and can't be just copied around.  @file{gmp.h} remains compiler
independent though, since all compilers for a particular ABI will be expected
to use the same limb type.

Currently no attempt is made to follow whatever conventions a system has for
installing library or header files built for a particular ABI.  This will
probably only matter when installing multiple builds of GMP, and it might be
as simple as configuring with a special @samp{libdir}, or it might require
more than that.  Note that builds for different ABIs need to done separately,
with a fresh @command{./configure} and @command{make} each.

@table @asis
@sp 1
@need 1000
@item HPPA 2.0 (@samp{hppa2.0*})

@table @asis
@item @samp{ABI=2.0w}

The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up
when using @command{cc}.  @command{gcc} support for this is in progress.
Applications must be compiled with

@example
cc  +DD64
@end example

@item @samp{ABI=2.0n}

The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit limb using
@code{long long}.  This is available on HP-UX 10 or up when using
@command{cc}.  No @command{gcc} support is planned for this.  Applications
must be compiled with

@example
cc  +DA2.0 +e
@end example

@item @samp{ABI=1.0}

HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI.
No special compiler options are needed for applications.
@end table

All three ABIs are available for CPUs @samp{hppa2.0w} and @samp{hppa2.0}, but
for CPU @samp{hppa2.0n} only 2.0n or 1.0 are allowed.

@sp 1
@need 1000
@item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]})

IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3 or better
CPU.  In both these ABIs GMP uses a 64-bit limb.  A new enough @command{gcc}
is required (2.95 for instance).

@table @asis
@item @samp{ABI=n32}

The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a
@code{long long}.  Applications must be compiled with

@example
gcc  -mabi=n32
cc   -n32
@end example

@item @samp{ABI=64}

The 64-bit ABI is 64-bit pointers and integers.  Applications must be compiled
with

@example
gcc  -mabi=64
cc   -64
@end example
@end table

Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary
support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.

@sp 1
@need 1000
@item PowerPC 64 (@samp{powerpc64*})

@table @asis
@item @samp{ABI=aix64}

The AIX 64 ABI uses 64-bit limbs and pointers and is available on systems
@samp{powerpc64*-*-aix*}.  Applications must be compiled (and linked) with

@example
gcc  -maix64
xlc  -q64
@end example

@item @samp{ABI=32L}

This uses the 32-bit ABI but a 64-bit limb using GCC @code{long long} in
64-bit registers.  Applications must be compiled with

@example
gcc  -mpowerpc64
@end example

@item @samp{ABI=32}

This is the basic 32-bit PowerPC ABI.  No special compiler options are needed
for applications.
@end table

@sp 1
@need 1000
@item Sparc V9 (@samp{sparcv9} and @samp{ultrasparc*})

@table @asis
@item @samp{ABI=64}

The 64-bit V9 ABI is available on Solaris 2.7 and up and GNU/Linux.  GCC 2.95
or up, or Sun @command{cc} is required.  Applications must be compiled with

@example
gcc  -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc   -xarch=v9
@end example

@item @samp{ABI=32}

On Solaris 2.6 and earlier only the plain V8 32-bit ABI can be used, since the
kernel doesn't save all registers.  GMP still uses as much of the V9 ISA as it
can in these circumstances.  No special compiler options are required for
applications, though using something like the following requesting V9 code
within the V8 ABI is recommended.

@example
gcc  -mv8plus
cc   -xarch=v8plus
@end example

@command{gcc} 2.8 and earlier only supports @samp{-mv8} though.
@end table

Don't be confused by the names of these sparc @samp{-m} and @samp{-x} options,
they're called @samp{arch} but they effectively control the ABI.
@end table


@need 2000
@node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing GMP
@section Notes for Package Builds
@cindex Build notes for binary packaging
@cindex Packaged builds

GMP should present no great difficulties for packaging in a binary
distribution.

@cindex Libtool versioning
@cindex Shared library versioning
Libtool is used to build the library and @samp{-version-info} is set
appropriately, having started from @samp{3:0:0} in GMP 3.0.  The GMP 4 series
will be upwardly binary compatible in each release and will be upwardly binary
compatible with all of the GMP 3 series.  Additional function interfaces may
be added in each release, so on systems where libtool versioning is not fully
checked by the loader an auxiliary mechanism may be needed to express that a
dynamic linked application depends on a new enough GMP.

An auxiliary mechanism may also be needed to express that @file{libgmpxx.la}
(from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libgmp.la}
from the same GMP version, since this is not done by the libtool versioning,
nor otherwise.  A mismatch will result in unresolved symbols from the linker,
or perhaps the loader.

When building a package for a CPU family, care should be taken to use
@samp{--host} (or @samp{--build}) to choose the least common denominator among
the CPUs which might use the package.  For example this might necessitate
@samp{i386} for x86s, or plain @samp{sparc} (meaning V7) for SPARCs.

Users who care about speed will want GMP built for their exact CPU type, to
make use of the available optimizations.  Providing a way to suitably rebuild
a package may be useful.  This could be as simple as making it possible for a
user to omit @samp{--build} (and @samp{--host}) so @samp{./config.guess} will
detect the CPU.  But a way to manually specify a @samp{--build} will be wanted
for systems where @samp{./config.guess} is inexact.

Note that @file{gmp.h} is a generated file, and will be architecture and ABI
dependent.


@need 2000
@node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing GMP
@section Notes for Particular Systems
@cindex Build notes for particular systems
@table @asis

@c This section is more or less meant for notes about performance or about
@c build problems that have been worked around but might leave a user
@c scratching their head.  Fun with different ABIs on a system belongs in the
@c above section.

@item AIX 3 and 4

On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since
some versions of the native @command{ar} fail on the convenience libraries
used.  A shared build can be attempted with

@example
./configure --enable-shared --disable-static
@end example

Note that the @samp{--disable-static} is necessary because in a shared build
libtool makes @file{libgmp.a} a symlink to @file{libgmp.so}, apparently for
the benefit of old versions of @command{ld} which only recognise @file{.a},
but unfortunately this is done even if a fully functional @command{ld} is
available.

@item ARM

On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a
bug in unsigned division, giving wrong results for some operands.  GMP
@samp{./configure} will demand GCC 2.95.4 or later.

@item Microsoft Windows
On systems @samp{*-*-cygwin*}, @samp{*-*-mingw*} and @samp{*-*-pw32*} by
default GMP builds only a static library, but a DLL can be built instead using

@example
./configure --disable-static --enable-shared
@end example

Static and DLL libraries can't both be built, since certain export directives
in @file{gmp.h} must be different.  @samp{--enable-cxx} cannot be used when
building a DLL, since libtool doesn't currently support C++ DLLs.  This might
change in the future.

GCC is recommended for compiling GMP, but the resulting DLL can be used with
any compiler.  On mingw only the standard Windows libraries will be needed, on
Cygwin the usual cygwin runtime will be required.

@item Motorola 68k CPU Types

@samp{m68k} is taken to mean 68000.  @samp{m68020} or higher will give a
performance boost on applicable CPUs.  @samp{m68360} can be used for CPU32
series chips.  @samp{m68302} can be used for ``Dragonball'' series chips,
though this is merely a synonym for @samp{m68000}.

@item OpenBSD 2.6

@command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it
unsuitable for @file{.asm} file processing.  @samp{./configure} will detect
the problem and either abort or choose another m4 in the @env{PATH}.  The bug
is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.

@item Power CPU Types

In GMP, CPU types @samp{power} and @samp{powerpc} will each use instructions
not available on the other, so it's important to choose the right one for the
CPU that will be used.  Currently GMP has no assembler code support for using
just the common instruction subset.  To get executables that run on both, the
current suggestion is to use the generic C code (CPU @samp{none}), possibly
with appropriate compiler options (like @samp{-mcpu=common} for
@command{gcc}).  CPU @samp{rs6000} (which is not a CPU but a family of
workstations) is accepted by @file{config.sub}, but is currently equivalent to
@samp{none}.

@item Sparc CPU Types

@samp{sparcv8} or @samp{supersparc} on relevant systems will give a
significant performance increase over the V7 code.

@item SunOS 4

@command{/usr/bin/m4} lacks various features needed to process @file{.asm}
files, and instead @samp{./configure} will automatically use
@command{/usr/5bin/m4}, which we believe is always available (if not then use
GNU m4).

@item x86 CPU Types

@samp{i386} selects generic code which will run reasonably well on all x86
chips.

@samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for the intended
P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
P-III)@.  @samp{i386} is a better choice when making binaries that must run on
both.

@samp{pentium4} and an SSE2 capable assembler are important for best results
on Pentium 4.  The specific code is for instance roughly a 2@cross{} to
3@cross{} speedup over the generic @samp{i386} code.

@item x86 MMX and SSE2 Code

If the CPU selected has MMX code but the assembler doesn't support it, a
warning is given and non-MMX code is used instead.  This will be an inferior
build, since the MMX code that's present is there because it's faster than the
corresponding plain integer code.  The same applies to SSE2.

Old versions of @samp{gas} don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 doesn't (and unfortunately
there's no newer assembler for that system).

Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register
to register @code{movq} instructions, and so can't be used for MMX code.
Install a recent @command{gas} if MMX code is wanted on these systems.

@item x86 GCC @samp{-march=pentiumpro}

GCC 2.95.2 and 2.95.3 miscompiled some versions of @file{mpz/powm.c} when
@samp{-march=pentiumpro} was used, so for relevant CPUs that option is only in
the default @env{CFLAGS} for GCC 2.95.4 and up.
@end table


@need 2000
@node Known Build Problems,  , Notes for Particular Systems, Installing GMP
@section Known Build Problems
@cindex Build problems known

@c This section is more or less meant for known build problems that are not
@c otherwise worked around and require some sort of manual intervention.

You might find more up-to-date information at @uref{http://swox.com/gmp/}.

@table @asis
@item DJGPP

The DJGPP port of @command{bash} 2.03 is unable to run the @samp{configure}
script, it exits silently, having died writing a preamble to
@file{config.log}.  Use @command{bash} 2.04 or higher.

@samp{make all} was found to run out of memory during the final
@file{libgmp.la} link on one system tested, despite having 64Mb available.  A
separate @samp{make libgmp.la} helped, perhaps recursing into the various
subdirectories uses up memory.

@item GNU binutils @command{strip}
@cindex Stripped libraries

GNU binutils @command{strip} should not be used on the static libraries
@file{libgmp.a} and @file{libmp.a}, neither directly nor via @samp{make
install-strip}.  It can be used on the shared libraries @file{libgmp.so} and
@file{libmp.so} though.

Currently (binutils 2.10.0), @command{strip} unpacks an archive then operates
on the files, but GMP contains multiple object files of the same name
(eg. three versions of @file{init.o}), and they overwrite each other, leaving
only the one that happens to be last.

If stripped static libraries are wanted, the suggested workaround is to build
normally, strip the separate object files, and do another @samp{make all} to
rebuild.  Alternately @samp{CFLAGS} with @samp{-g} omitted can always be used
if it's just debugging which is unwanted.

@item NeXT prior to 3.3

The system compiler on old versions of NeXT was a massacred and old GCC, even
if it called itself @file{cc}.  This compiler cannot be used to build GMP, you
need to get a real GCC, and install that.  (NeXT may have fixed this in
release 3.3 of their system.)

@item POWER and PowerPC

Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or
PowerPC.  If you want to use GCC for these machines, get GCC 2.7.2.1 (or
later).

@item Sequent Symmetry

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

@item Solaris 2.6

The system @command{sed} prints an error ``Output line too long'' when libtool
builds @file{libgmp.la}.  This doesn't seem cause any obvious ill effects, but
GNU @command{sed} is recommended, to avoid any doubt.

@item Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32

A shared library build of GMP seems to fail in this combination, it builds but
then fails the tests, apparently due to some incorrect data relocations within
@code{gmp_randinit_lc_2exp_size}.  The exact cause is unknown,
@samp{--disable-shared} is recommended.

@item Windows DLL test programs

When creating a DLL version of @file{libgmp}, libtool creates wrapper scripts
like @file{t-mul} for programs that would normally be @file{t-mul.exe}, in
order to setup the right library paths etc.  This works fine, but the absence
of @file{t-mul.exe} etc causes @command{make} to think they need recompiling
every time, which is an annoyance when re-running a @samp{make check}.
@end table


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

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

@example
#include <gmp.h>
@end example

Note however that prototypes for GMP functions with @code{FILE *} parameters
are only provided if @code{<stdio.h>} is included too.

@example
#include <stdio.h>
#include <gmp.h>
@end example

@strong{Using functions, macros, data types, etc.@: not documented in this
manual is strongly discouraged.  If you do so your application is guaranteed
to be incompatible with future versions of GMP.}

@menu
* Nomenclature and Types::      
* Function Classes::            
* Variable Conventions::        
* Parameter Conventions::       
* Memory Management::           
* Reentrancy::                  
* Useful Macros and Constants::  
* Compatibility with older versions::  
* Efficiency::                  
* Debugging::                   
* Profiling::                   
* Autoconf::                    
@end menu

@node Nomenclature and Types, Function Classes, GMP Basics, GMP Basics
@section Nomenclature and Types
@cindex Nomenclature
@cindex Types

@cindex Integer
@tindex @code{mpz_t}
@noindent
In this manual, @dfn{integer} usually means a multiple precision integer, as
defined by the GMP 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 a 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
machine word.  (We chose this word because a limb of the human body is
analogous to a digit, only larger, and containing several digits.)  Normally a
limb is 32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.


@node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics
@section Function Classes
@cindex Function classes

There are six classes of functions in the GMP library:

@enumerate
@item
Functions for signed integer arithmetic, with names beginning with
@code{mpz_}.  The associated type is @code{mpz_t}.  There are about 150
functions in this class.

@item
Functions for rational number arithmetic, with names beginning with
@code{mpq_}.  The associated type is @code{mpq_t}.  There are about 40
functions in this class, but the integer functions can be used for arithmetic
on the numerator and denominator separately.

@item
Functions for floating-point arithmetic, with names beginning with
@code{mpf_}.  The associated type is @code{mpf_t}.  There are about 60
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_}.  The associated type is array of @code{mp_limb_t}.  There are
about 30 (hard-to-use) functions in this class.

@item
Miscellaneous functions.  Functions for setting up custom allocation and
functions for generating random numbers.
@end enumerate


@node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics
@section Variable Conventions
@cindex Variable conventions
@cindex Conventions for variables

GMP functions generally have output arguments before input arguments.  This
notation is by analogy with the assignment operator.  The BSD MP compatibility
functions are exceptions, having the output arguments last.

GMP lets you use the same variable for both input and output in one call.  For
example, the main function for integer multiplication, @code{mpz_mul}, can be
used to square @code{x} and put the result back in @code{x} with

@example
mpz_mul (x, x, x);
@end example

Before you can assign to a GMP variable, you need to initialize it by calling
one of the special initialization functions.  When you're done with a
variable, you need to clear it out, using one of the functions for that
purpose.  Which function to use depends on the type of variable.  See the
chapters on integer functions, rational number functions, and floating-point
functions for details.

A variable should only be initialized once, or at least cleared between each
initialization.  After a variable has been initialized, it may be assigned to
any number of times.

For efficiency reasons, avoid excessive initializing and clearing.  In
general, initialize near the start of a function and clear near the end.  For
example,

@example
void
foo (void)
@{
  mpz_t  n;
  int    i;
  mpz_init (n);
  for (i = 1; i < 100; i++)
    @{
      mpz_mul (n, @dots{});
      mpz_fdiv_q (n, @dots{});
      @dots{}
    @}
  mpz_clear (n);
@}
@end example


@node Parameter Conventions, Memory Management, Variable Conventions, GMP Basics
@section Parameter Conventions
@cindex Parameter conventions
@cindex Conventions for parameters

When a GMP variable is used as a function parameter, it's effectively a
call-by-reference, meaning if the function stores a value there it will change
the original in the caller.

When a function is going to return a GMP result, it should designate a
parameter that it sets, like the library functions do.  More than one value
can be returned by having more than one output parameter, again like the
library functions.  A @code{return} of an @code{mpz_t} etc doesn't return the
object, only a pointer, and this is almost certainly not what's wanted.

Here's an example accepting an @code{mpz_t} parameter, doing a calculation,
and storing the result to the indicated parameter.

@example
void
foo (mpz_t result, mpz_t param, unsigned long n)
@{
  unsigned long  i;
  mpz_mul_ui (result, param, n);
  for (i = 1; i < n; i++)
    mpz_add_ui (result, result, i*7);
@}

int
main (void)
@{
  mpz_t  r, n;
  mpz_init (r);
  mpz_init_set_str (n, "123456", 0);
  foo (r, n, 20L);
  gmp_printf ("%Zd\n", r);
  return 0;
@}
@end example

@code{foo} works even if the mainline passes the same variable as both
@code{param} and @code{result}, just like the library functions.  But
sometimes this is tricky to arrange, and an application might not want to
bother supporting that sort of thing.

For interest, the GMP types @code{mpz_t} etc are implemented as one-element
arrays of certain structures.  This is why declaring a variable creates an
object with the fields GMP needs, but then using it as a parameter passes a
pointer to the object.  Note that the actual fields in each @code{mpz_t} etc
are for internal use only and should not be accessed directly by code that
expects to be compatible with future GMP releases.


@need 1000
@node Memory Management, Reentrancy, Parameter Conventions, GMP Basics
@section Memory Management
@cindex Memory Management

The GMP types like @code{mpz_t} are small, containing only a couple of sizes,
and pointers to allocated data.  Once a variable is initialized, GMP takes
care of all space allocation.  Additional space is allocated whenever a
variable doesn't have enough.

@code{mpz_t} and @code{mpq_t} variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular point
can use @code{mpz_realloc2}, or clear variables no longer needed.

@code{mpf_t} variables, in the current implementation, use a fixed amount of
space, determined by the chosen precision and allocated at initialization, so
their size doesn't change.

All memory is allocated using @code{malloc} and friends by default, but this
can be changed, see @ref{Custom Allocation}.  Temporary memory on the stack is
also used (via @code{alloca}), but this can be changed at build-time if
desired, see @ref{Build Options}.


@node Reentrancy, Useful Macros and Constants, Memory Management, GMP Basics
@section Reentrancy
@cindex Reentrancy
@cindex Thread safety
@cindex Multi-threading

GMP is reentrant and thread-safe, with some exceptions:

@itemize @bullet
@item
If configured with @option{--enable-alloca=malloc-notreentrant} (or with
@option{--enable-alloca=notreentrant} when @code{alloca} is not available),
then naturally GMP is not reentrant.

@item
@code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the
selected precision.  @code{mpf_init2} can be used instead.

@item
@code{mp_set_memory_functions} uses global variables to store the selected
memory allocation functions.

@item
@code{mpz_random} and the other old random number functions use a global
random state and are hence not reentrant.  The newer random number functions
that accept a @code{gmp_randstate_t} parameter can be used instead.

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

@item
If the standard I/O functions such as @code{fwrite} are not reentrant then the
GMP I/O functions using them will not be reentrant either.

@item
It's safe for two threads to read from the same GMP variable simultaneously,
but it's not safe for one to read while the another might be writing, nor for
two threads to write simultaneously.  It's not safe for two threads to
generate a random number from the same @code{gmp_randstate_t} simultaneously,
since this involves an update of that variable.

@item
On SCO systems the default @code{<ctype.h>} macros use per-file static
variables and may not be reentrant, depending whether the compiler optimizes
away fetches from them.  The GMP text-based input functions are affected.
@end itemize


@need 2000
@node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics
@section Useful Macros and Constants
@cindex Useful macros and constants
@cindex Constants

@deftypevr {Global Constant} {const int} mp_bits_per_limb
@findex mp_bits_per_limb
@cindex Bits per limb
@cindex Limb size
The number of bits per limb.
@end deftypevr

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

@deftypevr {Global Constant} {const char * const} gmp_version
@findex gmp_version
The GMP version number, as a null-terminated string, in the form ``i.j'' or
``i.j.k''.  This release is @nicode{"@value{VERSION}"}.
@end deftypevr


@node Compatibility with older versions, Efficiency, Useful Macros and Constants, GMP Basics
@section Compatibility with older versions
@cindex Compatibility with older versions
@cindex Upward compatibility

This version of GMP is upwardly binary compatible with all 3.x versions, and
upwardly compatible at the source level with all 2.x versions, with the
following exceptions.

@itemize @bullet
@item
@code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency
with other @code{mpn} functions.

@item
@code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and
3.0.1, but in 3.1 reverted to the 2.x style.
@end itemize

There are a number of compatibility issues between GMP 1 and GMP 2 that of
course also apply when porting applications from GMP 1 to GMP 4.  Please
see the GMP 2 manual for details.

The Berkeley MP compatibility library (@pxref{BSD Compatible Functions}) is
source and binary compatible with the standard @file{libmp}.

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

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

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

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

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

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

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

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

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

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

@c @end enumerate


@need 1000
@node Efficiency, Debugging, Compatibility with older versions, GMP Basics
@section Efficiency
@cindex Efficiency

@table @asis
@item Small operands
On small operands, the time for function call overheads and memory allocation
can be significant in comparison to actual calculation.  This is unavoidable
in a general purpose variable precision library, although GMP attempts to be
as efficient as it can on both large and small operands.

@item Static Linking
On some CPUs, in particular the x86s, the static @file{libgmp.a} should be
used for maximum speed, since the PIC code in the shared @file{libgmp.so} will
have a small overhead on each function call and global data address.  For many
programs this will be insignificant, but for long calculations there's a gain
to be had.

@item Initializing and clearing
Avoid excessive initializing and clearing of variables, since this can be
quite time consuming, especially in comparison to otherwise fast operations
like addition.

A language interpreter might want to keep a free list or stack of
initialized variables ready for use.  It should be possible to integrate
something like that with a garbage collector too.

@item Reallocations
An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing
values will have its memory repeatedly @code{realloc}ed, which could be quite
slow or could fragment memory, depending on the C library.  If an application
can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can
be called to allocate the necessary space from the beginning
(@pxref{Initializing Integers}).

It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2}
is too small, since all functions will do a further reallocation if necessary.
Badly overestimating memory required will waste space though.

@item @code{2exp} functions
It's up to an application to call functions like @code{mpz_mul_2exp} when
appropriate.  General purpose functions like @code{mpz_mul} make no attempt to
identify powers of two or other special forms, because such inputs will
usually be very rare and testing every time would be wasteful.

@item @code{ui} and @code{si} functions
The @code{ui} functions and the small number of @code{si} functions exist for
convenience and should be used where applicable.  But if for example an
@code{mpz_t} contains a value that fits in an @code{unsigned long} there's no
need extract it and call a @code{ui} function, just use the regular @code{mpz}
function.

@item In-Place Operations
@code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg}
and @code{mpf_neg} are fast when used for in-place operations like
@code{mpz_abs(x,x)}, since in the current implementation only a single field
of @code{x} needs changing.  On suitable compilers (GCC for instance) this is
inlined too.

@code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui}
benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since
usually only one or two limbs of @code{x} will need to be changed.  The same
applies to the full precision @code{mpz_add} etc if @code{y} is small.  If
@code{y} is big then cache locality may be helped, but that's all.

@code{mpz_mul} is currently the opposite, a separate destination is slightly
better.  A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one
limb, make a temporary copy of @code{x} before forming the result.  Normally
that copying will only be a tiny fraction of the time for the multiply, so
this is not a particularly important consideration.

@code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make
no attempt to recognise a copy of something to itself, so a call like
@code{mpz_set(x,x)} will be wasteful.  Naturally that would never be written
deliberately, but if it might arise from two pointers to the same object then
a test to avoid it might be desirable.

@example
if (x != y)
  mpz_set (x, y);
@end example

Note that it's never worth introducing extra @code{mpz_set} calls just to get
in-place operations.  If a result should go to a particular variable then just
direct it there and let GMP take care of data movement.

@item Divisibility Testing (Small Integers)

@code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions
for testing whether an @code{mpz_t} is divisible by an individual small
integer.  They use an algorithm which is faster than @code{mpz_tdiv_ui}, but
which gives no useful information about the actual remainder, only whether
it's zero (or a particular value).

However when testing divisibility by several small integers, it's best to take
a remainder modulo their product, to save multi-precision operations.  For
instance to test whether a number is divisible by any of 23, 29 or 31 take a
remainder modulo @ma{23@times{}29@times{}31 = 20677} and then test that.

The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well
as a remainder are generally a little slower than the remainder-only functions
like @code{mpz_tdiv_ui}.  If the quotient is only rarely wanted then it's
probably best to just take a remainder and then go back and calculate the
quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the
remainder is zero).

@item Rational Arithmetic
The @code{mpq} functions operate on @code{mpq_t} values with no common factors
in the numerator and denominator.  Common factors are checked-for and cast out
as necessary.  In general, cancelling factors every time is the best approach
since it minimizes the sizes for subsequent operations.

However, applications that know something about the factorization of the
values they're working with might be able to avoid some of the GCDs used for
canonicalization, or swap them for divisions.  For example when multiplying by
a prime it's enough to check for factors of it in the denominator instead of
doing a full GCD.  Or when forming a big product it might be known that very
little cancellation will be possible, and so canonicalization can be left to
the end.

The @code{mpq_numref} and @code{mpq_denref} macros give access to the
numerator and denominator to do things outside the scope of the supplied
@code{mpq} functions.  @xref{Applying Integer Functions}.

The canonical form for rationals allows mixed-type @code{mpq_t} and integer
additions or subtractions to be done directly with multiples of the
denominator.  This will be somewhat faster than @code{mpq_add}.  For example,

@example
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));

/* mpq += unsigned long */
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);

/* mpq -= mpz */
mpz_submul (mpq_numref(q), mpq_denref(q), z);
@end example

@item Number Sequences
Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui}
are designed for calculating isolated values.  If a range of values is wanted
it's probably best to call to get a starting point and iterate from there.
@end table


@node Debugging, Profiling, Efficiency, GMP Basics
@section Debugging
@cindex Debugging

@table @asis
@item Stack Overflow
Depending on the system, a segmentation violation or bus error might be the
only indication of stack overflow.  See @samp{--enable-alloca} choices in
@ref{Build Options}, for how to address this.

@item Heap Problems
The most likely cause of application problems with GMP is heap corruption.
Failing to @code{init} GMP variables will have unpredictable effects, and
corruption arising elsewhere in a program may well affect GMP.  Initializing
GMP variables more than once or failing to clear them will cause memory leaks.

In all such cases a malloc debugger is recommended.  On a GNU or BSD system
the standard C library @code{malloc} has some diagnostic facilities, see
@ref{Allocation Debugging,,,libc,The GNU C Library Reference Manual}, or
@samp{man 3 malloc}.  Other possibilities, in no particular order, include

@display
@uref{http://www.inf.ethz.ch/personal/biere/projects/ccmalloc}
@uref{http://quorum.tamu.edu/jon/gnu} @ (debauch)
@uref{http://dmalloc.com}
@uref{http://www.perens.com/FreeSoftware} @ (electric fence)
@uref{http://packages.debian.org/fda}
@uref{http://www.gnupdate.org/components/leakbug}
@uref{http://people.redhat.com/~otaylor/memprof}
@uref{http://www.cbmamiga.demon.co.uk/mpatrol}
@end display

@item Stack Backtraces
On some systems the compiler options GMP uses by default can interfere with
debugging.  In particular on x86 and 68k systems @samp{-fomit-frame-pointer}
is used and this generally inhibits stack backtracing.  Recompiling without
such options may help while debugging, though the usual caveats about it
potentially moving a memory problem or hiding a compiler bug will apply.

@item GNU Debugger
A sample @file{.gdbinit} is included in the distribution, showing how to call
some undocumented dump functions to print GMP variables from within GDB.  Note
that these functions shouldn't be used in final application code since they're
undocumented and may be subject to incompatible changes in future versions of
GMP.

@item Source File Paths
GMP has multiple source files with the same name, in different directories.
For example @file{mpz}, @file{mpq}, @file{mpf} and @file{mpfr} each have an
@file{init.c}.  If the debugger can't already determine the right one it may
help to build with absolute paths on each C file.  One way to do that is to
use a separate object directory with an absolute path to the source directory.

@example
cd /my/build/dir
/my/source/dir/gmp-@value{VERSION}/configure
@end example

This works via @code{VPATH}, and might require GNU @command{make}.
Alternately it might be possible to change the @code{.c.lo} rules
appropriately.

@item Assertion Checking
The build option @option{--enable-assert} is available to add some consistency
checks to the library (see @ref{Build Options}).  These are likely to be of
limited value to most applications.  Assertion failures are just as likely to
indicate memory corruption as a library or compiler bug.

Applications using the low-level @code{mpn} functions, however, will benefit
from @option{--enable-assert} since it adds checks on the parameters of most
such functions, many of which have subtle restrictions on their usage.  Note
however that only the generic C code has checks, not the assembler code, so
CPU @samp{none} should be used for maximum checking.

@item Temporary Memory Checking
The build option @option{--enable-alloca=debug} arranges that each block of
temporary memory in GMP is allocated with a separate call to @code{malloc} (or
the allocation function set with @code{mp_set_memory_functions}).

This can help a malloc debugger detect accesses outside the intended bounds,
or detect memory not released.  In a normal build, on the other hand,
temporary memory is allocated in blocks which GMP divides up for its own use,
or may be allocated with a compiler builtin @code{alloca} which will go
nowhere near any malloc debugger hooks.

@item Other Problems
Any suspected bug in GMP itself should be isolated to make sure it's not an
application problem, see @ref{Reporting Bugs}.
@end table


@node Profiling, Autoconf, Debugging, GMP Basics
@section Profiling
@cindex Profiling

Running a program under a profiler is a good way to find where it's spending
most time and where improvements can be best sought.

Depending on the system, it may be possible to get a flat profile, meaning
simple timer sampling of the program counter, with no special GMP build
options, just a @samp{-p} when compiling the mainline.  This is a good way to
ensure minimum interference with normal operation.  The necessary symbol type
and size information exists in most of the GMP assembler code.

The @samp{--enable-profiling} build option can be used to add suitable
compiler flags, either for @command{prof} (@samp{-p}) or @command{gprof}
(@samp{-pg}), see @ref{Build Options}.  Which of the two is available and what
they do will depend on the system, and possibly on support available in
@file{libc}.  For some systems appropriate corresponding @code{mcount} calls
are added to the assembler code too.

On x86 systems @command{prof} gives call counting, so that average time spent
in a function can be determined.  @command{gprof}, where supported, adds call
graph construction, so for instance calls to @code{mpn_add_n} from
@code{mpz_add} and from @code{mpz_mul} can be differentiated.

On x86 and 68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are
incompatible, so the latter is not used when @command{gprof} profiling is
selected, which may result in poorer code generation.  If @command{prof}
profiling is selected instead it should still be possible to use
@command{gprof}, but only the @samp{gprof -p} flat profile and call counts can
be expected to be valid, not the @samp{gprof -q} call graph.


@node Autoconf,  , Profiling, GMP Basics
@section Autoconf
@cindex Autoconf detections

Autoconf based applications can easily check whether GMP is installed.  The
only thing to be noted is that GMP library symbols from version 3 onwards have
prefixes like @code{__gmpz}.  The following therefore would be a simple test,

@example
AC_CHECK_LIB(gmp, __gmpz_init)
@end example

This just uses the default @code{AC_CHECK_LIB} actions for found or not found,
but an application that must have GMP would want to generate an error if not
found.  For example,

@example
AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR(
[GNU MP not found, see http://swox.com/gmp])])
@end example

If functions added in some particular version of GMP are required, then one of
those can be used when checking.  For example @code{mpz_mul_si} was added in
GMP 3.1,
    
@example
AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR(
[GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
@end example

An alternative would be to test the version number in @file{gmp.h} using say
@code{AC_EGREP_CPP}.  That would make it possible to test the exact version,
if some particular sub-minor release is known to be necessary.

An application that can use either GMP 2 or 3 will need to test for
@code{__gmpz_init} (GMP 3 and up) or @code{mpz_init} (GMP 2), and it's also
worth checking for @file{libgmp2} since Debian GNU/Linux systems used that
name in the past.  For example,

@example
AC_CHECK_LIB(gmp, __gmpz_init, ,
  [AC_CHECK_LIB(gmp, mpz_init, ,
    [AC_CHECK_LIB(gmp2, mpz_init)])])
@end example

In general it's suggested that applications should simply demand a new enough
GMP rather than trying to provide supplements for features not available in
past versions.

Occasionally an application will need or want to know the size of a type at
configuration or preprocessing time, not just with @code{sizeof} in the code.
This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or
up is best for this, since prior versions needed certain @samp{-D} defines on
systems using a @code{long long} limb.  The following would suit Autoconf 2.50
or up,

@example
AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
@end example

The optional @code{mpfr} functions are provided in a separate
@file{libmpfr.a}, and this might be from GMP with @option{--enable-mpfr} or
from MPFR installed separately.  Either way @file{libmpfr} depends on
@file{libgmp}, it doesn't stand alone.  Currently only a static
@file{libmpfr.a} will be available, not a shared library, since upward binary
compatibility is not guaranteed.

@example
AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR(
[Need MPFR either from GNU MP 4 or separate MPFR package.
See http://www.mpfr.org or http://swox.com/gmp])
@end example


@node Reporting Bugs, Integer Functions, GMP Basics, Top
@comment  node-name,  next,  previous,  up
@chapter Reporting Bugs
@cindex Reporting bugs
@cindex Bug reporting

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

Before you report a bug, check it's not already addressed in @ref{Known Build
Problems}, or perhaps @ref{Notes for Particular Systems}.  You may also want
to check @uref{http://swox.com/gmp/} for patches for this release.

Please include the following in any report,

@itemize @bullet
@item
The GMP version number, and if pre-packaged or patched then say so.

@item
A test program that makes it possible for us to reproduce the bug.  Include
instructions on how to run the program.

@item
A description of what is wrong.  If the results are incorrect, in what way.
If you get a crash, say so.

@item
If you get a crash, include a stack backtrace from the debugger if it's
informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).

@item
Please do not send core dumps, executables or @command{strace}s.

@item
The configuration options you used when building GMP, if any.

@item
The name of the compiler and its version.  For @command{gcc}, get the version
with @samp{gcc -v}, otherwise perhaps @samp{what `which cc`}, or similar.

@item
The output from running @samp{uname -a}.

@item
The output from running @samp{./config.guess}, and from running
@samp{./configfsf.guess} (might be the same).

@item
If the bug is related to @samp{configure}, then the contents of
@file{config.log}.

@item
If the bug is related to an @file{asm} file not assembling, then the contents
of @file{config.m4} and the offending line or lines from the temporary
@file{mpn/tmp-<file>.s}.
@end itemize

Please make an effort to produce a self-contained report, with something
definite that can be tested or debugged.  Vague queries or piecemeal messages
are difficult to act on and don't help the development effort.

It is not uncommon that an observed problem is actually due to a bug in the
compiler; the GMP code tends to explore interesting corners in compilers.

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

Send your report to: @email{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 GMP functions for performing integer arithmetic.
These functions start with the prefix @code{mpz_}.

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

@menu
* Initializing Integers::       
* Assigning Integers::          
* Simultaneous Integer Init & Assign::  
* Converting Integers::         
* Integer Arithmetic::          
* Integer Division::            
* Integer Exponentiation::      
* Integer Roots::               
* Number Theoretic Functions::  
* Integer Comparisons::         
* Integer Logic and Bit Fiddling::  
* I/O of Integers::             
* Integer Random Numbers::      
* Miscellaneous Integer Functions::  
@end menu

@node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Integer initialization functions
@cindex Initialization functions

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

@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

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

@deftypefun void mpz_init (mpz_t @var{integer})
Initialize @var{integer}, and set its value to 0.
@end deftypefun

@deftypefun void mpz_init2 (mpz_t @var{integer}, unsigned long @var{n})
Initialize @var{integer}, with space for @var{n} bits, and set its value to 0.

@var{n} is only the initial space, @var{integer} will grow automatically in
the normal way, if necessary, for subsequent values stored.  @code{mpz_init2}
makes it possible to avoid such reallocations if a maximum size is known in
advance.
@end deftypefun

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

@deftypefun void mpz_realloc2 (mpz_t @var{integer}, unsigned long @var{n})
Change the space allocated for @var{integer} to @var{n} bits.  The value in
@var{integer} is preserved if it fits, or is set to 0 if not.

This function can be used to increase the space for a variable in order to
avoid repeated automatic reallocations, or to decrease it to give memory back
to the heap.
@end deftypefun

@deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
This is a special type of initialization.  @strong{Fixed} space of
@var{fixed_num_bits} bits is allocated to each of the @var{array_size}
integers in @var{integer_array}.

The space will not be automatically increased, unlike the normal
@code{mpz_init}, but instead an application must ensure it's sufficient for
any value stored.  The following space requirements apply to various
functions,

@itemize @bullet
@item
@code{mpz_abs}, @code{mpz_neg}, @code{mpz_set}, @code{mpz_set_si} and
@code{mpz_set_ui} need room for the value they store.

@item
@code{mpz_add}, @code{mpz_add_ui}, @code{mpz_sub} and @code{mpz_sub_ui} need
room for the larger of the two operands, plus an extra
@code{mp_bits_per_limb}.

@item
@code{mpz_mul}, @code{mpz_mul_ui} and @code{mpz_mul_ui} need room for the sum
of the number of bits in their operands, but each rounded up to a multiple of
@code{mp_bits_per_limb}.

@item
@code{mpz_swap} can be used between two array variables, but not between an
array and a normal variable.
@end itemize

For other functions, or if in doubt, the suggestion is to calculate in a
regular @code{mpz_init} variable and copy the result to an array variable with
@code{mpz_set}.

@code{mpz_array_init} can reduce memory usage in algorithms that need large
arrays of integers, since it avoids allocating and reallocating lots of small
memory blocks.  There is no way to free the storage allocated by this
function.  Don't call @code{mpz_clear}!
@end deftypefun

@deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
Change the space for @var{integer} to @var{new_alloc} limbs.  The value in
@var{integer} is preserved if it fits, or is set to 0 if not.  The return
value is not useful to applications and should be ignored.

@code{mpz_realloc2} is the preferred way to accomplish allocation changes like
this.  @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that
@code{_mpz_realloc} takes the new size in limbs.
@end deftypefun


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

@code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to
make it an integer.
@end deftypefun

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

This function returns 0 if the entire string is a valid number in base
@var{base}.  Otherwise it returns @minus{}1.

[It turns out that it is not entirely true that this function ignores
white-space.  It does ignore it between digits, but not after a minus sign or
within or after ``0x''.  We are considering changing the definition of this
function, making it fail when there is any white-space in the input, since
that makes a lot of sense.  Send your opinion of this change to
@email{bug-gmp@@gnu.org}.  Do you really want it to accept @nicode{"3 14"} as
meaning 314 as it does now?]
@end deftypefun

@deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
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
@section Combined Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Integer init and assign

For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there.  These functions'
names have the form @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 GMP integers to standard C
types.  Functions for converting @emph{to} GMP integers are described in
@ref{Assigning Integers} and @ref{I/O of Integers}.

@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.  To find out if the value will fit, use
the function @code{mpz_fits_slong_p}.
@end deftypefun

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

@deftypefun double mpz_get_d_2exp (signed long int @var{exp}, mpz_t @var{op})
Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
raised to @var{exp}}, with @ma{0.5@le{}@GMPabs{@var{d}}<1}, is a good
approximation to @var{op}.
@end deftypefun

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

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
+ 2}.  The two extra bytes are for a possible minus sign, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun

@deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
Return limb number @var{n} from @var{op}.  The sign of @var{op} is ignored,
just the absolute value is used.  The least significant limb is number 0.

@code{mpz_size} can be used to find how many limbs make up @var{op}.
@code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to
@code{mpz_size(@var{op})-1}.
@end deftypefun


@need 2000
@node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@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})
Set @var{rop} to @ma{@var{op1} + @var{op2}}.
@end deftypefun

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

@deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_mul_si (mpz_t @var{rop}, mpz_t @var{op1}, long int @var{op2})
@deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @ma{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_addmul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @ma{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_submul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_submul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @ma{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@cindex Bit shift left
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.  This operation can also be defined as a left shift by @var{op2}
bits.
@end deftypefun

@deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
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


@need 2000
@node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions
@section Division Functions
@cindex Integer division functions
@cindex Division functions

Division is undefined if the divisor is zero.  Passing a zero divisor to the
division or modulo functions (including the modular powering functions
@code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
zero.  This lets a program handle arithmetic exceptions in these functions the
same way as for normal C @code{int} arithmetic.

@c  Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
@c  between each, and seem to let tex do a better job of page breaks than an
@c  @sp 1 in the middle of one big set.

@deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@end deftypefun

@deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@end deftypefun

@deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@cindex Bit shift right

@sp 1
Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
@var{r}.  For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}.
The rounding is in three styles, each suiting different applications.

@itemize @bullet
@item
@code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
have the opposite sign to @var{d}.  The @code{c} stands for ``ceil''.

@item
@code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
@var{r} will have the same sign as @var{d}.  The @code{f} stands for
``floor''.

@item
@code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
as @var{n}.  The @code{t} stands for ``truncate''.
@end itemize

In all cases @var{q} and @var{r} will satisfy
@m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and
@var{r} will satisfy @ma{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.

The @code{q} functions calculate only the quotient, the @code{r} functions
only the remainder, and the @code{qr} functions calculate both.  Note that for
@code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or
results will be unpredictable.

For the @code{ui} variants the return value is the remainder, and in fact
returning the remainder is all the @code{div_ui} functions do.  For
@code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
return value is the absolute value of the remainder.

The @code{2exp} functions are right shifts and bit masks, but of course
rounding the same as the other functions.  For positive @var{n} both
@code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp} are simple bitwise right
shifts.  For negative @var{n}, @code{mpz_fdiv_q_2exp} is effectively an
arithmetic right shift treating @var{n} as twos complement the same as the
bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp} effectively
treats @var{n} as sign and magnitude.
@end deftypefun

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

@code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
remainder as well as setting @var{r}.  See @code{mpz_fdiv_ui} above if only
the return value is wanted.
@end deftypefun

@deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long @var{d})
@cindex Exact division functions
Set @var{q} to @var{n}/@var{d}.  These functions produce correct results only
when it is known in advance that @var{d} divides @var{n}.

These routines are much faster than the other division functions, and are the
best choice when exact division is known to occur, for example reducing a
rational to lowest terms.
@end deftypefun

@deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d})
@deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, unsigned long int @var{d})
@deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, unsigned long int @var{b})
Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
@code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.
@end deftypefun

@deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d})
@deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d})
@deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, unsigned long int @var{b})
Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.
@end deftypefun


@need 2000
@node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
@section Exponentiation Functions
@cindex Integer exponentiation functions
@cindex Exponentiation functions
@cindex Powering functions

@deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
@deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod})
Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
modulo @var{mod}}.

Negative @var{exp} is supported if an inverse @ma{@var{base}^@W{-1} @bmod
@var{mod}} exists (see @code{mpz_invert} in @ref{Number Theoretic Functions}).
If an inverse doesn't exist then a divide by zero is raised.
@end deftypefun

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


@need 2000
@node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions
@section Root Extraction Functions
@cindex Integer root functions
@cindex Root extraction functions

@deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n})
Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor@C{},} the truncated integer
part of the @var{n}th root of @var{op}.  Return non-zero if the computation
was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power.
@end deftypefun

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

@deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{rop2} to the
remainder @m{(@var{op} - @var{rop1}^2),
@var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
perfect square.

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

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

Under this definition both 0 and 1 are considered to be perfect powers.
Negative values of @var{op} are accepted, but of course can only be odd
perfect powers.
@end deftypefun

@deftypefun int mpz_perfect_square_p (mpz_t @var{op})
Return non-zero if @var{op} is a perfect square, i.e., if the square root of
@var{op} is an integer.  Under this definition both 0 and 1 are considered to
be perfect squares.
@end deftypefun


@need 2000
@node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions
@section Number Theoretic Functions
@cindex Number theoretic functions

@deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
@cindex Prime testing functions
Determine whether @var{n} is prime.  Return 2 if @var{n} is definitely prime,
return 1 if @var{n} is probably prime (without being certain), or return 0 if
@var{n} is definitely composite.

This function does some trial divisions, then some Miller-Rabin probabilistic
primality tests.  @var{reps} controls how many such tests are done, 5 to 10 is
a reasonable number, more will reduce the chances of a composite being
returned as ``probably prime''.

Miller-Rabin and similar tests can be more properly called compositeness
tests.  Numbers which fail are known to be composite but those which pass
might be prime or might be composite.  Only a few composites pass, hence those
which pass are considered probably prime.
@end deftypefun

@deftypefun void mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the next prime greater than @var{op}.

This function uses a probabilistic algorithm to identify primes.  For
practical purposes it's adequate, the chance of a composite passing will be
extremely small.
@end deftypefun

@c mpz_prime_p not implemented as of gmp 3.0.

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

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

@deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@cindex Greatest common divisor functions
Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.
The result is always positive even if one 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 @code{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})
@cindex Extended GCD
Compute @var{g}, @var{s}, and @var{t}, such that
@ma{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g} =
@gcd{}(@var{a}, @var{b})}.  If @var{t} is @code{NULL}, that argument is
not computed.
@end deftypefun

@deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long @var{op2})
@cindex Least common multiple functions
Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
@var{rop} is always positive, irrespective of the signs of @var{op1} and
@var{op2}.  @var{rop} will be zero if either @var{op1} or @var{op2} is zero.
@end deftypefun

@deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@cindex Modular inverse functions
Compute the inverse of @var{op1} modulo @var{op2} and put the result in
@var{rop}.  If the inverse exists, the return value is non-zero and @var{rop}
will satisfy @ma{0 @le{} @var{rop} < @var{op2}}.  If an inverse doesn't exist
the return value is zero and @var{rop} is undefined.
@end deftypefun

@deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b})
@cindex Jacobi symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})}.  This is defined only for @var{b} odd.
@end deftypefun

@deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p})
Calculate the Legendre symbol @m{\left(a \over p\right),
(@var{a}/@var{p})}.  This is defined only for @var{p} an odd positive
prime, and for such @var{p} it's identical to the Jacobi symbol.
@end deftypefun

@deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b})
@deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b})
@deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b})
@deftypefunx int mpz_si_kronecker (long @var{a}, mpz_t @var{b})
@deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b})
@cindex Kronecker symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @ma{a} odd, or
@m{\left(a \over 2\right) = 0, (a/2)=0} when @ma{a} even.

When @var{b} is odd the Jacobi symbol and Kronecker symbol are
identical, so @code{mpz_kronecker_ui} etc can be used for mixed
precision Jacobi symbols too.

For more information see Henri Cohen section 1.4.2 (@pxref{References}),
or any number theory textbook.  See also the example program
@file{demos/qcn.c} which uses @code{mpz_kronecker_ui}.
@end deftypefun

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

@deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{op})
@cindex Factorial functions
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}, @w{unsigned long int @var{k}})
@cindex Binomial coefficient functions
Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over
@var{k}} and store the result in @var{rop}.  Negative values of @var{n} are
supported by @code{mpz_bin_ui}, using the identity
@m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6
part G.
@end deftypefun

@deftypefun void mpz_fib_ui (mpz_t @var{fn}, unsigned long int @var{n})
@deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, unsigned long int @var{n})
@cindex Fibonacci sequence functions
@code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci
number.  @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to
@m{F_{n-1},F[n-1]}.

These functions are designed for calculating isolated Fibonacci numbers.  When
a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and
iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
similar.
@end deftypefun

@deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, unsigned long int @var{n})
@deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, unsigned long int @var{n})
@cindex Lucas number functions
@code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas
number.  @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1}
to @m{L_{n-1},L[n-1]}.

These functions are designed for calculating isolated Lucas numbers.  When a
sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and
iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
similar.

The Fibonacci numbers and Lucas numbers are related sequences, so it's never
necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}.  The
formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers
Algorithm}, the reverse is straightforward too.
@end deftypefun


@node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Integer comparison functions
@cindex Comparison functions

@deftypefn Function int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefnx Function int mpz_cmp_d (mpz_t @var{op1}, double @var{op2})
@deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2})
@deftypefnx Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @ma{@var{op1} >
@var{op2}}, zero if @ma{@var{op1} = @var{op2}}, or a negative value if
@ma{@var{op1} < @var{op2}}.

Note that @code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate
their arguments more than once.
@end deftypefn

@deftypefn Function int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefnx Function int mpz_cmpabs_d (mpz_t @var{op1}, double @var{op2})
@deftypefnx Function int mpz_cmpabs_ui (mpz_t @var{op1}, unsigned long int @var{op2})
Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
value if @ma{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
@ma{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
@ma{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.

Note that @code{mpz_cmpabs_si} is a macro and will evaluate its arguments more
than once.
@end deftypefn

@deftypefn Macro int mpz_sgn (mpz_t @var{op})
@cindex Sign tests
@cindex Integer sign tests
Return @ma{+1} if @ma{@var{op} > 0}, 0 if @ma{@var{op} = 0}, and @ma{-1} if
@ma{@var{op} < 0}.

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


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

These functions behave as if twos complement arithmetic were used (although
sign-magnitude is the actual implementation).  The least significant bit is
number 0.

@deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} logical-and @var{op2}.
@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})
If @ma{@var{op}@ge{}0}, return the population count of @var{op}, which is the
number of 1 bits in the binary representation.  If @ma{@var{op}<0}, the number
of 1s is infinite, and the return value is @var{MAX_ULONG}, the largest
possible @code{unsigned long}.
@end deftypefun

@deftypefun {unsigned long int} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
If @var{op1} and @var{op2} are both @ma{@ge{}0} or both @ma{<0}, return the
hamming distance between the two operands, which is the number of bit
positions where @var{op1} and @var{op2} have different bit values.  If one
operand is @ma{@ge{}0} and the other @ma{<0} then the number of bits different
is infinite, and the return value is @var{MAX_ULONG}, the largest possible
@code{unsigned long}.
@end deftypefun

@deftypefun {unsigned long int} mpz_scan0 (mpz_t @var{op}, unsigned long int @var{starting_bit})
@deftypefunx {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})
Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
bits, until the first 0 or 1 bit (respectively) is found.  Return the index of
the found bit.

If the bit at @var{starting_bit} is already what's sought, then
@var{starting_bit} is returned.

If there's no bit found, then @var{MAX_ULONG} is returned.  This will happen
in @code{mpz_scan0} past the end of a positive number, or @code{mpz_scan1}
past the end of a negative.
@end deftypefun

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

@node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions
@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 @code{NULL} pointer for a @var{stream} argument to any of
these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
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 Integer Random Numbers, Miscellaneous Integer Functions, I/O of Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Random Number Functions
@cindex Integer random number functions
@cindex Random number functions

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

@deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
Generate a uniformly distributed random integer in the range 0 to @m{2^n-1,
2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

@deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, mpz_t @var{n})
Generate a uniform random integer in the range 0 to @ma{@var{n}-1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
Generate a random integer with long strings of zeros and ones in the
binary representation.  Useful for testing functions and algorithms,
since this kind of random numbers have proven to be more likely to
trigger corner-case bugs.  The random number will be in the range
0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_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


@need 2000
@node Miscellaneous Integer Functions,  , Integer Random Numbers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous integer functions
@cindex Integer miscellaneous functions

@deftypefun int mpz_fits_ulong_p (mpz_t @var{op})
@deftypefunx int mpz_fits_slong_p (mpz_t @var{op})
@deftypefunx int mpz_fits_uint_p (mpz_t @var{op})
@deftypefunx int mpz_fits_sint_p (mpz_t @var{op})
@deftypefunx int mpz_fits_ushort_p (mpz_t @var{op})
@deftypefunx int mpz_fits_sshort_p (mpz_t @var{op})
Return non-zero iff the value of @var{op} fits in an @code{unsigned long int},
@code{signed long int}, @code{unsigned int}, @code{signed int}, @code{unsigned
short int}, or @code{signed short int}, respectively.  Otherwise, return zero.
@end deftypefun

@deftypefn Macro int mpz_odd_p (mpz_t @var{op})
@deftypefnx Macro int mpz_even_p (mpz_t @var{op})
Determine whether @var{op} is odd or even, respectively.  Return non-zero if
yes, zero if no.  These macros evaluate their argument more than once.
@end deftypefn

@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 sign of @var{op} is ignored, just the
absolute value is used.  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 null-terminator).
@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 GMP 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.

@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::      
* Rational Conversions::        
* Rational Arithmetic::         
* Comparing Rationals::         
* Applying Integer Functions::  
* I/O of Rationals::            
@end menu

@node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Rational init and assign

@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

@deftypefun int mpq_set_str (mpq_t @var{rop}, char *@var{str}, int @var{base})
Set @var{rop} from a null-terminated string @var{str} in the given @var{base}.

The string can be an integer like "41" or a fraction like "41/152".  The
fraction must be in canonical form (@pxref{Rational Number Functions}), or if
not then @code{mpq_canonicalize} must be called.

The numerator and optional denominator are parsed the same as in
@code{mpz_set_str} (@pxref{Assigning Integers}).  White space is allowed in
the string, and is simply ignored.  The @var{base} can vary from 2 to 36, or
if @var{base} is 0 then the leading characters are used: @code{0x} for hex,
@code{0} for octal, or decimal otherwise.  Note that this is done separately
for the numerator and denominator, so for instance @code{0xEF/100} is 239/100,
whereas @code{0xEF/0x100} is 239/256.

The return value is 0 if the entire string is a valid number, or @minus{}1 if
not.
@end deftypefun

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


@need 2000
@node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Rational conversion functions
@cindex Conversion functions

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

@deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op})
@deftypefunx void mpq_set_f (mpq_t @var{rop}, mpf_t @var{op})
Set @var{rop} to the value of @var{op}, without rounding.
@end deftypefun

@deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, mpq_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base may vary
from 2 to 36.  The string will be of the form @samp{num/den}, or if the
denominator is 1 then just @samp{num}.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being

@example
mpz_sizeinbase (mpq_numref(@var{op}), @var{base})
+ mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3
@end example

The three extra bytes are for a possible minus sign, possible slash, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun


@node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Rational arithmetic functions
@cindex Arithmetic functions

@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})
Set @var{product} to @ma{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
@end deftypefun

@deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

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

@deftypefun void mpq_div_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
@var{op2}}.
@end deftypefun

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

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

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

@node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Rational comparison functions
@cindex Comparison functions

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

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

@deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
@deftypefnx Macro int mpq_cmp_si (mpq_t @var{op1}, long int @var{num2}, unsigned long int @var{den2})
Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
@ma{@var{op1} > @var{num2}/@var{den2}}, zero if @ma{@var{op1} =
@var{num2}/@var{den2}}, and a negative value if @ma{@var{op1} <
@var{num2}/@var{den2}}.

@var{num2} and @var{den2} are allowed to have common factors.

These functions are implemented as a macros and evaluate their arguments
multiple times.
@end deftypefn

@deftypefn Macro int mpq_sgn (mpq_t @var{op})
@cindex Sign tests
@cindex Rational sign tests
Return @ma{+1} if @ma{@var{op} > 0}, 0 if @ma{@var{op} = 0}, and @ma{-1} if
@ma{@var{op} < 0}.

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, I/O of Rationals, Comparing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Applying Integer Functions to Rationals
@cindex Rational numerator and denominator
@cindex Numerator and denominator

The set of @code{mpq} functions is quite small.  In particular, there are few
functions for either input or output.  The following functions give direct
access to the numerator and denominator of an @code{mpq_t}.

Note that if an assignment to the numerator and/or denominator could take an
@code{mpq_t} out of the canonical form described at the start of this chapter
(@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be
called before any other @code{mpq} functions are applied to that @code{mpq_t}.

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

@deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
@deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
@deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
@deftypefunx void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
Get or set the numerator or denominator of a rational.  These functions are
equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
@code{mpq_denref}.  Direct use of @code{mpq_numref} or @code{mpq_denref} is
recommended instead of these functions.
@end deftypefun


@need 2000
@node I/O of Rationals,  , Applying Integer Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Rational input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

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

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

@deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base may vary from 2 to 36.  Output is in the form
@samp{num/den} or if the denominator is 1 then just @samp{num}.

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

@deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string of digits from @var{stream} and convert them to a rational in
@var{rop}.  Any initial white-space characters are read and discarded.  Return
the number of characters read (including white space), or 0 if a rational
could not be read.

The input can be a fraction like @samp{17/63} or just an integer like
@samp{123}.  Reading stops at the first character not in this form, and white
space is not permitted within the string.  If the input might not be in
canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational
Number Functions}).

The @var{base} can be between 2 and 36, or can be 0 in which case the leading
characters of the string determine the base, @samp{0x} or @samp{0X} for
hexadecimal, @samp{0} for octal, or decimal otherwise.  The leading characters
are examined separately for the numerator and denominator of a fraction, so
for instance @samp{0x10/11} is 16/11, whereas @samp{0x10/0x11} is 16/17.
@end deftypefun


@node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions
@cindex User-defined precision
@cindex Precision of floats

GMP floating point numbers are stored in objects of type @code{mpf_t} and
functions operating on them have an @code{mpf_} prefix.

The mantissa of each float has a user-selectable precision, limited only by
available memory.  Each variable has its own precision, and that can be
increased or decreased at any time.

The exponent of each float is a fixed precision, one machine word on most
systems.  In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
@ma{2^@W{-68719476768}} to @ma{2^@W{68719476736}}, or on a 64-bit system this
will be greater.  Note however @code{mpf_get_str} can only return an exponent
which fits an @code{mp_exp_t} and currently @code{mpf_set_str} doesn't accept
exponents bigger than a @code{long}.

Each variable keeps a size for the mantissa data actually in use.  This means
that if a float is exactly represented in only a few bits then only those bits
will be used in a calculation, even if the selected precision is high.

All calculations are performed to the precision of the destination variable.
Each function is defined to calculate with ``infinite precision'' followed by
a truncation to the destination precision, but of course the work done is only
what's needed to determine a result under that definition.

The precision selected for a variable is a minimum value, GMP may increase it
a little to facilitate efficient calculation.  Currently this means rounding
up to a whole limb, and then sometimes having a further partial limb,
depending on the high limb of the mantissa.  But applications shouldn't be
concerned by such details.

@code{mpf} functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the exponent or
results will be unpredictable.  This might change in a future release.

Note that the @code{mpf} functions are @emph{not} intended as a smooth
extension to IEEE P754 arithmetic.  In particular results obtained on one
computer often differ from the results on a computer with a different word
size.

@menu
* Initializing Floats::         
* Assigning Floats::            
* 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, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Float initialization functions
@cindex Initialization 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

@deftypefun {unsigned long int} mpf_get_default_prec (void)
Return the default default precision actually used.
@end deftypefun

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

@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 {unsigned long int} mpf_get_prec (mpf_t @var{op})
Return the current precision of @var{op}, in bits.
@end deftypefun

@deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  The
value in @var{rop} will be truncated to the new precision.

This function requires a call to @code{realloc}, and so should not be used in
a tight loop.
@end deftypefun

@deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
without changing the memory allocated.

@var{prec} must be no more than the allocated precision for @var{rop}, that
being the precision when @var{rop} was initialized, or in the most recent
@code{mpf_set_prec}.

The value in @var{rop} is unchanged, and in particular if it had a higher
precision than @var{prec} it will retain that higher precision.  New values
written to @var{rop} will use the new @var{prec}.

Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another
@code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original
allocated precision.  Failing to do so will have unpredictable results.

@code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the
original allocated precision.  After @code{mpf_set_prec_raw} it reflects the
@var{prec} value set.

@code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at
different precisions during a calculation, perhaps to gradually increase
precision in an iteration, or just to use various different precisions for
different purposes during a calculation.
@end deftypefun


@need 2000
@node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Assignment Functions
@cindex Float assignment functions
@cindex 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 decimal point expected is taken from
the current locale, on systems providing @code{localeconv}.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
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 is not
really true; white-space is ignored in the beginning of the string and within
the mantissa, but not in other places, such as after a minus sign or in the
exponent.  We are considering changing the definition of this function, making
it fail when there is any white-space in the input, since that makes a lot of
sense.  Please tell us your opinion about this change.  Do you really want it
to accept @nicode{"3 14"} as meaning 314 as it does now?]

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

@deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
Swap @var{rop1} and @var{rop2} efficiently.  Both the values and the
precisions of the two variables are swapped.
@end deftypefun


@node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Combined Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Float init and assign functions

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

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

@deftypefun double mpf_get_d_2exp (signed long int @var{exp}, mpf_t @var{op})
Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
raised to @var{exp}}, with @ma{0.5@le{}@GMPabs{@var{d}}<1}, is a good
approximation to @var{op}.  This is similar to the standard C function
@code{frexp}.
@end deftypefun

@deftypefun long mpf_get_si (mpf_t @var{op})
@deftypefunx {unsigned long} mpf_get_ui (mpf_t @var{op})
Convert @var{op} to a @code{long} or @code{unsigned long}, truncating any
fraction part.  If @var{op} is too big for the return type, the result is
undefined.

See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p}
(@pxref{Miscellaneous Float Functions}).
@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}.  @var{base} can be
2 to 36.  Up to @var{n_digits} digits will be generated.  Trailing zeros are
not returned.  No more digits than can be accurately represented by @var{op}
are ever generated.  If @var{n_digits} is 0 then that accurate maximum number
of digits are generated.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of
@ma{@var{n\_digits} + 2} bytes, that being enough for the mantissa, a possible
minus sign, and a null-terminator.  When @var{n_digits} is 0 to get all
significant digits, an application won't be able to know the space required,
and @var{str} should be @code{NULL} in that case.

The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit.  The applicable exponent is written through
the @var{expptr} pointer.  For example, the number 3.1416 would be returned as
string @nicode{"31416"} and exponent 1.

When @var{op} is zero, an empty string is produced and the exponent returned
is 0.

A pointer to the result string is returned, being either the allocated block
or the given @var{str}.
@end deftypefun


@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})
Set @var{rop} to @ma{@var{op1} + @var{op2}}.
@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})
Set @var{rop} to @ma{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

Division is undefined if the divisor is zero, and passing a zero divisor to the
divide functions will make these functions intentionally divide by zero.  This
lets the user handle arithmetic exceptions in these functions in the same
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})
@cindex Division functions
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})
@cindex Root extraction functions
Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
@end deftypefun

@deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Exponentiation functions
@cindex Powering functions
Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}.
@end deftypefun

@deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})
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})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

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

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

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

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

Caution: Currently only whole limbs are compared, and only in an exact
fashion.  In the future values like 1000 and 0111 may be considered the same
to 3 bits (on the basis that their difference is that small).
@end deftypefun

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

@deftypefn Macro int mpf_sgn (mpf_t @var{op})
@cindex Sign tests
@cindex Float sign tests
Return @ma{+1} if @ma{@var{op} > 0}, 0 if @ma{@var{op} = 0}, and @ma{-1} if
@ma{@var{op} < 0}.

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

When using any of these functions, it is a good idea to include @file{stdio.h}
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})
Print @var{op} to @var{stream}, as a string of digits.  Return the number of
bytes written, or if an error occurred, return 0.

The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
which may vary from 2 to 36.  An exponent then printed, separated by an
@samp{e}, or if @var{base} is greater than 10 then by an @samp{@@}.  The
exponent is always in decimal.  The decimal point follows the current locale,
on systems providing @code{localeconv}.

Up to @var{n_digits} will be printed from the mantissa, except that no more
digits than are accurately representable by @var{op} will be printed.
@var{n_digits} can be 0 to select that accurate maximum.
@end deftypefun

@deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string in base @var{base} from @var{stream}, and put the read float in
@var{rop}.  The string is of the form @samp{M@@N} or, if the base is 10 or
less, alternatively @samp{MeN}.  @samp{M} is the mantissa and @samp{N} is the
exponent.  The mantissa is always in the specified base.  The exponent is
either in the specified base or, if @var{base} is negative, in decimal.  The
decimal point expected is taken from the current locale, on systems providing
@code{localeconv}.

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

@deftypefun void mpf_ceil (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_floor (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_trunc (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to @var{op} rounded to an integer.  @code{mpf_ceil} rounds to the
next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc}
to the integer towards zero.
@end deftypefun

@deftypefun int mpf_integer_p (mpf_t @var{op})
Return non-zero if @var{op} is an integer.
@end deftypefun

@deftypefun int mpf_fits_ulong_p (mpf_t @var{op})
@deftypefunx int mpf_fits_slong_p (mpf_t @var{op})
@deftypefunx int mpf_fits_uint_p (mpf_t @var{op})
@deftypefunx int mpf_fits_sint_p (mpf_t @var{op})
@deftypefunx int mpf_fits_ushort_p (mpf_t @var{op})
@deftypefunx int mpf_fits_sshort_p (mpf_t @var{op})
Return non-zero if @var{op} would fit in the respective C data type, when
truncated to an integer.
@end deftypefun

@deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{nbits})
Generate a uniformly distributed random float in @var{rop}, such that @ma{0
@le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

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


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

This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.

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.  Unless otherwise stated, in-place
operations are allowed where source and destination are the same, but not where
they only partly overlap.

The @code{mpn} functions are the base for the implementation of the
@code{mpz_}, @code{mpf_}, and @code{mpq_} functions.

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

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

@noindent
In the notation used here, a source operand is identified by the pointer to
the least significant limb, and the limb count in braces.  For example,
@{@var{s1p}, @var{s1n}@}.

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

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

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

@deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return carry,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun

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

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

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

@deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return borrow,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to
@var{s2n}.
@end deftypefun

@deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
2*@var{n}-limb result to @var{rp}.

The destination has to have space for 2*@var{n} limbs, even if the product's
most significant limb is zero.
@end deftypefun

@deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
significant limbs of the product to @var{rp}.  Return the most significant
limb of the product.  @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
allowed to overlap provided @ma{@var{rp} @le{} @var{s1p}}.

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written in assembly
for most CPUs.

Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
@end deftypefun

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

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written in assembly
for most CPUs.
@end deftypefun

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

This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.  It is written
in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
result to @var{rp}.  Return the most significant limb of the result.

The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
result might be one limb smaller.

This function requires that @var{s1n} is greater than or equal to
@var{s2n}.  The destination must be distinct from both input operands.
@end deftypefun

@deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn})
Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
@var{dn}@}.  The quotient is rounded towards 0.

No overlap is permitted between arguments.  @var{nn} must be greater than or
equal to @var{dn}.  The most significant limb of @var{dp} must be non-zero.
The @var{qxn} operand must be zero.
@comment FIXME: Relax overlap requirements!
@end deftypefun

@deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
[This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
performance.]

Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
quotient at @var{r1p}, with the exception of the most significant limb, which
is returned.  The remainder replaces the dividend at @var{rs2p}; it will be
@var{s3n} limbs long (i.e., as many limbs as the divisor).

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

It is required that @var{rs2n} is greater than or equal to @var{s3n}.  It is
required that the most significant bit of the divisor is set.

If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}.  Aside
from that special case, no overlap between arguments is permitted.

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

The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
limbs large.
@end deftypefun

@deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb})
@deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}})
Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
@var{r1p}.  Return the remainder.

The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in
addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
@var{qxn}@}.  Either or both @var{s2n} and @var{qxn} can be zero.  For most
usages, @var{qxn} will be zero.

@code{mpn_divmod_1} exists for upward source compatibility and is simply a
macro calling @code{mpn_divrem_1} with a @var{qxn} of 0.

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

@deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
[This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
performance.]
@end deftypefun

@deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
@deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry})
Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
the result to @{@var{rp}, @var{n}@}.  If 3 divides exactly, the return value is
zero and the result is the quotient.  If not, the return value is non-zero and
the result won't be anything useful.

@code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
return value from a previous call, so a large calculation can be done piece by
piece from low to high.  @code{mpn_divexact_by3} is simply a macro calling
@code{mpn_divexact_by3c} with a 0 carry parameter.

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

The source @ma{a}, result @ma{q}, size @ma{n}, initial carry @ma{i}, and
return value @ma{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
@m{b=2\GMPraise{@code{mp\_bits\_per\_limb}}, b=2^mp_bits_per_limb}.  The
return @ma{c} is always 0, 1 or 2, and the initial carry @ma{i} must also be
0, 1 or 2 (these are both borrows really).  When @ma{c=0} clearly
@ma{q=(a-i)/3}.  When @m{c \neq 0, c!=0}, the remainder @ma{(a-i) @bmod{} 3}
is given by @ma{3-c}, because @ma{b @equiv{} 1 @bmod{} 3} (when
@code{mp_bits_per_limb} is even, which is always so currently).
@end deftypefn

@deftypefun mp_limb_t mpn_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder.
@var{s1n} can be zero.
@end deftypefun

@deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}, unsigned long int @var{d})
This function puts the low
@ma{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of @var{q} =
@{@var{s1p}, @var{s1n}@}/@{@var{s2p}, @var{s2n}@} mod @m{2^d,2^@var{d}} at
@var{rp}, and returns the high @var{d} mod @code{mp_bits_per_limb} bits of
@var{q}.

@{@var{s1p}, @var{s1n}@} - @var{q} * @{@var{s2p}, @var{s2n}@} mod @m{2
\GMPraise{@var{s1n}*@code{mp\_bits\_per\_limb}},
2^(@var{s1n}*@nicode{mp\_bits\_per\_limb})} is placed at @var{s1p}.  Since the
low @ma{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of this
difference are zero, it is possible to overwrite the low limbs at @var{s1p}
with this difference, provided @ma{@var{rp} @le{} @var{s1p}}.

This function requires that @ma{@var{s1n} * @nicode{mp\_bits\_per\_limb}
@ge{} @var{D}}, and that @{@var{s2p}, @var{s2n}@} 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{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the left are returned in the
least significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@ma{@var{rp} @ge{} @var{sp}}.

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

@deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the right are returned in the
most significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@ma{@var{rp} @le{} @var{sp}}.

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

@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
positive value if @ma{@var{s1} > @var{s2}}, 0 if they are equal, or a negative
value if @ma{@var{s1} < @var{s2}}.
@end deftypefun

@deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{s1p},
@var{s1n}@} and @{@var{s2p}, @var{s2n}@}.  The result can be up to @var{s2n}
limbs, the return value is the actual number produced.  Both source operands
are destroyed.

@{@var{s1p}, @var{s1n}@} must have at least as many bits as @{@var{s2p},
@var{s2n}@}.  @{@var{s2p}, @var{s2n}@} must be odd.  Both operands must have
non-zero most significant limbs.
@end deftypefun

@deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
Return the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
@var{s2limb}.  Both operands must be non-zero.
@end deftypefun

@deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, mp_size_t *@var{r2n}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Calculate the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
@{@var{s2p}, @var{s2n}@}.  Store the gcd at @{@var{r1p}, @var{retval}@} and
the first cofactor at @{@var{r2p}, *@var{r2n}@}, with *@var{r2n} negative if
the cofactor is negative.  @var{r1p} and @var{r2p} should each have room for
@ma{@var{s1n}+1} limbs, but the return value and value stored through
@var{r2n} indicate the actual number produced.

@ma{@{@var{s1p}, @var{s1n}@} @ge{} @{@var{s2p}, @var{s2n}@}} is required, and
both must be non-zero.  The regions @{@var{s1p}, @ma{@var{s1n}+1}@} and
@{@var{s2p}, @ma{@var{s2n}+1}@} are destroyed (i.e. the operands plus an extra
limb past the end of each).

The cofactor @var{r1} will satisfy @m{r_2 s_1 + k s_2 = r_1, @var{r2}*@var{s1}
+ @var{k}*@var{s2} = @var{r1}}.  The second cofactor @var{k} is not calculated
but can easily be obtained from @m{(r_1 - r_2 s_1) / s_2, (@var{r1} -
@var{r2}*@var{s1}) / @var{s2}}.
@end deftypefun

@deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Compute the square root of @{@var{sp}, @var{n}@} and put the result at
@{@var{r1p}, @ma{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p},
@var{retval}@}.  @var{r2p} needs space for @var{n} limbs, but the return value
indicates how many are produced.

The most significant limb of @{@var{sp}, @var{n}@} must be non-zero.  The
areas @{@var{r1p}, @ma{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must
be completely separate.  The areas @{@var{r2p}, @var{n}@} and @{@var{sp},
@var{n}@} must be either identical or completely separate.

If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
case the return value is zero or non-zero according to whether the remainder
would have been zero or non-zero.

A return value of zero indicates a perfect square.  See also
@code{mpz_perfect_square_p}.
@end deftypefun

@deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n})
Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in
base @var{base}, and return the number of characters produced.  There may be
leading zeros in the string.  The string is not in ASCII; to convert it to
printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on
the base and range.

The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be
non-zero.  The area @{@var{s1p}, @var{s1n}+1@} is clobbered.

The area at @var{str} has to have space for the largest possible number
represented by a @var{s1n} long limb array, plus one extra character.
@end deftypefun

@deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{r1p}, const char *@var{str}, size_t @var{strsize}, int @var{base})
Convert the raw unsigned char array at @var{str} of length @var{strsize} to a
limb array.  The base of @var{str} is @var{base}.  @var{strsize} must be at
least 1.

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.
@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.
@end deftypefun

@deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
@deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
Generate a random number of length @var{r1n} and store it at @var{r1p}.  The
most significant limb is always non-zero.  @code{mpn_random} generates
uniformly distributed limb data, @code{mpn_random2} generates long strings of
zeros and ones in the binary representation.

@code{mpn_random2} is intended for testing the correctness of the @code{mpn}
routines.
@end deftypefun

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

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

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


@node Random Number Functions, Formatted Output, Low-level Functions, Top
@chapter Random Number Functions
@cindex Random number functions

Sequences of pseudo-random numbers in GMP are generated using a variable of
type @code{gmp_randstate_t}, which holds an algorithm selection and a current
state.  Such a variable must be initialized by a call to one of the
@code{gmp_randinit} functions, and can be seeded with one of the
@code{gmp_randseed} functions.

The functions actually generating random numbers are described in @ref{Integer
Random Numbers}, and @ref{Miscellaneous Float Functions}.

The older style random number functions don't accept a @code{gmp_randstate_t}
parameter but instead share a global variable of that type.  They use a
default algorithm and are currently not seeded (though perhaps that will
change in the future).  The new functions accepting a @code{gmp_randstate_t}
are recommended for applications that care about randomness.

@menu
* Random State Initialization::
* Random State Seeding::
@end menu

@node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
@section Random State Initialization
@cindex Random number state

@deftypefun void gmp_randinit_default (gmp_randstate_t @var{state})
Initialize @var{state} with a default algorithm.  This will be a compromise
between speed and randomness, and is recommended for applications with no
special requirements.
@end deftypefun

@deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a}, @w{unsigned long @var{c}}, @w{unsigned long @var{m2exp}})
Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X +
@var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}.

The low bits of @ma{X} in this algorithm are not very random.  The least
significant bit will have a period no more than 2, and the second bit no more
than 4, etc.  For this reason only the high half of each @ma{X} is actually
used.

When a random number of more than @ma{@var{m2exp}/2} bits is to be generated,
multiple iterations of the recurrence are used and the results concatenated.
@end deftypefun

@deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, unsigned long @var{size})
Initialize @var{state} for a linear congruential algorithm as per
@code{gmp_randinit_lc_2exp}.  @var{a}, @var{c} and @var{m2exp} are selected
from a table, chosen so that @var{size} bits (or more) of each @ma{X} will be
used, ie. @ma{@var{m2exp} @ge{} @var{size}/2}.

If successful the return value is non-zero.  If @var{size} is bigger than the
table data provides then the return value is zero.  The maximum @var{size}
currently supported is 128.
@end deftypefun

@deftypefun void gmp_randinit (gmp_randstate_t @var{state}, @w{gmp_randalg_t @var{alg}}, ...)
@strong{This function is obsolete.}

Initialize @var{state} with an algorithm selected by @var{alg}.  The only
choice is @code{GMP_RAND_ALG_LC}, which is @code{gmp_randinit_lc_2exp_size}.
A third parameter of type @code{unsigned long} is required, this is the
@var{size} for that function.  @code{GMP_RAND_ALG_DEFAULT} or 0 are the same
as @code{GMP_RAND_ALG_LC}.

@code{gmp_randinit} sets bits in @code{gmp_errno} to indicate an error.
@code{GMP_ERROR_UNSUPPORTED_ARGUMENT} if @var{alg} is unsupported, or
@code{GMP_ERROR_INVALID_ARGUMENT} if the @var{size} parameter is too big.
@end deftypefun

@c  Not yet in the library.
@ignore
@deftypefun void gmp_randinit_lc (gmp_randstate_t @var{state}, mpz_t @var{a}, unsigned long int @var{c}, mpz_t @var{m})
Initialize @var{state} for a linear congruential scheme @m{X = (@var{a}X +
@var{c}) @bmod @var{m}, X = (@var{a}*X + @var{c}) mod 2^@var{m}}.
@end deftypefun
@end ignore

@deftypefun void gmp_randclear (gmp_randstate_t @var{state})
Free all memory occupied by @var{state}.
@end deftypefun


@node Random State Seeding,  , Random State Initialization, Random Number Functions
@section Random State Seeding
@cindex Random number seeding

@deftypefun void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed})
@deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, @w{unsigned long int @var{seed}})
Set an initial seed value into @var{state}.

The size of a seed determines how many different sequences of random numbers
that it's possible to generate.  The ``quality'' of the seed is the randomness
of a given seed compared to the previous seed used, and this affects the
randomness of separate number sequences.  The method for choosing a seed is
critical if the generated numbers are to be used for important applications,
such as generating cryptographic keys.

Traditionally the system time has been used to seed, but care needs to be
taken with this.  If an application seeds often and the resolution of the
system clock is low, then the same sequence of numbers might be repeated.
Also, the system time is quite easy to guess, so if unpredictability is
required then it should definitely not be the only source for the seed value.
On some systems there's a special device @file{/dev/random} which provides
random data better suited for use as a seed.
@end deftypefun


@node Formatted Output, Formatted Input, Random Number Functions, Top
@chapter Formatted Output
@cindex Formatted output
@cindex @code{printf} formatted output

@menu
* Formatted Output Strings::    
* Formatted Output Functions::  
* C++ Formatted Output::        
@end menu

@node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
@section Format Strings

@code{gmp_printf} and friends accept format strings similar to the standard C
@code{printf} (@pxref{Formatted Output,,,libc,The GNU C Library Reference
Manual}).  A format specification is of the form

@example
% [flags] [width] [.[precision]] [type] conv
@end example

GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
and @code{mpf_t} respectively.  @samp{Z} and @samp{Q} behave like integers.
@samp{Q} will print a @samp{/} and a denominator, if needed.  @samp{F} behaves
like a float.  For example,

@example
mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);

mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);

mpf_t f;
int   n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
@end example

All the standard C @code{printf} types behave the same as the C library
@code{printf}, and can be freely intermixed with the GMP extensions.  In the
current implementation the standard parts of the format string are simply
handed to @code{printf} and only the GMP extensions handled directly.

The flags accepted are as follows.  GLIBC style @nisamp{'}
(@pxref{Locales,,Locales and Internationalization,libc,The GNU C Library
Reference Manual}) is only for the standard C types (not the GMP types), and
only if the C library supports it.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{0} @tab pad with zeros (rather than spaces)
@item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0}
@item @nicode{+} @tab always show a sign
@item (space)    @tab show a space or a @samp{-} sign
@item @nicode{'} @tab group digits, GLIBC style (not GMP types)
@end multitable
@end quotation

The standard types accepted are as follows.  @samp{h} and @samp{l} are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the output.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{h}  @tab @nicode{short}
@item @nicode{hh} @tab @nicode{char}
@item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
@item @nicode{l}  @tab @nicode{long} or @nicode{wchar_t}
@item @nicode{ll} @tab same as @nicode{L}
@item @nicode{L}  @tab @nicode{long long} or @nicode{long double}
@item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
@item @nicode{t}  @tab @nicode{ptrdiff_t}
@item @nicode{z}  @tab @nicode{size_t}
@end multitable
@end quotation

@noindent
The GMP types are

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{F}  @tab @nicode{mpf_t}, float conversions
@item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
@item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
@end multitable
@end quotation

The conversions accepted are as follows.  @samp{a} and @samp{A} are always
supported for @code{mpf_t} but depend on the C library for standard C float
types.  @samp{m} and @samp{p} depend on the C library.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{a} @nicode{A} @tab hex floats, GLIBC style
@item @nicode{c}            @tab character
@item @nicode{d}            @tab decimal integer
@item @nicode{e} @nicode{E} @tab scientific format float
@item @nicode{f}            @tab fixed point float
@item @nicode{i}            @tab same as @nicode{d}
@item @nicode{g} @nicode{G} @tab fixed or scientific float
@item @nicode{m}            @tab @code{strerror} string, GLIBC style
@item @nicode{n}            @tab characters written so far
@item @nicode{o}            @tab octal integer
@item @nicode{p}            @tab pointer
@item @nicode{s}            @tab string
@item @nicode{u}            @tab unsigned integer
@item @nicode{x} @nicode{X} @tab hex integer
@end multitable
@end quotation

@samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for
@samp{Z} and @samp{Q} a sign is included.  @samp{u} is not meaningful for
@code{Z} and @code{Q}.

@samp{n} can be used with any of the types, even the GMP types.

Other types or conversions that might be accepted by the C library
@code{printf} cannot be used through @code{gmp_printf}, this includes for
instance extensions registered with GLIBC @code{register_printf_function}.
Also currently there's no support for POSIX @samp{$} style numbered arguments
(perhaps this will be added in the future).

The precision field has it's usual meaning for integer @samp{Z} and float
@samp{F} types, but is currently undefined for @samp{Q} and should not be used
with that.

@code{mpf_t} conversions only ever generate as many digits as can be
accurately represented by the operand, the same as @code{mpf_get_str} does.
Zeros will be used if necessary to pad to the requested precision.  This
happens even for an @samp{f} conversion of an @code{mpf_t} which is an
integer, for instance @ma{2^@W{1024}} in an @code{mpf_t} of 128 bits precision
will only produce about 20 digits, then pad with zeros to the decimal point.
An empty precision field like @samp{%.Fe} or @samp{%.Ff} can be used to
specifically request all significant digits.

The decimal point character (or string) is taken from the current locale
settings on systems which provide @code{localeconv} (@pxref{Locales,,Locales
and Internationalization,libc,The GNU C Library Reference Manual}).  The C
library will normally do the same for standard float output.


@node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output
@section Functions

Each of the following functions is similar to the corresponding C library
function.  The basic @code{printf} forms take a variable argument list.  The
@code{vprintf} forms take an argument pointer, see @ref{Variadic
Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3
va_start}.

It should be emphasised that if a format string is invalid, or the arguments
don't match what the format specifies, then the behaviour of any of these
functions will be unpredictable.  GCC format string checking is not available,
since it doesn't recognise the GMP extensions.

The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return
@ma{-1} to indicate a write error.  All the functions can return @ma{-1} if
the C library @code{printf} variant in use returns @ma{-1}, but this shouldn't
normally occur.

@deftypefun int gmp_printf (const char *@var{fmt}, ...)
@deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap})
Print to the standard output @code{stdout}.  Return the number of characters
written, or @ma{-1} if an error occurred.
@end deftypefun

@deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
Print to the stream @var{fp}.  Return the number of characters written, or
@ma{-1} if an error occurred.
@end deftypefun

@deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in @var{buf}.  Return the number of characters
written, excluding the terminating null.

No overlap is permitted between the space at @var{buf} and the string
@var{fmt}.

These functions are not recommended, since there's no protection against
exceeding the space available at @var{buf}.
@end deftypefun

@deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in @var{buf}.  No more than @var{size} bytes
will be written.  To get the full output, @var{size} must be enough for the
string and null-terminator.

The return value is the total number of characters which ought to have been
produced, excluding the terminating null.  If @ma{@var{retval} >= @var{size}}
then the actual output has been truncated to the first @ma{@var{size}-1}
characters, and a null appended.

No overlap is permitted between the region @{@var{buf},@var{size}@} and the
@var{fmt} string.

Notice the return value is in ISO C99 @code{snprintf} style.  This is so even
if the C library @code{vsnprintf} is the older GLIBC 2.0.x style.
@end deftypefun

@deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vasprintf (char *@var{pp}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in a block of memory obtained from the current
memory allocation function (@pxref{Custom Allocation}).  The block will be the
size of the string and null-terminator.  Put the address of the block in
*@var{pp}.  Return the number of characters produced, excluding the
null-terminator.

Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return
@ma{-1} if there's no more memory available, it lets the current allocation
function handle that.
@end deftypefun

@deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, ...)
@deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap})
Append to the current obstack object, in the same style as
@code{obstack_printf}.  Return the number of characters written.  A
null-terminator is not written.

@var{fmt} cannot be within the current obstack object, since the object might
move as it grows.

These functions are available only when the C library provides the obstack
feature, which probably means only on GNU systems, see
@ref{Obstacks,,,libc,The GNU C Library Reference Manual}.
@end deftypefun


@node C++ Formatted Output,  , Formatted Output Functions, Formatted Output
@section C++ Formatted Output
@cindex C++ @code{ostream} output
@cindex @code{ostream} output

The following functions are provided in @file{libgmpxx}, which is built if C++
support is enabled (@pxref{Build Options}).  Prototypes are available from
@code{<gmp.h>}.

@deftypefun ostream& operator<< (ostream& @var{stream}, mpz_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.

In hex or octal, @var{op} is printed as a signed number, the same as for
decimal.  This is unlike the standard @code{operator<<} routines on @code{int}
etc, which instead give twos complement.
@end deftypefun

@deftypefun ostream& operator<< (ostream& @var{stream}, mpq_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.

Output will be a fraction like @samp{5/9}, or if the denominator is 1 then
just a plain integer like @samp{123}.

In hex or octal, @var{op} is printed as a signed value, the same as for
decimal.  If @code{ios::showbase} is set then a base indicator is shown on
both the numerator and denominator (if the denominator is required).
@end deftypefun

@deftypefun ostream& operator<< (ostream& @var{stream}, mpf_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.  The decimal point follows the current
locale, on systems providing @code{localeconv}.

Hex and octal are supported, unlike the standard @code{operator<<} routines on
@code{double} etc.  The mantissa will be in hex or octal, the exponent will be
in decimal.  For hex the exponent delimiter is an @samp{@@}.  This is as per
@code{mpf_out_str}.  @code{ios::showbase} is supported, and will put a base on
the mantissa.
@end deftypefun

These operators mean that GMP types can be printed in the usual C++ way, for
example,

@example
mpz_t  z;
int    n;
...
cout << "iteration " << n << " value " << z << "\n";
@end example

But note that @code{ostream} output (and @code{istream} input, @pxref{C++
Formatted Input}) is the only overloading available and using for instance
@code{+} with an @code{mpz_t} will have unpredictable results.


@node Formatted Input, C++ Class Interface, Formatted Output, Top
@chapter Formatted Input
@cindex Formatted input
@cindex @code{scanf} formatted input

@menu
* Formatted Input Strings::     
* Formatted Input Functions::   
* C++ Formatted Input::         
@end menu


@node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input
@section Formatted Input Strings

@code{gmp_scanf} and friends accept format strings similar to the standard C
@code{scanf} (@pxref{Formatted Input,,,libc,The GNU C Library Reference
Manual}).  A format specification is of the form

@example
% [flags] [width] [type] conv
@end example

GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
and @code{mpf_t} respectively.  @samp{Z} and @samp{Q} behave like integers.
@samp{Q} will read a @samp{/} and a denominator, if present.  @samp{F} behaves
like a float.

GMP variables don't require an @code{&} when passed to @code{gmp_scanf}, since
they're already ``call-by-reference''.  For example,

@example
/* to read say "a(5) = 1234" */
int   n;
mpz_t z;
gmp_scanf ("a(%d) = %Zd\n", &n, z);

mpq_t q1, q2;
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);

/* to read say "topleft (1.55,-2.66)" */
mpf_t x, y;
char  buf[32];
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
@end example

All the standard C @code{scanf} types behave the same as in the C library
@code{scanf}, and can be freely intermixed with the GMP extensions.  In the
current implementation the standard parts of the format string are simply
handed to @code{scanf} and only the GMP extensions handled directly.

The flags accepted are as follows.  @samp{a} and @samp{'} will depend on
support from the C library, and @samp{'} cannot be used with GMP types.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{*} @tab read but don't store
@item @nicode{a} @tab allocate a buffer (string conversions)
@item @nicode{'} @tab group digits, GLIBC style (not GMP types)
@end multitable
@end quotation

The standard types accepted are as follows.  @samp{h} and @samp{l} are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the input.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{h}  @tab @nicode{short}
@item @nicode{hh} @tab @nicode{char}
@item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
@item @nicode{l}  @tab @nicode{long} or @nicode{wchar_t}
@item @nicode{ll} @tab same as @nicode{L}
@item @nicode{L}  @tab @nicode{long long} or @nicode{long double}
@item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
@item @nicode{t}  @tab @nicode{ptrdiff_t}
@item @nicode{z}  @tab @nicode{size_t}
@end multitable
@end quotation

@noindent
The GMP types are

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{F}  @tab @nicode{mpf_t}, float conversions
@item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
@item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
@end multitable
@end quotation

The conversions accepted are as follows.  @samp{p} and @samp{[} will depend on
support from the C library, the rest are standard.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{c}            @tab character or characters
@item @nicode{d}            @tab decimal integer
@item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G}
                            @tab float
@item @nicode{i}            @tab integer with base indicator
@item @nicode{n}            @tab characters written so far
@item @nicode{o}            @tab octal integer
@item @nicode{p}            @tab pointer
@item @nicode{s}            @tab string of non-whitespace characters
@item @nicode{u}            @tab decimal integer
@item @nicode{x} @nicode{X} @tab hex integer
@item @nicode{[}            @tab string of characters in a set
@end multitable
@end quotation

@samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all
read either fixed point or scientific format, and either @samp{e} or @samp{E}
for the exponent in scientific format.

@samp{x} and @samp{X} are identical, both accept both upper and lower case
hexadecimal.

@samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative
values.  For the standard C types these are described as ``unsigned''
conversions, but that merely affects certain overflow handling, negatives are
still allowed (see @code{strtoul}, @ref{Parsing of Integers,,,libc,The GNU C
Library Reference Manual}).  For GMP types there are no overflows, and
@samp{d} and @samp{u} are identical.

@samp{Q} type reads the numerator and (optional) denominator as given.  If the
value might not be in canonical form then @code{mpq_canonicalize} must be
called before using it in any calculations (@pxref{Rational Number
Functions}).

@samp{Qi} will read a base specification separately for the numerator and
denominator.  For example @samp{0x10/11} would be 16/11, whereas
@samp{0x10/0x11} would be 16/17.

@samp{n} can be used with any of the types above, even the GMP types.
@samp{*} to suppress assignment is allowed, though the field would then do
nothing at all.

Other conversions or types that might be accepted by the C library
@code{scanf} cannot be used through @code{gmp_scanf}.

Whitespace is read and discarded before a field, except for @samp{c} and
@samp{[} conversions.

For float conversions, the decimal point character (or string) expected is
taken from the current locale settings on systems which provide
@code{localeconv} (@pxref{Locales,,Locales and Internationalization,libc,The
GNU C Library Reference Manual}).  The C library will normally do the same for
standard float input.


@node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input
@section Formatted Input Functions

Each of the following functions is similar to the corresponding C library
function.  The plain @code{scanf} forms take a variable argument list.  The
@code{vscanf} forms take an argument pointer, see @ref{Variadic
Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3
va_start}.

It should be emphasised that if a format string is invalid, or the arguments
don't match what the format specifies, then the behaviour of any of these
functions will be unpredictable.  GCC format string checking is not available,
since it doesn't recognise the GMP extensions.

No overlap is permitted between the @var{fmt} string and any of the results
produced.

@deftypefun int gmp_scanf (const char *@var{fmt}, ...)
@deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap})
Read from the standard input @code{stdin}.
@end deftypefun

@deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
Read from the stream @var{fp}.
@end deftypefun

@deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, ...)
@deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap})
Read from a null-terminated string @var{s}.
@end deftypefun

The return value from each of these functions is the same as the standard C99
@code{scanf}, namely the number of fields successfully parsed and stored.
@samp{%n} fields and fields read but suppressed by @samp{*} don't count
towards the return value.

If end of file or file error, or end of string, is reached when a match is
required, and when no previous non-suppressed fields have matched, then the
return value is EOF instead of 0.  A match is required for a literal character
in the format string or a field other than @samp{%n}.  Whitespace in the
format string is only an optional match and won't induce an EOF in this
fashion.  Leading whitespace read and discarded for a field doesn't count as a
match.


@node C++ Formatted Input,  , Formatted Input Functions, Formatted Input
@section C++ Formatted Input
@cindex C++ @code{istream} input
@cindex @code{istream} input

The following functions are provided in @file{libgmpxx}, which is built only
if C++ support is enabled (@pxref{Build Options}).  Prototypes are available
from @code{<gmp.h>}.

@deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop})
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop})
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.

An integer like @samp{123} will be read, or a fraction like @samp{5/9}.  If
the fraction is not in canonical form then @code{mpq_canonicalize} must be
called (@pxref{Rational Number Functions}).
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop})
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.

Hex or octal floats are not supported, but might be in the future.
@end deftypefun

These operators mean that GMP types can be read in the usual C++ way, for
example,

@example
mpz_t  z;
...
cin >> z;
@end example

But note that @code{istream} input (and @code{ostream} output, @pxref{C++
Formatted Output}) is the only overloading available and using for instance
@code{+} with an @code{mpz_t} will have unpredictable results.


@node C++ Class Interface, BSD Compatible Functions, Formatted Input, Top
@chapter C++ Class Interface
@cindex C++ Interface

This chapter describes the C++ class based interface to GMP.

All GMP C language types and functions can be used in C++ programs, since
@file{gmp.h} has @code{extern "C"} qualifiers, but the class interface offers
overloaded functions and operators which may be more convenient.

Due to the implementation of this interface, a reasonably recent C++ compiler
is required, one supporting namespaces, partial specialization of templates
and member templates.  For GCC this means version 2.91 or later.

@strong{Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen difficulty
reveals itself.}

@menu
* C++ Interface General::       
* C++ Interface Integers::      
* C++ Interface Rationals::     
* C++ Interface Floats::        
* C++ Interface MPFR::          
* C++ Interface Random Numbers::  
* C++ Interface Limitations::   
@end menu


@node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface
@section C++ Interface General

@noindent
All the C++ classes and functions are available with

@example
#include <gmpxx.h>
@end example

@noindent
The classes defined are

@deftp Class mpz_class
@deftpx Class mpq_class
@deftpx Class mpf_class
@end deftp

The standard operators and various standard functions are overloaded to allow
arithmetic with these classes.  For example,

@example
int
main (void)
@{
  mpz_class a, b, c;

  a = 1234;
  b = "-5678";
  c = a+b;
  cout << "sum is " << c << "\n";
  cout << "absolute value is " << abs(c) << "\n";

  return 0;
@}
@end example

An important feature of the implementation is that an expression like
@code{a=b+c} results in a single call to the corresponding @code{mpz_add},
without using a temporary for the @code{b+c} part.  Expressions which by their
nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries
though.

The classes can be freely intermixed in expressions, as can the classes and
the standard C++ types.

Conversions back from the classes to standard C++ types aren't done
automatically, instead member functions like @code{get_si} are provided (see
the following sections for details).

Also there are no automatic conversions from the classes to the corresponding
GMP C types, instead a reference to the underlying C object can be obtained
with the following functions,

@deftypefun mpz_t mpz_class::get_mpz_t ()
@deftypefunx mpq_t mpq_class::get_mpq_t ()
@deftypefunx mpf_t mpf_class::get_mpf_t ()
@end deftypefun

These can be used to call a C function which doesn't have a C++ class
interface.  For example to set @code{a} to the GCD of @code{b} and @code{c},

@example
mpz_class a, b, c;
...
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
@end example

In the other direction, a class can be initialized from the corresponding GMP
C type, or assigned to if an explicit constructor is used.  In both cases this
makes a copy of the value, it doesn't create any sort of association.  For
example,

@example
mpz_t z;
// ... init and calculate z ...
mpz_class x(z);
mpz_class y;
y = mpz_class (z);
@end example

There are no namespace setups in @file{gmpxx.h}, all types and functions are
simply put into the global namespace.  This is what @file{gmp.h} has done in
the past, and continues to do for compatibility.  The extras provided by
@file{gmpxx.h} follow GMP naming conventions and are unlikely to clash with
anything.


@node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface
@section C++ Interface Integers

@deftypefun void mpz_class::mpz_class (type @var{n})
Construct an @code{mpz_class}.  All the standard C++ types may be used, except
@code{long long} and @code{long double}, and all the GMP C++ classes can be
used.  Any necessary conversion follows the corresponding C function, for
example @code{double} follows @code{mpz_set_d} (@pxref{Assigning Integers}).
@end deftypefun

@deftypefun void mpz_class::mpz_class (mpz_t @var{z})
Construct an @code{mpz_class} from an @code{mpz_t}.  The value in @var{z} is
copied into the new @code{mpz_class}, there won't be any permanent association
between it and @var{z}.
@end deftypefun

@deftypefun void mpz_class::mpz_class (const char *@var{s})
@deftypefunx void mpz_class::mpz_class (const char *@var{s}, int base)
@deftypefunx void mpz_class::mpz_class (const string& @var{s})
@deftypefunx void mpz_class::mpz_class (const string& @var{s}, int base)
Construct an @code{mpz_class} converted from a string using
@code{mpz_set_str}, (@pxref{Assigning Integers}).  If the @var{base} is not
given then 0 is used.
@end deftypefun

@deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d})
@deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d})
Divisions involving @code{mpz_class} round towards zero, as per the
@code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}).
This corresponds to the rounding used for plain @code{int} calculations on
most machines.

The @code{mpz_fdiv...} or @code{mpz_cdiv...} functions can always be called
directly if desired.  For example,

@example
mpz_class q, a, d;
...
mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
@end example
@end deftypefun

@deftypefun mpz_class abs (mpz_class @var{op1})
@deftypefunx int cmp (mpz_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpz_class @var{op2})
@deftypefunx double mpz_class::get_d (void)
@deftypefunx long mpz_class::get_si (void)
@deftypefunx {unsigned long} mpz_class::get_ui (void)
@maybepagebreak
@deftypefunx bool mpz_class::fits_sint_p (void)
@deftypefunx bool mpz_class::fits_slong_p (void)
@deftypefunx bool mpz_class::fits_sshort_p (void)
@maybepagebreak
@deftypefunx bool mpz_class::fits_uint_p (void)
@deftypefunx bool mpz_class::fits_ulong_p (void)
@deftypefunx bool mpz_class::fits_ushort_p (void)
@maybepagebreak
@deftypefunx int sgn (mpz_class @var{op})
@deftypefunx mpz_class sqrt (mpz_class @var{op})
These functions provide a C++ class interface to the corresponding GMP C
routines.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.
@end deftypefun

@sp 1
Overloaded operators for combinations of @code{mpz_class} and @code{double}
are provided for completeness, but it should be noted that if the given
@code{double} is not an integer then the way any rounding is done is currently
unspecified.  The rounding might take place at the start, in the middle, or at
the end of the operation, and it might change in the future.

Conversions between @code{mpz_class} and @code{double}, however, are defined
to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}.
And comparisons are always made exactly, as per @code{mpz_cmp_d}.


@node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface
@section C++ Interface Rationals

In all the following constructors, if a fraction is given then it should be in
canonical form, or if not then @code{mpq_class::canonicalize} called.

@deftypefun void mpq_class::mpq_class (type @var{op})
@deftypefunx void mpq_class::mpq_class (integer @var{num}, integer @var{den})
Construct an @code{mpq_class}.  The initial value can be a single value of any
type, or a pair of integers (@code{mpz_class} or standard C++ integer types)
representing a fraction, except that @code{long long} and @code{long double}
are not supported.  For example,

@example
mpq_class q (99);
mpq_class q (1.75);
mpq_class q (1, 3);
@end example
@end deftypefun

@deftypefun void mpq_class::mpq_class (mpq_t @var{q})
Construct an @code{mpq_class} from an @code{mpq_t}.  The value in @var{q} is
copied into the new @code{mpq_class}, there won't be any permanent association
between it and @var{q}.
@end deftypefun

@deftypefun void mpq_class::mpq_class (const char *@var{s})
@deftypefunx void mpq_class::mpq_class (const char *@var{s}, int base)
@deftypefunx void mpq_class::mpq_class (const string& @var{s})
@deftypefunx void mpq_class::mpq_class (const string& @var{s}, int base)
Construct an @code{mpq_class} converted from a string using
@code{mpq_set_str}, (@pxref{Initializing Rationals}).  If the @var{base} is
not given then 0 is used.
@end deftypefun

@deftypefun void mpq_class::canonicalize ()
Put an @code{mpq_class} into canonical form, as per @ref{Rational Number
Functions}.  All arithmetic operators require their operands in canonical
form, and will return results in canonical form.
@end deftypefun

@deftypefun mpq_class abs (mpq_class @var{op})
@deftypefunx int cmp (mpq_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpq_class @var{op2})
@maybepagebreak
@deftypefunx double mpq_class::get_d (void)
@deftypefunx int sgn (mpq_class @var{op})
These functions provide a C++ class interface to the corresponding GMP C
routines.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.
@end deftypefun

@deftypefun {mpz_class&} mpq_class::get_num ()
@deftypefunx {mpz_class&} mpq_class::get_den ()
Get a reference to an @code{mpz_class} which is the numerator or denominator
of an @code{mpq_class}.  This can be used both for read and write access.  If
the object returned is modified, it modifies the original @code{mpq_class}.

If direct manipulation might produce a non-canonical value, then
@code{mpq_class::canonicalize} must be called before further operations.
@end deftypefun

@deftypefun mpz_t mpq_class::get_num_mpz_t ()
@deftypefunx mpz_t mpq_class::get_den_mpz_t ()
Get a reference to the underlying @code{mpz_t} numerator or denominator of an
@code{mpq_class}.  This can be passed to C functions expecting an
@code{mpz_t}.  Any modifications made to the @code{mpz_t} will modify the
original @code{mpq_class}.

If direct manipulation might produce a non-canonical value, then
@code{mpq_class::canonicalize} must be called before further operations.
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop});
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings,
the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}).

If the @var{rop} read might not be in canonical form then
@code{mpq_class::canonicalize} must be called.
@end deftypefun


@node C++ Interface Floats, C++ Interface MPFR, C++ Interface Rationals, C++ Class Interface
@section C++ Interface Floats

When an expression requires the use of temporary intermediate @code{mpf_class}
values, like @code{f=g*h+x*y}, those temporaries will have the same precision
as the destination @code{f}.  Explicit constructors can be used if this
doesn't suit.

@deftypefun {} mpf_class::mpf_class (type @var{op})
@deftypefunx {} mpf_class::mpf_class (type @var{op}, unsigned long @var{prec})
Construct an @code{mpf_class}.  Any standard C++ type can be used, except
@code{long long} and @code{long double}, and any of the GMP C++ classes can be
used.

If @var{prec} is given, the initial precision is that value, in bits.  If
@var{prec} is not given, then the initial precision is determined by the type
of @var{op} given.  An @code{mpz_class}, @code{mpq_class}, string, or C++
builtin type will give the default @code{mpf} precision (@pxref{Initializing
Floats}).  An @code{mpf_class} or expression will give the precision of that
value.  The precision of a binary expression is the higher of the two
operands.

@example
mpf_class f(1.5);        // default precision
mpf_class f(1.5, 500);   // 500 bits (at least)
mpf_class f(x);          // precision of x
mpf_class f(abs(x));     // precision of x
mpf_class f(-g, 1000);   // 1000 bits (at least)
mpf_class f(x+y);        // greater of precisions of x and y
@end example
@end deftypefun

@deftypefun mpf_class abs (mpf_class @var{op})
@deftypefunx mpf_class ceil (mpf_class @var{op})
@deftypefunx int cmp (mpf_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpf_class @var{op2})
@maybepagebreak
@deftypefunx mpf_class floor (mpf_class @var{op})
@deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2})
@deftypefunx double mpf_class::get_d (void)
@deftypefunx long mpf_class::get_si (void)
@deftypefunx {unsigned long} mpf_class::get_ui (void)
@maybepagebreak
@deftypefunx bool mpf_class::fits_sint_p (void)
@deftypefunx bool mpf_class::fits_slong_p (void)
@deftypefunx bool mpf_class::fits_sshort_p (void)
@maybepagebreak
@deftypefunx bool mpf_class::fits_uint_p (void)
@deftypefunx bool mpf_class::fits_ulong_p (void)
@deftypefunx bool mpf_class::fits_ushort_p (void)
@maybepagebreak
@deftypefunx int sgn (mpf_class @var{op})
@deftypefunx mpf_class sqrt (mpf_class @var{op})
@deftypefunx mpf_class trunc (mpf_class @var{op})
These functions provide a C++ class interface to the corresponding GMP C
routines.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.

The accuracy provided by @code{hypot} is not currently guaranteed.
@end deftypefun

@deftypefun {unsigned long int} mpf_class::get_prec ()
@deftypefunx void mpf_class::set_prec (unsigned long @var{prec})
@deftypefunx void mpf_class::set_prec_raw (unsigned long @var{prec})
Get or set the current precision of an @code{mpf_class}.

The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing
Floats}) apply to @code{mpf_class::set_prec_raw}.  Note in particular that the
@code{mpf_class} must be restored to it's allocated precision before being
destroyed.  This must be done by application code, there's no automatic
mechanism for it.
@end deftypefun


@node C++ Interface MPFR, C++ Interface Random Numbers, C++ Interface Floats, C++ Class Interface
@section C++ Interface MPFR

The C++ class interface to MPFR is provided if MPFR is enabled (@pxref{Build
Options}).  This interface must be regarded as preliminary and possibly
subject to incompatible changes in the future, since MPFR itself is
preliminary.  All definitions can be obtained with

@example
#include <mpfrxx.h>
@end example

@noindent
This defines

@deftp Class mpfr_class
@end deftp

@noindent
which behaves similarly to @code{mpf_class} (@pxref{C++ Interface Floats}).


@node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface MPFR, C++ Class Interface
@section C++ Interface Random Numbers

@deftp Class gmp_randclass
The C++ class interface to the GMP random number functions uses
@code{gmp_randclass} to hold an algorithm selection and current state, as per
@code{gmp_randstate_t}.
@end deftp

@deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, ...), ...)
Construct a @code{gmp_randclass}, using a call to the given @var{randinit}
function (@pxref{Random State Initialization}).  The arguments expected are
the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}.
For example,

@example
gmp_randclass r1 (gmp_randinit_default);
gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
@end example

@code{gmp_randinit_lc_2exp_size} can fail if the size requested is too big,
the behaviour of @code{gmp_randclass::gmp_randclass} is undefined in this case
(perhaps this will change in the future).
@end deftypefun

@deftypefun {} gmp_randclass::gmp_randclass (gmp_randalg_t @var{alg}, ...)
Construct a @code{gmp_randclass} using the same parameters as
@code{gmp_randinit} (@pxref{Random State Initialization}).  This function is
obsolete and the above @var{randinit} style should be preferred.
@end deftypefun

@deftypefun void gmp_randclass::seed (unsigned long int @var{s})
@deftypefunx void gmp_randclass::seed (mpz_class @var{s})
Seed a random number generator.  See @pxref{Random Number Functions}, for how
to choose a good seed.
@end deftypefun

@deftypefun mpz_class gmp_randclass::get_z_bits (unsigned long @var{bits})
@deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits})
Generate a random integer with a specified number of bits.
@end deftypefun

@deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n})
Generate a random integer in the range 0 to @ma{@var{n}-1} inclusive.
@end deftypefun

@deftypefun mpf_class gmp_randclass::get_f ()
@deftypefunx mpf_class gmp_randclass::get_f (unsigned long @var{prec})
Generate a random float @var{f} in the range @ma{0 <= @var{f} < 1}.  @var{f}
will be to @var{prec} bits precision, or if @var{prec} is not given then to
the precision of the destination.  For example,

@example
gmp_randclass  r;
...
mpf_class  f (0, 512);   // 512 bits precision
f = r.get_f();           // random number, 512 bits
@end example
@end deftypefun



@node C++ Interface Limitations,  , C++ Interface Random Numbers, C++ Class Interface
@section C++ Interface Limitations

@table @asis
@item @code{mpq_class} and Templated Reading
A generic piece of template code probably won't know that @code{mpq_class}
requires a @code{canonicalize} call if inputs read with @code{operator>>}
might be non-canonical.  This can lead to incorrect results.

@code{operator>>} behaves as it does for reasons of efficiency.  A
canonicalize can be quite time consuming on large operands, and is best
avoided if it's not necessary.

But this potential difficulty reduces the usefulness of @code{mpq_class}.
Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in
the future, maybe a preprocessor define, a global flag, or an @code{ios} flag
pressed into service.  Or maybe, at the risk of inconsistency, the
@code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t}
@code{operator>>} not doing so, for use on those occasions when that's
acceptable.  Send feedback or alternate ideas to @email{bug-gmp@@gnu.org}.

@item Subclassing
Subclassing the GMP C++ classes works, but is not currently recommended.

Expressions involving subclasses resolve correctly (or seem to), but in normal
C++ fashion the subclass doesn't inherit constructors and assignments.
There's many of those in the GMP classes, and a good way to reestablish them
in a subclass is not yet provided.

@item Templated Expressions

A subtle difficulty exists when using expressions together with
application-defined template functions.  Consider the following, with @code{T}
intended to be some numeric type,

@example
template <class T>
T fun (const T &, const T &);
@end example

@noindent
When used with, say, plain @code{mpz_class} variables, it works fine: @code{T}
is resolved as @code{mpz_class}.

@example
mpz_class f(1), g(2);
fun (f, g);    // Good
@end example

@noindent
But when one of the arguments is an expression, it doesn't work.

@example
mpz_class f(1), g(2), h(3);
fun (f, g+h);  // Bad
@end example

This is because @code{g+h} ends up being a certain expression template type
internal to @code{gmpxx.h}, which the C++ template resolution rules are unable
to automatically convert to @code{mpz_class}.  The workaround is simply to add
an explicit cast.

@example
mpz_class f(1), g(2), h(3);
fun (f, mpz_class(g+h));  // Good
@end example

Similarly, within @code{fun} it may be necessary to cast an expression to type
@code{T} when calling a templated @code{fun2}.

@example
template <class T>
void fun (T f, T g)
@{
  fun2 (f, f+g);     // Bad
@}

template <class T>
void fun (T f, T g)
@{
  fun2 (f, T(f+g));  // Good
@}
@end example
@end table


@node BSD Compatible Functions, Custom Allocation, C++ Class Interface, Top
@comment  node-name,  next,  previous,  up
@chapter Berkeley MP Compatible Functions
@cindex Berkeley MP compatible functions
@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
@samp{--enable-mpbsd} option must be used when building GNU MP to make these
available (@pxref{Installing GMP}).

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

It is not recommended that new programs are written using these functions.
Apart from the incomplete set of functions, the interface for initializing
@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 your program.
This means that you probably need to give the @samp{-I<dir>} option to the
compiler, where @samp{<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,
null-terminated 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{op}, MINT *@var{root}, MINT *@var{remainder})
Set @var{root} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{remainder} to
@m{(@var{op} - @var{root}^2), @var{op}@minus{}@var{root}*@var{root}}, i.e.
zero if @var{op} is a perfect square.

If @var{root} and @var{remainder} are the same variable, the results are
undefined.
@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{op1}, MINT *@var{op2}, MINT *@var{res})
Set @var{res} to the greatest common divisor of @var{op1} and @var{op2}.
@end deftypefun

@deftypefun int mcmp (MINT *@var{op1}, MINT *@var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end deftypefun

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

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


@node Custom Allocation, Language Bindings, BSD Compatible Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Custom Allocation
@cindex Custom allocation
@cindex Memory allocation
@cindex Allocation of memory

By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory
allocation, and if they fail GMP prints a message to the standard error output
and terminates the program.

Alternate functions can be specified to allocate memory in a different way or
to have a different error action on running out of memory.

This feature is available in the Berkeley compatibility library (@pxref{BSD
Compatible Functions}) as well as the main GMP library.

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

These functions will be used for all memory allocation done by GMP, apart from
temporary space from @code{alloca} if that function is available and GMP is
configured to use it (@pxref{Build Options}).

@strong{Be sure to call @code{mp_set_memory_functions} only when there are no
active GMP objects allocated using the previous memory functions!  Usually
that means calling it before any other GMP function.}
@end deftypefun

The functions supplied should fit the following declarations:

@deftypefun {void *} allocate_function (size_t @var{alloc_size})
Return a pointer to newly allocated space with at least @var{alloc_size}
bytes.
@end deftypefun

@deftypefun {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})
Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be
@var{new_size} bytes.

The block may be moved if necessary or if desired, and in that case the
smaller of @var{old_size} and @var{new_size} bytes must be copied to the new
location.  The return value is a pointer to the resized block, that being the
new location if moved or just @var{ptr} if not.

@var{ptr} is never @code{NULL}, it's always a previously allocated block.
@var{new_size} may be bigger or smaller than @var{old_size}.
@end deftypefun

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

@var{ptr} is never @code{NULL}, it's always a previously allocated block of
@var{size} bytes.
@end deftypefun

A @dfn{byte} here means the unit used by the @code{sizeof} operator.

The @var{old_size} parameters to @var{reallocate_function} and
@var{deallocate_function} are passed for convenience, but of course can be
ignored if not needed.  The default functions using @code{malloc} and friends
for instance don't use them.

No error return is allowed from any of these functions, if they return then
they must have performed the specified operation.  In particular note that
@var{allocate_function} or @var{reallocate_function} mustn't return
@code{NULL}.

Getting a different fatal error action is a good use for custom allocation
functions, for example giving a graphical dialog rather than the default print
to @code{stderr}.  How much is possible when genuinely out of memory is
another question though.

There's currently no defined way for the allocation functions to recover from
an error such as out of memory, they must terminate program execution.  A
@code{longjmp} or throwing a C++ exception will have undefined results.  This
may change in the future.

GMP may use allocated blocks to hold pointers to other allocated blocks.  This
will limit the assumptions a conservative garbage collection scheme can make.

Since the default GMP allocation uses @code{malloc} and friends, those
functions will be linked in even if the first thing a program does is an
@code{mp_set_memory_functions}.  It's necessary to change the GMP sources if
this is a problem.


@node Language Bindings, Algorithms, Custom Allocation, Top
@chapter Language Bindings

The following packages and projects offer access to GMP from languages other
than C, though perhaps with varying levels of functionality and efficiency.

@c  GNUstep Base Library @uref{http://www.gnustep.org} (version 0.9.1) is
@c  intending to use GMP for its NSDecimal class, which would be an Objective
@c  C binding for GMP.  Has some configure stuff ready, but no code.

@c  @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces
@c  in tex, just to separate the URL from the preceding text a bit.
@iftex
@macro spaceuref {U}
@ @ @uref{\U\}
@end macro
@end iftex
@ifnottex
@macro spaceuref {U}
@uref{\U\}
@end macro
@end ifnottex

@sp 1
@table @asis
@item C++
@itemize @bullet
@item
GMP C++ class interface, @pxref{C++ Class Interface} @* Straightforward
interface, expression templates to eliminate temporaries.
@item
ALP @spaceuref{http://www.inria.fr/saga/logiciels/ALP} @* Linear algebra and
polynomials using templates.
@item
CLN @spaceuref{http://clisp.cons.org/~haible/packages-cln.html"} @* High level
classes for arithmetic.
@item
LiDIA @spaceuref{http://www.informatik.tu-darmstadt.de/TI/LiDIA} @* A C++
library for computational number theory.
@item
NTL @spaceuref{http://www.shoup.net/ntl} @* A C++ number theory library.
@end itemize

@item Fortran
@itemize @bullet
@item
Omni F77 @spaceuref{http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html} @*
Arbitrary precision floats.
@end itemize

@item Haskell
@itemize @bullet
@item
Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc}
@end itemize

@item Java
@itemize @bullet
@item
Kaffe @spaceuref{http://www.kaffe.org}
@item
Kissme @spaceuref{http://kissme.sourceforge.net}
@end itemize

@item Lisp
@itemize @bullet
@item
GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html} @* In the
process of switching to GMP for bignums.
@item
Librep @spaceuref{http://librep.sourceforge.net}
@end itemize

@item M4
@itemize @bullet
@item
GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu} @* Optionally provides
an arbitrary precision @code{mpeval}.
@end itemize

@item ML
@itemize @bullet
@item
MLton compiler @spaceuref{http://www.sourcelight.com/MLton}
@end itemize

@item Oz
@itemize @bullet
@item
Mozart @spaceuref{http://www.mozart-oz.org}
@end itemize

@item Perl
@itemize @bullet
@item
GMP module, see @file{demos/perl} in the GMP sources.
@item
Math::GMP @spaceuref{http://www.cpan.org} @* Compatible with Math::BigInt, but
not as many functions as the GMP module above.
@end itemize

@need 1000
@item Pike
@itemize @bullet
@item
mpz module in the standard distribution, @uref{http://pike.idonex.com}
@end itemize

@need 500
@item Prolog
@itemize @bullet
@item
SWI Prolog @spaceuref{http://www.swi.psy.uva.nl/projects/SWI-Prolog} @*
Arbitrary precision floats.
@end itemize

@item Python
@itemize @bullet
@item
mpz module in the standard distribution, @uref{http://www.python.org}
@end itemize

@item Scheme
@itemize @bullet
@item
RScheme @spaceuref{http://www.rscheme.org}
@end itemize

@item Other
@itemize @bullet
@item
DrGenius @spaceuref{http://drgenius.seul.org} @* Geometry system and
mathematical programming language.
@item
GiNaC @spaceuref{http://www.ginac.de} @* C++ computer algebra using CLN.
@item
Maxima @uref{http://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma
computer algebra using GCL.
@item
Q @spaceuref{http://www.musikwissenschaft.uni-mainz.de/~ag/q} @* Equational
programming system.
@item
Yacas @spaceuref{http://www.xs4all.nl/~apinkus/yacas.html} @* Computer algebra
system.
@end itemize

@end table


@node Algorithms, Internals, Language Bindings, Top
@chapter Algorithms
@cindex Algorithms

This chapter is an introduction to some of the algorithms used for various GMP
operations.  The code is likely to be hard to understand without knowing
something about the algorithms.

Some GMP internals are mentioned, but applications that expect to be
compatible with future GMP releases should take care to use only the
documented functions.

@menu
* Multiplication Algorithms::   
* Division Algorithms::         
* Greatest Common Divisor Algorithms::  
* Powering Algorithms::         
* Root Extraction Algorithms::  
* Radix Conversion Algorithms::  
* Other Algorithms::            
* Assembler Coding::            
@end menu


@node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms
@section Multiplication
@cindex Multiplication algorithms

N@cross{}N limb multiplications and squares are done using one of four
algorithms, as the size N increases.

@quotation
@multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item Algorithm @tab Threshold
@item Basecase  @tab (none)
@item Karatsuba @tab @code{KARATSUBA_MUL_THRESHOLD}
@item Toom-3    @tab @code{TOOM3_MUL_THRESHOLD}
@item FFT       @tab @code{FFT_MUL_THRESHOLD}
@end multitable
@end quotation

Similarly for squaring, with the @code{SQR} thresholds.  Note though that the
FFT is only used if GMP is configured with @samp{--enable-fft}, @pxref{Build
Options}.

N@cross{}M multiplications of operands with different sizes above
@code{KARATSUBA_MUL_THRESHOLD} are currently done by splitting into M@cross{}M
pieces.  The Karatsuba and Toom-3 routines then operate only on equal size
operands.  This is not very efficient, and is slated for improvement in the
future.

@menu
* Basecase Multiplication::     
* Karatsuba Multiplication::    
* Toom-Cook 3-Way Multiplication::  
* FFT Multiplication::          
* Other Multiplication::        
@end menu


@node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms
@subsection Basecase Multiplication

Basecase N@cross{}M multiplication is a straightforward rectangular set of
cross-products, the same as long multiplication done by hand and for that
reason sometimes known as the schoolbook or grammar school method.  This is an
@m{O(NM),O(N*M)} algorithm.  See Knuth section 4.3.1 algorithm M
(@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code.

Assembler implementations of @code{mpn_mul_basecase} are essentially the same
as the generic C code, but have all the usual assembler tricks and
obscurities introduced for speed.

A square can be done in roughly half the time of a multiply, by using the fact
that the cross products above and below the diagonal are the same.  A triangle
of products below the diagonal is formed, doubled (left shift by one bit), and
then the products on the diagonal added.  This can be seen in
@file{mpn/generic/sqr_basecase.c}.  Again the assembler implementations take
essentially the same approach.

@tex
\def\GMPline#1#2#3#4#5#6{%
  \hbox {%
    \vrule height 2.5ex depth 1ex
           \hbox to 2em {\hfil{#2}\hfil}%
    \vrule \hbox to 2em {\hfil{#3}\hfil}%
    \vrule \hbox to 2em {\hfil{#4}\hfil}%
    \vrule \hbox to 2em {\hfil{#5}\hfil}%
    \vrule \hbox to 2em {\hfil{#6}\hfil}%
    \vrule}}
\GMPdisplay{
  \hbox{%
    \vbox{%
      \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}%
      \vfill}%
    \vbox{%
      \hbox{%
        \hbox to 2em {\hfil u0\hfil}%
        \hbox to 2em {\hfil u1\hfil}%
        \hbox to 2em {\hfil u2\hfil}%
        \hbox to 2em {\hfil u3\hfil}%
        \hbox to 2em {\hfil u4\hfil}}%
      \vskip 0.7ex
      \hrule
      \GMPline{u0}{d}{}{}{}{}%
      \hrule
      \GMPline{u1}{}{d}{}{}{}%
      \hrule
      \GMPline{u2}{}{}{d}{}{}%
      \hrule
      \GMPline{u3}{}{}{}{d}{}%
      \hrule
      \GMPline{u4}{}{}{}{}{d}%
      \hrule}}}
@end tex
@ifnottex
@example
@group
     u0  u1  u2  u3  u4
   +---+---+---+---+---+
u0 | d |   |   |   |   |
   +---+---+---+---+---+
u1 |   | d |   |   |   |
   +---+---+---+---+---+
u2 |   |   | d |   |   |
   +---+---+---+---+---+
u3 |   |   |   | d |   |
   +---+---+---+---+---+
u4 |   |   |   |   | d |
   +---+---+---+---+---+
@end group
@end example
@end ifnottex

In practice squaring isn't a full 2@cross{} faster than multiplying, it's
usually around 1.5@cross{}.  Less than 1.5@cross{} probably indicates
@code{mpn_sqr_basecase} wants improving on that CPU.

On some CPUs @code{mpn_mul_basecase} can be faster than the generic C
@code{mpn_sqr_basecase}.  @code{BASECASE_SQR_THRESHOLD} is the size at which
to use @code{mpn_sqr_basecase}, this will be zero if that routine should be
used always.


@node Karatsuba Multiplication, Toom-Cook 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms
@subsection Karatsuba Multiplication

The Karatsuba multiplication algorithm is described in Knuth section 4.3.3
part A, and various other textbooks.  A brief description is given here.

The inputs @ma{x} and @ma{y} are treated as each split into two parts of equal
length (or the most significant part one limb shorter if N is odd).

@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\GMPbox#1#2{%
  \vbox {%
    \hrule
    \hbox{%
      \vrule height 2ex depth 1ex
      \hbox to \GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
      \vrule
      \hbox to \GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
      \vrule}
    \hrule
}}
\GMPdisplay{%
\vbox{%
  \hbox to 2\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPbox{x_1}{x_0}
  \vskip 0.5ex
  \GMPbox{y_1}{y_0}
}}
%}
%\moveright \lispnarrowing 
%\vskip 0.5 ex
%\vskip 0.5 ex
@end tex
@ifnottex
@example
@group
 high              low
+----------+----------+
|    x1    |    x0    |
+----------+----------+

+----------+----------+
|    y1    |    y0    |
+----------+----------+
@end group
@end example
@end ifnottex

Let @ma{b} be the power of 2 where the split occurs, ie.@: if @ms{x,0} is
@ma{k} limbs (@ms{y,0} the same) then
@m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the
following holds,

@display
@m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0,
  x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0}
@end display

This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs,
whereas a basecase multiply of N@cross{}N limbs is equivalent to four
multiplies of (N/2)@cross{}(N/2).  The factors @ma{(b^2+b)} etc represent the
positions where the three products must be added.

@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\GMPboxA#1#2{%
  \vbox to \GMPboxheight{%
    \hrule \vfil
    \hbox{%
      \strut \vrule
      \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
      \vrule
      \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
      \vrule}
    \vfil \hrule}}
\def\GMPboxB#1#2{%
  \hbox{%
    \vbox to \GMPboxheight{%
      \vfil \hbox to \GMPboxwidth {\hfil #1} \vfil }
    \vbox to \GMPboxheight{%
      \hrule \vfil
      \hbox{%
        \strut \vrule
        \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}
        \vrule}
      \vfil \hrule}}}
\GMPdisplay{%
\vbox{%
  \hbox to 4\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPboxA{x_1y_1}{x_0y_0}
  \vskip 0.5ex
  \GMPboxB{$+$}{x_1y_1}
  \vskip 0.5ex
  \GMPboxB{$+$}{x_0y_0}
  \vskip 0.5ex
  \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)}
}}
@end tex
@ifnottex
@example
@group
 high                              low
+--------+--------+ +--------+--------+
|      x1*y1      | |      x0*y0      |
+--------+--------+ +--------+--------+
          +--------+--------+
      add |      x1*y1      |
          +--------+--------+
          +--------+--------+
      add |      x0*y0      |
          +--------+--------+
          +--------+--------+
      sub | (x1-x0)*(y1-y0) |
          +--------+--------+
@end group
@end example
@end ifnottex

The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an
absolute value, and the sign used to choose to add or subtract.  Notice the
sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1),
high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb
additions, rather than @m{6k,6*k}, but in GMP extra function call overheads
outweigh the saving.

Squaring is similar to multiplying, but with @ma{x=y} the formula reduces to
an equivalent with three squares,

@display
@m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2,
   x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2}
@end display

The final result is accumulated from those three squares the same way as for
the three multiplies above.  The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now
always positive.

A similar formula for both multiplying and squaring can be constructed with a
middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}.  But those sums can exceed
@ma{k} limbs, leading to more carry handling and additions than the form
above.

Karatsuba multiplication is asymptotically an @ma{O(N^@W{1.585})} algorithm,
the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies
each 1/2 the size of the inputs.  This is a big improvement over the basecase
multiply at @ma{O(N^2)} and the advantage soon overcomes the extra additions
Karatsuba performs.

@code{KARATSUBA_MUL_THRESHOLD} can be as little as 10 limbs.  The @code{SQR}
threshold is usually about twice the @code{MUL}.  The basecase algorithm will
take a time of the form @m{M(N) = aN^2 + bN + c, M(N) = a*N^2 + b*N + c} and
the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N +
e}.  Clearly per-crossproduct speedups in the basecase code reduce @ma{a} and
decrease the threshold, but linear style speedups reducing @ma{b} will
actually increase the threshold.  The latter can be seen for instance when
adding an optimized @code{mpn_sqr_diagonal} to @code{mpn_sqr_basecase}.  Of
course all speedups reduce total time, and in that sense the algorithm
thresholds are merely of academic interest.


@node Toom-Cook 3-Way Multiplication, FFT Multiplication, Karatsuba Multiplication, Multiplication Algorithms
@subsection Toom-Cook 3-Way Multiplication

The Karatsuba formula is the simplest case of a general approach to splitting
inputs that leads to both Toom-Cook and FFT algorithms.  A description of
Toom-Cook can be found in Knuth section 4.3.3, with an example 3-way
calculation after Theorem A.  The 3-way form used in GMP is described here.

The operands are each considered split into 3 pieces of equal length (or the
most significant part 1 or 2 limbs shorter than the others).

@iftex
@global@newdimen@GMPboxwidth @GMPboxwidth=5em
@global@newdimen@GMPboxheight @GMPboxheight=3ex
@end iftex
@tex
\def\GMPbox#1#2#3{%
  \vbox to \GMPboxheight{%
    \hrule \vfil
    \hbox{%
      \strut \vrule
      \hbox to \GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
      \vrule
      \hbox to \GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
      \vrule
      \hbox to \GMPboxwidth {\hfil\hbox{$#3$}\hfil}%
      \vrule}
    \vfil \hrule
}}
\GMPdisplay{%
\vbox{%
  \hbox to 3\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPbox{x_2}{x_1}{x_0}
  \vskip 0.5ex
  \GMPbox{y_2}{y_1}{y_0}
  \vskip 0.5ex
}}
@end tex
@ifnottex
@example
@group
 high                         low
+----------+----------+----------+
|    x2    |    x1    |    x0    |
+----------+----------+----------+

+----------+----------+----------+
|    y2    |    y1    |    y0    |
+----------+----------+----------+
@end group
@end example
@end ifnottex

@noindent
These parts are treated as the coefficients of two polynomials

@display
@group
@m{X(t) = x_2t^2 + x_1t + x_0,
   X(t) = x2*t^2 + x1*t + x0}
@m{Y(t) = y_2t^2 + y_1t + y_0,
   Y(t) = y2*t^2 + y1*t + y0}
@end group
@end display

Again let @ma{b} equal the power of 2 which is the size of the @ms{x,0},
@ms{x,1}, @ms{y,0} and @ms{y,1} pieces, ie.@: if they're @ma{k} limbs each
then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}},
b=2^(k*mp_bits_per_limb)}.  With this @ma{x=X(b)} and @ma{y=Y(b)}.

Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients
are

@display
@m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0,
   W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}
@end display

@noindent
The @m{w_i,w[i]} are going to be determined, and when they are they'll give
the final result using @ma{w=W(b)}, since @m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}.
The coefficients will be roughly @ma{b^2} each, and the final @ma{W(b)} will
be an addition like,

@tex
\def\GMPbox#1#2{%
  \moveright #1\GMPboxwidth
  \vbox to \GMPboxheight{%
    \hrule \vfil
    \hbox{%
      \strut \vrule
      \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
      \vrule}
    \vfil \hrule
}}
\GMPdisplay{%
\vbox{%
  \hbox to 6\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPbox{0}{w_4}
  \vskip 0.5ex
  \GMPbox{1}{w_3}
  \vskip 0.5ex
  \GMPbox{2}{w_2}
  \vskip 0.5ex
  \GMPbox{3}{w_1}
  \vskip 0.5ex
  \GMPbox{4}{w_1}
}}
@end tex
@ifnottex
@example
@group
 high                                        low
+-------+-------+
|       w4      |
+-------+-------+
       +--------+-------+
       |        w3      |
       +--------+-------+
               +--------+-------+
               |        w2      |
               +--------+-------+
                       +--------+-------+
                       |        w1      |
                       +--------+-------+
                                +-------+-------+
                                |       w0      |
                                +-------+-------+
@end group
@end example
@end ifnottex

The @m{w_i,w[i]} coefficients could be formed by a simple set of cross
products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2},
@m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all
nine @m{x_iy_j,x[i]*y[j]} for @ma{i,j=0,1,2}, and would be equivalent merely
to a basecase multiply.  Instead the following approach is used.

@ma{X(t)} and @ma{Y(t)} are evaluated and multiplied at 5 points, giving
values of @ma{W(t)} at those points.  The points used can be chosen in
various ways, but in GMP the following are used

@quotation
@multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item Point                 @tab Value
@item @ma{t=0}              @tab @m{x_0y_0,x0*y0}, which gives @ms{w,0} immediately
@item @ma{t=2}              @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0)*(4*y2+2*y1+y0)}
@item @ma{t=1}              @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0)*(y2+y1+y0)}
@item @m{t={1\over2},t=1/2} @tab @m{(x_2+2x_1+4x_0)(y_2+2y_1+4y_0),(x2+2*x1+4*x0)*(y2+2*y1+4*y0)}
@item @m{t=\infty,t=inf}    @tab @m{x_2y_2,x2*y2}, which gives @ms{w,4} immediately
@end multitable
@end quotation

At @m{t={1\over2},t=1/2} the value calculated is actually
@m{16X({1\over2})Y({1\over2}), 16*X(1/2)*Y(1/2)}, giving a value for
@m{16W({1\over2}),16*W(1/2)}, and this is always an integer.  At
@m{t=\infty,t=inf} the value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over
t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much
easier to think of as simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately
(much like @m{x_0y_0,x0*y0} at @ma{t=0} gives @ms{w,0} immediately).

Now each of the points substituted into
@m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+@dots{}+w0} gives a linear combination
of the @m{w_i,w[i]} coefficients, and the value of those combinations has just
been calculated.

@tex
\GMPdisplay{%
$\matrix{%
W(0)           & = &       &   &      &   &      &   &      &   &   w_0 \cr
16W({1\over2}) & = &   w_4 & + & 2w_3 & + & 4w_2 & + & 8w_1 & + & 16w_0 \cr
W(1)           & = &   w_4 & + &  w_3 & + &  w_2 & + &  w_1 & + &   w_0 \cr
W(2)           & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + &   w_0 \cr
W(\infty)      & = &   w_4 \cr
}$}
@end tex
@ifnottex
@example
@group
   W(0)   =                                 w0
16*W(1/2) =    w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0
   W(1)   =    w4 +   w3 +   w2 +   w1 +    w0
   W(2)   = 16*w4 + 8*w3 + 4*w2 + 2*w1 +    w0
   W(inf) =    w4
@end group
@end example
@end ifnottex

This is a set of five equations in five unknowns, and some elementary linear
algebra quickly isolates each @m{w_i,w[i]}, by subtracting multiples of one
equation from another.

In the code the set of five values @ma{W(0)},@dots{},@m{W(\infty),W(inf)} will
represent those certain linear combinations.  By adding or subtracting one
from another as necessary, values which are each @m{w_i,w[i]} alone are
arrived at.  This involves only a few subtractions of small multiples (some of
which are powers of 2), and so is fast.  A couple of divisions remain by
powers of 2 and one division by 3 (or by 6 rather), and that last uses the
special @code{mpn_divexact_by3} (@pxref{Exact Division}).

In the code the values @ms{w,4}, @ms{w,2} and @ms{w,0} are formed in the
destination with pointers @code{E}, @code{C} and @code{A}, and @ms{w,3} and
@ms{w,1} in temporary space @code{D} and @code{B} are added to them.  There
are extra limbs @code{tD}, @code{tC} and @code{tB} at the high end of
@ms{w,3}, @ms{w,2} and @ms{w,1} which are handled separately.  The final
addition then is as follows.

@tex
\def\GMPboxT#1{%
  \vbox to \GMPboxheight{%
    \hrule
    \hbox {\strut \vrule{} #1 \vrule}%
    \hrule
}}
\GMPdisplay{%
\advance\baselineskip by 1ex
\vbox{%
  \hbox to 6\GMPboxwidth {high \hfil low}
  \vbox to \GMPboxheight{%
    \hrule \vfil
    \hbox{%
      \strut \vrule
      \hbox to 2\GMPboxwidth {\hfil@code{E}\hfil}
      \vrule
      \hbox to 2\GMPboxwidth {\hfil@code{C}\hfil}
      \vrule
      \hbox to 2\GMPboxwidth {\hfil@code{A}\hfil}
      \vrule}
    \vfil \hrule
  }%
  \moveright \GMPboxwidth
  \vbox to \GMPboxheight{%
    \hrule \vfil
    \hbox{%
      \strut \vrule
      \hbox to 2\GMPboxwidth {\hfil@code{D}\hfil}
      \vrule
      \hbox to 2\GMPboxwidth {\hfil@code{B}\hfil}
      \vrule}
    \vfil \hrule
  }%
  \hbox{%
    \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tD}}}%
    \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tC}}}%
    \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tB}}}}
}}
@end tex
@ifnottex
@example
@group
 high                                        low
+-------+-------+-------+-------+-------+-------+
|       E       |       C       |       A       |
+-------+-------+-------+-------+-------+-------+
         +------+-------++------+-------+
         |      D       ||      B       |
         +------+-------++------+-------+
      --      --      --
     |tD|    |tC|    |tB|        
      --      --      --
@end group
@end example
@end ifnottex

The conversion of @ma{W(t)} values to the coefficients is interpolation.  A
polynomial of degree 4 like @ma{W(t)} is uniquely determined by values known
at 5 different points.  The points can be chosen to make the linear equations
come out with a convenient set of steps for isolating the @m{w_i,w[i]}.

In @file{mpn/generic/mul_n.c} the @code{interpolate3} routine performs the
interpolation.  The open-coded one-pass version may be a bit hard to
understand, the steps performed can be better seen in the @code{USE_MORE_MPN}
version.

Squaring follows the same procedure as multiplication, but there's only one
@ma{X(t)} and it's evaluated at 5 points, and those values squared to give
values of @ma{W(t)}.  The interpolation is then identical, and in fact the
same @code{interpolate3} subroutine is used for both squaring and multiplying.

Toom-3 is asymptotically @ma{O(N^@W{1.465})}, the exponent being
@m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the
original size.  This is an improvement over Karatsuba at @ma{O(N^@W{1.585})},
though Toom-Cook does more work in the evaluation and interpolation and so it
only realizes its advantage above a certain size.

Near the crossover between Toom-3 and Karatsuba there's generally a range of
sizes where the difference between the two is small.
@code{TOOM3_MUL_THRESHOLD} is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to small
variations in measuring.  A graph of time versus size for the two shows the
effect, see @file{tune/README}.

At the fairly small sizes where the Toom-3 thresholds occur it's worth
remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be
expected to make accurate predictions, due of course to the big influence of
all sorts of overheads, and the fact that only a few recursions of each are
being performed.  Even at large sizes there's a good chance machine dependent
effects like cache architecture will mean actual performance deviates from
what might be predicted.

The formula given above for the Karatsuba algorithm has an equivalent for
Toom-3 involving only five multiplies, but this would be complicated and
unenlightening.

An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using
a vector to represent the @ma{x} and @ma{y} splits and a matrix multiplication
for the evaluation and interpolation stages.  The matrix inverses are not
meant to be actually used, and they have elements with values much greater
than in fact arise in the interpolation steps.  The diagram shown for the
3-way is attractive, but again doesn't have to be implemented that way and for
example with a bit of rearrangement just one division by 6 can be done.


@node FFT Multiplication, Other Multiplication, Toom-Cook 3-Way Multiplication, Multiplication Algorithms
@subsection FFT Multiplication

At large to very large sizes a Fermat style FFT multiplication is used,
following Sch@"onhage and Strassen (@pxref{References}).  Descriptions of FFTs
in various forms can be found in many textbooks, for instance Knuth section
4.3.3 part C or Lipson chapter IX.  A brief description of the form used in
GMP is given here.

The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given
@ma{N}.  A full product @m{xy,x*y} is obtained by choosing @m{N \ge
\mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding
@ma{x} and @ma{y} with high zero limbs.  The modular product is the native
form for the algorithm, so padding to get a full product is unavoidable.

The algorithm follows a split, evaluate, pointwise multiply, interpolate and
combine similar to that described above for Karatsuba and Toom-3.  A @ma{k}
parameter controls the split, with an FFT-@ma{k} splitting into @ma{2^k}
pieces of @ma{M=N/2^k} bits each.  @ma{N} must be a multiple of
@m{2^k\times@code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so
the split falls on limb boundaries, avoiding bit shifts in the split and
combine stages.

The evaluations, pointwise multiplications, and interpolation, are all done
modulo @m{2^{N'}+1, 2^N'+1} where @ma{N'} is @ma{2M+k+3} rounded up to a
multiple of @ma{2^k} and of @code{mp_bits_per_limb}.  The results of
interpolation will be the following negacyclic convolution of the input
pieces, and the choice of @ma{N'} ensures these sums aren't truncated.
@tex
$$ w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j $$
@end tex
@ifnottex

@example
           ---
           \         b
w[n] =     /     (-1) * x[i] * y[j]
           ---
       i+j==b*2^k+n
          b=0,1
@end example

@end ifnottex
The points used for the evaluation are @ma{g^i} for @ma{i=0} to @ma{2^k-1}
where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}.  @ma{g} is a @m{2^k,2^k'}th root of
unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary cancellations at the
interpolation stage, and it's also a power of 2 so the fast fourier transforms
used for the evaluation and interpolation do only shifts, adds and negations.

The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either
recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or
basecase), whichever is optimal at the size @ma{N'}.  The interpolation is an
inverse fast fourier transform.  The resulting set of sums of @m{x_iy_j,
x[i]*y[j]} are added at appropriate offsets to give the final result.

Squaring is the same, but @ma{x} is the only input so it's one transform at
the evaluate stage and the pointwise multiplies are squares.  The
interpolation is the same.

For a mod @ma{2^N+1} product, an FFT-@ma{k} is an @m{O(N^{k/(k-1)}),
O(N^(k/(k-1)))} algorithm, the exponent representing @ma{2^k} recursed modular
multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original.  Each
successive @ma{k} is an asymptotic improvement, but overheads mean each is
only faster at bigger and bigger sizes.  In the code, @code{FFT_MUL_TABLE} and
@code{FFT_SQR_TABLE} are the thresholds where each @ma{k} is used.  Each new
@ma{k} effectively swaps some multiplying for some shifts, adds and overheads.

A mod @ma{2^N+1} product can be formed with a normal
@ma{N@cross{}N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT and
Toom-3 etc can be compared directly.  A @ma{k=4} FFT at @ma{O(N^@W{1.333})}
can be expected to be the first faster than Toom-3 at @ma{O(N^@W{1.465})}.  In
practice this is what's found, with @code{FFT_MODF_MUL_THRESHOLD} and
@code{FFT_MODF_SQR_THRESHOLD} being between 300 and 1000 limbs, depending on
the CPU.  So far it's been found that only very large FFTs recurse into
pointwise multiplies above these sizes.

When an FFT is to give a full product, the change of @ma{N} to @ma{2N} doesn't
alter the theoretical complexity for a given @ma{k}, but for the purposes of
considering where an FFT might be first used it can be assumed that the FFT is
recursing into a normal multiply and that on that basis it's doing @ma{2^k}
recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of the inputs,
making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}.  This would mean @ma{k=7} at
@ma{O(N^@W{1.4})} would be the first FFT faster than Toom-3.  In practice
@code{FFT_MUL_THRESHOLD} and @code{FFT_SQR_THRESHOLD} have been found to be in
the @ma{k=8} range, somewhere between 3000 and 10000 limbs.

The way @ma{N} is split into @ma{2^k} pieces and then @ma{2M+k+3} is rounded
up to a multiple of @ma{2^k} and @code{mp_bits_per_limb} means that when
@ma{2^k@ge{}@nicode{mp\_bits\_per\_limb}} the effective @ma{N} is a multiple
of @m{2^{2k-1},2^(2k-1)} bits.  The @ma{+k+3} means some values of @ma{N} just
under such a multiple will be rounded to the next.  The complexity
calculations above assume that a favourable size is used, meaning one which
isn't padded through rounding, and it's also assumed that the extra @ma{+k+3}
bits are negligible at typical FFT sizes.

The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a
step-effect into measured speeds.  For example @ma{k=8} will round @ma{N} up
to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb groups
of sizes for which @code{mpn_mul_n} runs at the same speed.  Or for @ma{k=9}
groups of 2048 limbs, @ma{k=10} groups of 8192 limbs, etc.  In practice it's
been found each @ma{k} is used at quite small multiples of its size constraint
and so the step effect is quite noticeable in a time versus size graph.

The threshold determinations currently measure at the mid-points of size
steps, but this is sub-optimal since at the start of a new step it can happen
that it's better to go back to the previous @ma{k} for a while.  Something
more sophisticated for @code{FFT_MUL_TABLE} and @code{FFT_SQR_TABLE} will be
needed.


@node Other Multiplication,  , FFT Multiplication, Multiplication Algorithms
@subsection Other Multiplication

The 3-way Toom-Cook algorithm described above (@pxref{Toom-Cook 3-Way
Multiplication}) generalizes to split into an arbitrary number of pieces, as
per Knuth section 4.3.3 algorithm C.  This is not currently used, though it's
possible a Toom-4 might fit in between Toom-3 and the FFTs.  The notes here
are merely for interest.

In general a split into @ma{r+1} pieces is made, and evaluations and pointwise
multiplications done at @m{2r+1,2*r+1} points.  A 4-way split does 7 pointwise
multiplies, 5-way does 9, etc.  Asymptotically an @ma{(r+1)}-way algorithm is
@m{O(N^{log(2r+1)/log(r+1)}, O(N^(log(2*r+1)/log(r+1)))}.  Only the pointwise
multiplications count towards big-@ma{O} complexity, but the time spent in the
evaluate and interpolate stages grows with @ma{r} and has a significant
practical impact, with the asymptotic advantage of each @ma{r} realized only
at bigger and bigger sizes.  The overheads grow as @m{O(Nr),O(N*r)}, whereas
in an @ma{r=2^k} FFT they grow only as @m{O(N \log r), O(N*log(r))}.

Knuth algorithm C evaluates at points 0,1,2,@dots{},@m{2r,2*r}, but exercise 4
uses @ma{-r},@dots{},0,@dots{},@ma{r} and the latter saves some small
multiplies in the evaluate stage (or rather trades them for additions), and
has a further saving of nearly half the interpolate steps.  The idea is to
separate odd and even final coefficients and then perform algorithm C steps C7
and C8 on them separately.  The divisors at step C7 become @ma{j^2} and the
multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}.

Splitting odd and even parts through positive and negative points can be
thought of as using @ma{-1} as a square root of unity.  If a 4th root of unity
was available then a further split and speedup would be possible, but no such
root exists for plain integers.  Going to complex integers with
@m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in cartesian
form it takes three real multiplies to do a complex multiply.  The existence
of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast
fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}.


@node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms
@section Division Algorithms
@cindex Division algorithms

@menu
* Single Limb Division::        
* Basecase Division::           
* Divide and Conquer Division::  
* Exact Division::              
* Exact Remainder::             
* Small Quotient Division::     
@end menu


@node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms
@subsection Single Limb Division

N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from
high to low, either with a hardware divide instruction or a multiplication by
inverse, whichever is best on a given CPU.

The multiply by inverse follows section 8 of ``Division by Invariant Integers
using Multiplication'' by Granlund and Montgomery (@pxref{References}) and is
implemented as @code{udiv_qrnnd_preinv} in @file{gmp-impl.h}.  The idea is to
have a fixed-point approximation to @ma{1/d} (see @code{invert_limb}) and then
multiply by the high limb (plus one bit) of the dividend to get a quotient
@ma{q}.  With @ma{d} normalized (high bit set), @ma{q} is no more than 1 too
small.  Subtracting @m{qd,q*d} from the dividend gives a remainder, and
reveals whether @ma{q} or @ma{q-1} is correct.

The result is a division done with two multiplications and four or five
arithmetic operations.  On CPUs with low latency multipliers this can be much
faster than a hardware divide, though the cost of calculating the inverse at
the start may mean it's only better on inputs bigger than say 4 or 5 limbs.

When a divisor must be normalized, either for the generic C
@code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is
actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @ma{a} is the dividend and
@ma{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set.  The
bit shifts for the dividend are usually accomplished ``on the fly'' meaning by
extracting the appropriate bits at each step.  Done this way the quotient
limbs come out aligned ready to store.  When only the remainder is wanted, an
alternative is to take the dividend limbs unshifted and calculate @m{r = a
\bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k \bmod
d2^k, r*2^k mod d*2^k}.  This can help on CPUs with poor bit shifts or few
registers.

The multiply by inverse can be done two limbs at a time.  The calculation is
basically the same, but the inverse is two limbs and the divisor treated as if
padded with a low zero limb.  This means more work, since the inverse will
need a 2@cross{}2 multiply, but the four 1@cross{}1s to do that are
independent and can therefore be done partly or wholly in parallel.  Likewise
for a 2@cross{}1 calculating @m{qd,q*d}.  The net effect is to process two
limbs with roughly the same two multiplies worth of latency that one limb at a
time gives.  This extends to 3 or 4 limbs at a time, though the extra work to
apply the inverse will almost certainly soon reach the limits of multiplier
throughput.

A similar approach in reverse can be taken to process just half a limb at a
time if the divisor is only a half limb.  In this case the 1@cross{}1 multiply
for the inverse effectively becomes two @m{1\over2@cross{}1, (1/2)x1} for each
limb, which can be a saving on CPUs with a fast half limb multiply, or in fact
if the only multiply is a half limb, and especially if it's not pipelined.


@node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms
@subsection Basecase Division

Basecase N@cross{}M division is like long division done by hand, but in base
@m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}.  See Knuth
section 4.3.1 algorithm D, and @file{mpn/generic/sb_divrem_mn.c}.

Briefly stated, while the dividend remains larger than the divisor, a high
quotient limb is formed and the N@cross{}1 product @m{qd,q*d} subtracted at
the top end of the dividend.  With a normalized divisor (most significant bit
set), each quotient limb can be formed with a 2@cross{}1 division and a
1@cross{}1 multiplication plus some subtractions.  The 2@cross{}1 division is
by the high limb of the divisor and is done either with a hardware divide or a
multiply by inverse (the same as in @ref{Single Limb Division}) whichever is
faster.  Such a quotient is sometimes one too big, requiring an addback of the
divisor, but that happens rarely.

With Q=N@minus{}M being the number of quotient limbs, this is an
@m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase
Q@cross{}M multiplication, differing in fact only in the extra multiply and
divide for each of the Q quotient limbs.


@node Divide and Conquer Division, Exact Division, Basecase Division, Division Algorithms
@subsection Divide and Conquer Division

For divisors larger than @code{DC_THRESHOLD}, division is done by dividing.
Or to be precise by a recursive divide and conquer algorithm based on work by
Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}).

The algorithm consists essentially of recognising that a 2N@cross{}N division
can be done with the basecase division algorithm (@pxref{Basecase Division}),
but using N/2 limbs as a base, not just a single limb.  This way the
multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of
Karatsuba and higher multiplication algorithms (@pxref{Multiplication
Algorithms}).  The ``digits'' of the quotient are formed by recursive
N@cross{}(N/2) divisions.

If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication
then the work is about the same as a basecase division, but with more function
call overheads and with some subtractions separated from the multiplies.
These overheads mean that it's only when N/2 is above
@code{KARATSUBA_MUL_THRESHOLD} that divide and conquer is of use.

@code{DC_THRESHOLD} is based on the divisor size N, so it will be somewhere
above twice @code{KARATSUBA_MUL_THRESHOLD}, but how much above depends on the
CPU.  An optimized @code{mpn_mul_basecase} can lower @code{DC_THRESHOLD} a
little by offering a ready-made advantage over repeated @code{mpn_submul_1}
calls.

Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where
@ma{M(N)} is the time for an N@cross{}N multiplication done with FFTs.  The
actual time is a sum over multiplications of the recursed sizes, as can be
seen near the end of section 2.2 of Burnikel and Ziegler.  For example, within
the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}.  With higher
algorithms the @ma{M(N)} term improves and the multiplier tends to @m{\log N,
log(N)}.  In practice, at moderate to large sizes, a 2N@cross{}N division is
about 2 to 4 times slower than an N@cross{}N multiplication.

Newton's method used for division is asymptotically @ma{O(M(N))} and should
therefore be superior to divide and conquer, but it's believed this would only
be for large to very large N.


@node Exact Division, Exact Remainder, Divide and Conquer Division, Division Algorithms
@subsection Exact Division

A so-called exact division is when the dividend is known to be an exact
multiple of the divisor.  Jebelean's exact division algorithm uses this
knowledge to make some significant optimizations (@pxref{References}).

The idea can be illustrated in decimal for example with 368154 divided by
543.  Because the low digit of the dividend is 4, the low digit of the
quotient must be 8.  This is arrived at from @m{4 \mathord{\times} 7 \bmod 10,
4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of
the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7
@equiv{} 1 mod 10}.  So @m{8\mathord{\times}543 = 4344,8*543=4344} can be
subtracted from the dividend leaving 363810.  Notice the low digit has become
zero.

The procedure is repeated at the second digit, with the next quotient digit 7
(@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting
@m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800.  And finally at
the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7
mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0.
So the quotient is 678.

Notice however that the multiplies and subtractions don't need to extend past
the low three digits of the dividend, since that's enough to determine the
three quotient digits.  For the last quotient digit no subtraction is needed
at all.  On a 2N@cross{}N division like this one, only about half the work of
a normal basecase division is necessary.

For an N@cross{}M exact division producing Q=N@minus{}M quotient limbs, the
saving over a normal basecase division is in two parts.  Firstly, each of the
Q quotient limbs needs only one multiply, not a 2@cross{}1 divide and
multiply.  Secondly, the crossproducts are reduced when @ma{Q>M} to
@m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @ma{Q@le{}M} to @m{Q(Q-1)/2,
Q*(Q-1)/2}.  Notice the savings are complementary.  If Q is big then many
divisions are saved, or if Q is small then the crossproducts reduce to a small
number.

The modular inverse used is calculated efficiently by @code{modlimb_invert} in
@file{gmp-impl.h}.  This does four multiplies for a 32-bit limb, or six for a
64-bit limb.  @file{tune/modlinv.c} has some alternate implementations that
might suit processors better at bit twiddling than multiplying.

The sub-quadratic exact division described by Jebelean in ``Exact Division
with Karatsuba Complexity'' is not currently implemented.  It uses a
rearrangement similar to the divide and conquer for normal division
(@pxref{Divide and Conquer Division}), but operating from low to high.  A
further possibility not currently implemented is ``Bidirectional Exact Integer
Division'' by Krandick and Jebelean which forms quotient limbs from both the
high and low ends of the dividend, and can halve once more the number of
crossproducts needed in a 2N@cross{}N division.

A special case exact division by 3 exists in @code{mpn_divexact_by3},
supporting Toom-3 multiplication and @code{mpq} canonicalizations.  It forms
quotient digits with a multiply by the modular inverse of 3 (which is
@code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next
limb.  The multiplications don't need to be on the dependent chain, as long as
the effect of the borrows is applied.  Only a few optimized assembler
implementations currently exist.


@node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms
@subsection Exact Remainder

If the exact division algorithm is done with a full subtraction at each stage
and the dividend isn't a multiple of the divisor, then low zero limbs are
produced but with a remainder in the high limbs.  For dividend @ma{a}, divisor
@ma{d}, quotient @ma{q}, and @m{b = 2 \GMPraise{@code{mp\_bits\_per\_limb}}, b
= 2^mp_bits_per_limb}, then this remainder @ma{r} is of the form
@tex
$$ a = qd + r b^n $$
@end tex
@ifnottex

@example
a = q*d + r*b^n
@end example

@end ifnottex
@ma{n} represents the number of zero limbs produced by the subtractions, that
being the number of limbs produced for @ma{q}.  @ma{r} will be in the range
@ma{0@le{}r<d} and can be viewed as a remainder, but one shifted up by a
factor of @ma{b^n}.

Carrying out full subtractions at each stage means the same number of cross
products must be done as a normal division, but there's still some single limb
divisions saved.  When @ma{d} is a single limb some simplifications arise,
providing good speedups on a number of processors.

@code{mpn_bdivmod}, @code{mpn_divexact_by3}, @code{mpn_modexact_1_odd} and the
@code{redc} function in @code{mpz_powm} differ subtly in how they return
@ma{r}, leading to some negations in the above formula, but all are
essentially the same.

Clearly @ma{r} is zero when @ma{a} is a multiple of @ma{d}, and this leads to
divisibility or congruence tests which are potentially more efficient than a
normal division.

The factor of @ma{b^n} on @ma{r} can be ignored in a GCD when @ma{d} is odd,
hence the use of @code{mpn_bdivmod} in @code{mpn_gcd}, and the use of
@code{mpn_modexact_1_odd} by @code{mpn_gcd_1} and @code{mpz_kronecker_ui} etc
(@pxref{Greatest Common Divisor Algorithms}).

Montgomery's REDC method for modular multiplications uses operands of the form
of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n})
(yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of @ma{b^n} in the exact
remainder to reach a product in the same form @m{(xy)b^{-n},
(x*y)*b^-n} (@pxref{Modular Powering Algorithm}).

Notice that @ma{r} generally gives no useful information about the ordinary
remainder @ma{a @bmod d} since @ma{b^n @bmod d} could be anything.  If however
@ma{b^n @equiv{} 1 @bmod d}, then @ma{r} is the negative of the ordinary
remainder.  This occurs whenever @ma{d} is a factor of @ma{b^n-1}, as for
example with 3 in @code{mpn_divexact_by3}.  Other such factors include 5, 17
and 257, but no particular use has been found for this.


@node Small Quotient Division,  , Exact Remainder, Division Algorithms
@subsection Small Quotient Division

An N@cross{}M division where the number of quotient limbs Q=N@minus{}M is
small can be optimized somewhat.

An ordinary basecase division normalizes the divisor by shifting it to make
the high bit set, shifting the dividend accordingly, and shifting the
remainder back down at the end of the calculation.  This is wasteful if only a
few quotient limbs are to be formed.  Instead a division of just the top
@m{\rm2Q,2*Q} limbs of the dividend by the top Q limbs of the divisor can be
used to form a trial quotient.  This requires only those limbs normalized, not
the whole of the divisor and dividend.

A multiply and subtract then applies the trial quotient to the M@minus{}Q
unused limbs of the divisor and N@minus{}Q dividend limbs (which includes Q
limbs remaining from the trial quotient division).  The starting trial
quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1
too big are detected by first comparing the most significant limbs that will
arise from the subtraction.  An addback is done if the quotient still turns
out to be 1 too big.

This whole procedure is essentially the same as one step of the basecase
algorithm done in a Q limb base, though with the trial quotient test done only
with the high limbs, not an entire Q limb ``digit'' product.  The correctness
of this weaker test can be established by following the argument of Knuth
section 4.3.1 exercise 20 but with the @m{v_2 \GMPhat q > b \GMPhat r
+ u_2, v2*q>b*r+u2} condition appropriately relaxed.


@need 1000
@node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms
@section Greatest Common Divisor
@cindex Greatest common divisor algorithms

@menu
* Binary GCD::                  
* Accelerated GCD::             
* Extended GCD::                
* Jacobi Symbol::               
@end menu


@node Binary GCD, Accelerated GCD, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms
@subsection Binary GCD

At small sizes GMP uses an @ma{O(N^2)} binary style GCD.  This is described in
many textbooks, for example Knuth section 4.5.2 algorithm B.  It simply
consists of successively reducing operands @ma{a} and @ma{b} using
@ma{@gcd{}(a,b) = @gcd{}(@min{}(a,b),@abs{}(a-b))}, and also that if @ma{a}
and @ma{b} are first made odd then @ma{@abs{}(a-b)} is even and factors of two
can be discarded.

Variants like letting @ma{a-b} become negative and doing a different next step
are of interest only as far as they suit particular CPUs, since on small
operands it's machine dependent factors that determine performance.

The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using
@ma{a @bmod b} but this has so far been found to be slower everywhere.  One
reason the binary method does well is that the implied quotient at each step
is usually small, so often only one or two subtractions are needed to get the
same effect as a division.  Quotients 1, 2 and 3 for example occur 67.7% of
the time, see Knuth section 4.5.3 Theorem E.

When the implied quotient is large, meaning @ma{b} is much smaller than
@ma{a}, then a division is worthwhile.  This is the basis for the initial
@ma{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter
for both N@cross{}1 and 1@cross{}1 cases).  But after that initial reduction,
big quotients occur too rarely to make it worth checking for them.


@node Accelerated GCD, Extended GCD, Binary GCD, Greatest Common Divisor Algorithms
@subsection Accelerated GCD

For sizes above @code{GCD_ACCEL_THRESHOLD}, GMP uses the Accelerated GCD
algorithm described independently by Weber and Jebelean (the latter as the
``Generalized Binary'' algorithm), @pxref{References}.  This algorithm is
still @ma{O(N^2)}, but is much faster than the binary algorithm since it does
fewer multi-precision operations.  It consists of alternating the @ma{k}-ary
reduction by Sorenson, and a ``dmod'' exact remainder reduction.

For operands @ma{u} and @ma{v} the @ma{k}-ary reduction replaces @ma{u} with
@m{nv-du,n*v-d*u} where @ma{n} and @ma{d} are single limb values chosen to
give two trailing zero limbs on that value, which can be stripped.  @ma{n} and
@ma{d} are calculated using an algorithm similar to half of a two limb GCD
(see @code{find_a} in @file{mpn/generic/gcd.c}).

When @ma{u} and @ma{v} differ in size by more than a certain number of bits, a
dmod is performed to zero out bits at the low end of the larger.  It consists
of an exact remainder style division applied to an appropriate number of bits
(@pxref{Exact Division}, and @pxref{Exact Remainder}).  This is faster than a
@ma{k}-ary reduction but useful only when the operands differ in size.
There's a dmod after each @ma{k}-ary reduction, and if the dmod leaves the
operands still differing in size then it's repeated.

The @ma{k}-ary reduction step can introduce spurious factors into the GCD
calculated, and these are eliminated at the end by taking GCDs with the
original inputs @ma{@gcd{}(u,@gcd{}(v,g))} using the binary algorithm.  Since
@ma{g} is almost always small this takes very little time.

At small sizes the algorithm needs a good implementation of @code{find_a}.  At
larger sizes it's dominated by @code{mpn_addmul_1} applying @ma{n} and @ma{d}.


@node Extended GCD, Jacobi Symbol, Accelerated GCD, Greatest Common Divisor Algorithms
@subsection Extended GCD

The extended GCD calculates @ma{@gcd{}(a,b)} and also cofactors @ma{x} and
@ma{y} satisfying @m{ax+by=\gcd(a@C{}b), a*x+b*y=gcd(a@C{}b)}.  Lehmer's
multi-step improvement of the extended Euclidean algorithm is used.  See Knuth
section 4.5.2 algorithm L, and @file{mpn/generic/gcdext.c}.  This is an
@ma{O(N^2)} algorithm.

The multipliers at each step are found using single limb calculations for
sizes up to @code{GCDEXT_THRESHOLD}, or double limb calculations above that.
The single limb code is faster but doesn't produce full-limb multipliers,
hence not making full use of the @code{mpn_addmul_1} calls.

When a CPU has a data-dependent multiplier, meaning one which is faster on
operands with fewer bits, the extra work in the double-limb calculation might
only save some looping overheads, leading to a large @code{GCDEXT_THRESHOLD}.

Currently the single limb calculation doesn't optimize for the small quotients
that often occur, and this can lead to unusually low values of
@code{GCDEXT_THRESHOLD}, depending on the CPU.

An analysis of double-limb calculations can be found in ``A Double-Digit
Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}).  The code in GMP
was developed independently.

It should be noted that when a double limb calculation is used, it's used for
the whole of that GCD, it doesn't fall back to single limb part way through.
This is because as the algorithm proceeds, the inputs @ma{a} and @ma{b} are
reduced, but the cofactors @ma{x} and @ma{y} grow, so the multipliers at each
step are applied to a roughly constant total number of limbs.


@node Jacobi Symbol,  , Extended GCD, Greatest Common Divisor Algorithms
@subsection Jacobi Symbol

@code{mpz_jacobi} and @code{mpz_kronecker} are currently implemented with a
simple binary algorithm similar to that described for the GCDs (@pxref{Binary
GCD}).  They're not very fast when both inputs are large.  Lehmer's multi-step
improvement or a binary based multi-step algorithm is likely to be better.

When one operand fits a single limb, and that includes @code{mpz_kronecker_ui}
and friends, an initial reduction is done with either @code{mpn_mod_1} or
@code{mpn_modexact_1_odd}, followed by the binary algorithm on a single limb.
The binary algorithm is well suited to a single limb, and the whole
calculation in this case is quite efficient.

In all the routines sign changes for the result are accumulated using some bit
twiddling, avoiding table lookups or conditional jumps.


@need 1000
@node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms
@section Powering Algorithms
@cindex Powering algorithms

@menu
* Normal Powering Algorithm::   
* Modular Powering Algorithm::  
@end menu


@node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms
@subsection Normal Powering

Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is seen in
the exponent, as per Knuth section 4.6.3.  The ``left to right''
variant described there is used rather than algorithm A, since it's just as
easy and can be done with somewhat less temporary memory.


@node Modular Powering Algorithm,  , Normal Powering Algorithm, Powering Algorithms
@subsection Modular Powering

Modular powering is implemented using a @ma{2^k}-ary sliding window algorithm,
as per ``Handbook of Applied Cryptography'' algorithm 14.85
(@pxref{References}).  @ma{k} is chosen according to the size of the exponent.
Larger exponents use larger values of @ma{k}, the choice being made to
minimize the average number of multiplications that must supplement the
squaring.

The modular multiplies and squares use either a simple division or the REDC
method by Montgomery (@pxref{References}).  REDC is a little faster,
essentially saving N single limb divisions in a fashion similar to an exact
remainder (@pxref{Exact Remainder}).  The current REDC has some limitations.
It's only @ma{O(N^2)} so above @code{POWM_THRESHOLD} division becomes faster
and is used.  It doesn't attempt to detect small bases, but rather always uses
a REDC form, which is usually a full size operand.  And lastly it's only
applied to odd moduli.


@node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms
@section Root Extraction Algorithms
@cindex Root extraction algorithms

@menu
* Square Root Algorithm::       
* Nth Root Algorithm::          
* Perfect Square Algorithm::    
* Perfect Power Algorithm::     
@end menu


@node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms
@subsection Square Root

Square roots are taken using the ``Karatsuba Square Root'' algorithm by Paul
Zimmermann (@pxref{References}).  This is expressed in a divide and conquer
form, but as noted in the paper it can also be viewed as a discrete variant of
Newton's method.

In the Karatsuba multiplication range this is an @m{O({3\over2}
M(N/2)),O(1.5*M(N/2))} algorithm, where @ma{M(n)} is the time to multiply two
numbers of @ma{n} limbs.  In the FFT multiplication range this grows to a
bound of @m{O(6 M(N/2)),O(6*M(N/2))}.  In practice a factor of about 1.5 to
1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT
range.

The algorithm does all its calculations in integers and the resulting
@code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}.
The extended precision given by @code{mpf_sqrt_ui} is obtained by
padding with zero limbs.


@node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms
@subsection Nth Root

Integer Nth roots are taken using Newton's method with the following
iteration, where @ma{A} is the input and @ma{n} is the root to be taken.
@tex
$$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$
@end tex
@ifnottex

@example
         1         A
a[i+1] = - * ( --------- + (n-1)*a[i] )
         n     a[i]^(n-1)
@end example

@end ifnottex
The initial approximation @m{a_1,a[1]} is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just above the
true root.  The iteration converges quadratically when started from a good
approximation.  When @ma{n} is large more initial bits are needed to get good
convergence.  The current implementation is not particularly well optimized.


@node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms
@subsection Perfect Square

@code{mpz_perfect_square_p} is able to quickly exclude most non-squares by
checking whether the input is a quadratic residue modulo some small integers.

The first test is modulo 256 which means simply examining the least
significant byte.  Only 44 different values occur as the low byte of a square,
so 82.8% of non-squares can be immediately excluded.  Similar tests modulo
primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits
then primes up to 53 are used, excluding 99.99%.  A single N@cross{}1
remainder using @code{PP} from @file{gmp-impl.h} quickly gives all these
remainders.

A square root must still be taken for any value that passes the residue tests,
to verify it's really a square and not one of the 0.086% (or 0.000156% for 64
bits) non-squares that get through.  @xref{Square Root Algorithm}.


@node Perfect Power Algorithm,  , Perfect Square Algorithm, Root Extraction Algorithms
@subsection Perfect Power

Detecting perfect powers is required by some factorization algorithms.
Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(@xref{Nth Root Algorithm}.)

If a prime divisor @ma{p} with multiplicity @ma{e} can be found, then only
roots which are divisors of @ma{e} need to be considered, much reducing the
work necessary.  To this end divisibility by a set of small primes is checked.


@node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms
@section Radix Conversion
@cindex Radix conversion algorithms

Radix conversions are less important than other algorithms.  A program
dominated by conversions should probably use a different data representation.

@menu
* Binary to Radix::             
* Radix to Binary::             
@end menu


@node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms
@subsection Binary to Radix

Conversions from binary to a power-of-2 radix use a simple and fast @ma{O(N)}
bit extraction algorithm.

Conversions from binary to other radices use repeated divisions, first by the
biggest power of the radix that fits in a single limb, then by the radix on
the remainders.  This is an @ma{O(N^2)} algorithm and can be quite
time-consuming on large inputs.


@node Radix to Binary,  , Binary to Radix, Radix Conversion Algorithms
@subsection Radix to Binary

Conversions from a power-of-2 radix into binary use a simple and fast
@ma{O(N)} bitwise concatenation algorithm.

Conversions from other radices use repeated multiplications, first
accumulating as many digits as fit in a limb, then doing an N@cross{}1
multi-precision multiplication.  This is @ma{O(N^2)} and is certainly
sub-optimal on sizes above the Karatsuba multiply threshold.


@need 1000
@node Other Algorithms, Assembler Coding, Radix Conversion Algorithms, Algorithms
@section Other Algorithms

@menu
* Factorial Algorithm::         
* Binomial Coefficients Algorithm::  
* Fibonacci Numbers Algorithm::  
* Lucas Numbers Algorithm::     
@end menu


@node Factorial Algorithm, Binomial Coefficients Algorithm, Other Algorithms, Other Algorithms
@subsection Factorial

Factorials @ma{n!} are calculated by a simple product from @ma{1} to @ma{n},
but arranged into certain sub-products.

First as many factors as fit in a limb are accumulated, then two of those
multiplied to give a 2-limb product.  When two 2-limb products are ready
they're multiplied to a 4-limb product, and when two 4-limbs are ready they're
multiplied to an 8-limb product, etc.  A stack of outstanding products is
built up, with two of the same size multiplied together when ready.

Arranging for multiplications to have operands the same (or nearly the same)
size means the Karatsuba and higher multiplication algorithms can be used.
And even on sizes below the Karatsuba threshold an N@cross{}N multiply will
give a basecase multiply more to work on.

An obvious improvement not currently implemented would be to strip factors of
2 from the products and apply them at the end with a bit shift.  Another
possibility would be to determine the prime factorization of the result (which
can be done easily), and use a powering method, at each stage squaring then
multiplying in those primes with a 1 in their exponent at that point.  The
advantage would be some multiplies turned into squares.


@node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms
@subsection Binomial Coefficients

Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated
by first arranging @ma{k @le{} n/2} using @m{\left({n}\atop{k}\right) =
\left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then
evaluating the following product simply from @ma{i=2} to @ma{i=k}.
@tex
$$ \left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i} $$
@end tex
@ifnottex

@example
                      k  (n-k+i)
C(n,k) =  (n-k+1) * prod -------
                     i=2    i
@end example

@end ifnottex
It's easy to show that each denominator @ma{i} will divide the product so far,
so the exact division algorithm is used (@pxref{Exact Division}).

The numerators @ma{n-k+i} and denominators @ma{i} are first accumulated into
as many fit a limb, to save multi-precision operations, though for
@code{mpz_bin_ui} this applies only to the divisors, since @ma{n} is an
@code{mpz_t} and @ma{n-k+i} in general won't fit in a limb at all.

An obvious improvement would be to strip factors of 2 from each multiplier and
divisor and count them separately, to be applied with a bit shift at the end.
Factors of 3 and perhaps 5 could even be handled similarly.  Another
possibility, if @ma{n} is not too big, would be to determine the prime
factorization of the result based on the factorials involved, and power up
those primes appropriately.  This would help most when @ma{k} is near
@ma{n/2}.


@node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms
@subsection Fibonacci Numbers

The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed
for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]}
values efficiently.

For small @ma{n}, a table of single limb values in @code{__gmp_fib_table} is
used.  On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb
up to @m{F_{93},F[93]}.  For convenience the table starts at @m{F_{-1},F[-1]}.

Beyond the table, values are generated with a binary powering algorithm,
calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to
low across the bits of @ma{n}.  The formulas used are
@tex
$$\eqalign{
  F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr
  F_{2k-1} &=  F_k^2 + F_{k-1}^2           \cr
  F_{2k}   &= F_{2k+1} - F_{2k-1}
}$$
@end tex
@ifnottex

@example
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
F[2k-1] =   F[k]^2 + F[k-1]^2

F[2k] = F[2k+1] - F[2k-1]
@end example

@end ifnottex
At each step, @ma{k} is the high @ma{b} bits of @ma{n}.  If the next bit of
@ma{n} is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if it's a 1
then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process repeated
until all bits of @ma{n} are incorporated.  Notice these formulas require just
two squares per bit of @ma{n}.

It'd be possible to handle the first few @ma{n} above the single limb table
with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} =
F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually
turns out to be faster for only about 10 or 20 values of @ma{n}, and including
a block of code for just those doesn't seem worthwhile.  If they really
mattered it'd be better to extend the data table.

Using a table avoids lots of calculations on small numbers, and makes small
@ma{n} go fast.  A bigger table would make more small @ma{n} go fast, it's
just a question of balancing size against desired speed.  For GMP the code is
kept compact, with the emphasis primarily on a good powering algorithm.

@code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but
@code{mpz_fib_ui} is only interested in @m{F_n,F[n]}.  In this case the last
step of the algorithm can become one multiply instead of two squares.  One of
the following two formulas is used, according as @ma{n} is odd or even.
@tex
$$\eqalign{
  F_{2k}   &= F_k (F_k + 2F_{k-1}) \cr
  F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k
}$$
@end tex
@ifnottex

@example
F[2k]   = F[k]*(F[k]+2F[k-1])

F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
@end example

@end ifnottex
@m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a
multiply.  For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above
can be applied just to the low limb of the calculation, without a carry or
borrow into further limbs, which saves some code size.  See comments with
@code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done.


@node Lucas Numbers Algorithm,  , Fibonacci Numbers Algorithm, Other Algorithms
@subsection Lucas Numbers

@code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci
numbers with the following simple formulas.
@tex
$$\eqalign{
  L_k     &=  F_k + 2F_{k-1} \cr
  L_{k-1} &= 2F_k -  F_{k-1}
}$$
@end tex
@ifnottex

@example
L[k]   =   F[k] + 2*F[k-1]
L[k-1] = 2*F[k] -   F[k-1]
@end example

@end ifnottex
@code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be
saved.  Trailing zero bits on @ma{n} can be handled with a single square each.
@tex
$$ L_{2k} = L_k^2 - 2(-1)^k $$
@end tex
@ifnottex

@example
L[2k] = L[k]^2 - 2*(-1)^k
@end example

@end ifnottex
And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
numbers, similar to what @code{mpz_fib_ui} does.
@tex
$$ L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k $$
@end tex
@ifnottex

@example
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
@end example

@end ifnottex


@node Assembler Coding,  , Other Algorithms, Algorithms
@section Assembler Coding

The assembler subroutines in GMP are the most significant source of speed at
small to moderate sizes.  At larger sizes algorithm selection becomes more
important, but of course speedups in low level routines will still speed up
everything proportionally.

Carry handling and widening multiplies that are important for GMP can't be
easily expressed in C.  GCC @code{asm} blocks help a lot and are provided in
@file{longlong.h}, but hand coding low level routines invariably offers a
speedup over generic C by a factor of anything from 2 to 10.

@menu
* Assembler Code Organisation::  
* Assembler Basics::            
* Assembler Carry Propagation::  
* Assembler Cache Handling::    
* Assembler Floating Point::    
* Assembler SIMD Instructions::  
* Assembler Software Pipelining::  
* Assembler Loop Unrolling::    
@end menu


@node Assembler Code Organisation, Assembler Basics, Assembler Coding, Assembler Coding
@subsection Code Organisation

The various @file{mpn} subdirectories contain machine-dependent code, written
in C or assembler.  The @file{mpn/generic} subdirectory contains default code,
used when there's no machine-specific version of a particular file.

Each @file{mpn} subdirectory is for an ISA family.  Generally 32-bit and
64-bit variants in a family cannot share code and will have separate
directories.  Within a family further subdirectories may exist for CPU
variants.


@node Assembler Basics, Assembler Carry Propagation, Assembler Code Organisation, Assembler Coding
@subsection Assembler Basics

@code{mpn_addmul_1} and @code{mpn_submul_1} are the most important routines
for overall GMP performance.  All multiplications and divisions come down to
repeated calls to these.  @code{mpn_add_n}, @code{mpn_sub_n},
@code{mpn_lshift} and @code{mpn_rshift} are next most important.

On some CPUs assembler versions of the internal functions
@code{mpn_mul_basecase} and @code{mpn_sqr_basecase} give significant speedups,
mainly through avoiding function call overheads.  They can also potentially
make better use of a wide superscalar processor.

The restrictions on overlaps between sources and destinations
(@pxref{Low-level Functions}) are designed to facilitate a variety of
implementations.  For example, knowing @code{mpn_add_n} won't have partly
overlapping sources and destination means reading can be done far ahead of
writing on superscalar processors, and loops can be vectorized on a vector
processor, depending on the carry handling.


@node Assembler Carry Propagation, Assembler Cache Handling, Assembler Basics, Assembler Coding
@subsection Carry Propagation

The problem that presents most challenges in GMP is propagating carries from
one limb to the next.  In functions like @code{mpn_addmul_1} and
@code{mpn_add_n}, carries are the only dependencies between limb operations.

On processors with carry flags, a straightforward CISC style @code{adc} is
generally best.  AMD K6 @code{mpn_addmul_1} however is an example of an
unusual set of circumstances where a branch works out better.

On RISC processors generally an add and compare for overflow is used.  This
sort of thing can be seen in @file{mpn/generic/aors_n.c}.  Some carry
propagation schemes require 4 instructions, meaning at least 4 cycles per
limb, but other schemes may use just 1 or 2.  On wide superscalar processors
performance may be completely determined by the number of dependent
instructions between carry-in and carry-out for each limb.

On vector processors good use can be made of the fact that a carry bit only
very rarely propagates more than one limb.  When adding a single bit to a
limb, there's only a carry out if that limb was @code{0xFF...FF} which on
random data will be only 1 in @m{2\GMPraise{@code{mp\_bits\_per\_limb}},
2^mp_bits_per_limb}.  @file{mpn/cray/add_n.c} is an example of this, it adds
all limbs in parallel, adds one set of carry bits in parallel and then only
rarely needs to fall through to a loop propagating further carries.

On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
for the RISC style idioms that are necessary to handle carry bits in
C.  Often conditional jumps are generated where @code{adc} or @code{sbb} forms
would be better.  And so unfortunately almost any loop involving carry bits
needs to be coded in assembler for best results.


@node Assembler Cache Handling, Assembler Floating Point, Assembler Carry Propagation, Assembler Coding
@subsection Cache Handling

GMP aims to perform well both on operands that fit entirely in L1 cache and
those that don't.  In the assembler subroutines this means prefetching, either
always or when large enough operands are presented.

Pre-fetching sources combines well with loop unrolling, since a prefetch can
be initiated once per unrolled loop (or more than once if the loop processes
more than one cache line).

Pre-fetching destinations won't be necessary if the CPU has a big enough store
queue.  Older processors without a write-allocate L1 however will want
destination prefetching, to avoid repeated write-throughs, unless they can
keep up with the rate at which destination limbs are produced.

The distance ahead to prefetch will be determined by the rate data is
processed versus the time it takes to bring a line up to L1.  Naturally the
net data rate from L2 or RAM will always limit the rate of data processing.
Prefetch distance may also be limited by the number of prefetches the
processor can have in progress at any one time.

If a special prefetch instruction doesn't exist then a plain load can be used,
so long as the CPU supports out-of-order loads.  But this may mean having a
second copy of a loop so that the last few limbs can be processed without
prefetching, since reading past the end of an operand must be avoided.


@node Assembler Floating Point, Assembler SIMD Instructions, Assembler Cache Handling, Assembler Coding
@subsection Floating Point

Floating point arithmetic is used in GMP for multiplications on CPUs with poor
integer multipliers.  Floating point generally doesn't suit other operations
like additions or shifts, due to difficulties implementing carry handling.

With IEEE 53-bit double precision floats, integer multiplications producing up
to 53 bits will give exact results.  Breaking a multiplication into
16@cross{}@ma{32@rightarrow{}48} bit pieces is convenient.  With some care
though three 21@cross{}@ma{32@rightarrow{}53} bit products can be used to do a
64@cross{}32 multiply, if one of those 21@cross{}32 parts uses the sign bit.

Generally limbs want to be treated as unsigned, but on some CPUs floating
point conversions only treat integers as signed.  Copying through a zero
extended memory region or testing and adjusting for a sign bit may be
necessary.

Currently floating point FFTs aren't used for large multiplications.  On some
processors they probably have a good chance of being worthwhile, if great care
is taken with precision control.


@node Assembler SIMD Instructions, Assembler Software Pipelining, Assembler Floating Point, Assembler Coding
@subsection SIMD Instructions

The single-instruction multiple-data support in current microprocessors is
aimed at signal processing algorithms where each data point can be treated
more or less independently.  There's generally not much support for
propagating the sort of carries that arise in GMP.

SIMD multiplications of say four 16@cross{}16 bit multiplies only do as much
work as one 32@cross{}32 from GMP's point of view, and need some shifts and
adds besides.  But of course if say the SIMD form is fully pipelined and uses
less instruction decoding then it may still be worthwhile.

On the 80x86 chips, MMX has so far found a use in @code{mpn_rshift} and
@code{mpn_lshift} since it allows 64-bit operations, and is used in a special
case for 16-bit multipliers in the P55 @code{mpn_mul_1}.  3DNow and SSE
haven't found a use so far.


@node Assembler Software Pipelining, Assembler Loop Unrolling, Assembler SIMD Instructions, Assembler Coding
@subsection Software Pipelining

Software pipelining consists of scheduling instructions around the branch
point in a loop.  For example a loop taking a checksum of an array of limbs
might have a load and an add, but the load wouldn't be for that add, rather
for the one next time around the loop.  Each load then is effectively
scheduled back in the previous iteration, allowing latency to be hidden.

Naturally this is wanted only when doing things like loads or multiplies that
take a few cycles to complete, and only where a CPU has multiple functional
units so that other work can be done while waiting.

A pipeline with several stages will have a data value in progress at each
stage and each loop iteration moves them along one stage.  This is like
juggling.

Within the loop some moves between registers may be necessary to have the
right values in the right places for each iteration.  Loop unrolling can help
this, with each unrolled block able to use different registers for different
values, even if some shuffling is still needed just before going back to the
top of the loop.


@node Assembler Loop Unrolling,  , Assembler Software Pipelining, Assembler Coding
@subsection Loop Unrolling

Loop unrolling consists of replicating code so that several limbs are
processed in each loop.  At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for example
alternately using one register combination and then another.  Judicious use of
@command{m4} macros can help avoid lots of duplication in the source code.

Unrolling is commonly done to a power of 2 multiple so the number of unrolled
loops and the number of remaining limbs can be calculated with a shift and
mask.  But other multiples can be used too, just by subtracting each @var{n}
limbs processed from a counter and waiting for less than @var{n} remaining (or
offsetting the counter by @var{n} so it goes negative when there's less than
@var{n} remaining).

The limbs not a multiple of the unrolling can be handled in various ways, for
example

@itemize @bullet
@item
A simple loop at the end (or the start) to process the excess.  Care will be
wanted that it isn't too much slower than the unrolled part.

@item
A set of binary tests, for example after an 8-limb unrolling, test for 4 more
limbs to process, then a further 2 more or not, and finally 1 more or not.
This will probably take more code space than a simple loop.

@item
A @code{switch} statement, providing separate code for each possible excess,
for example an 8-limb unrolling would have separate code for 0 remaining, 1
remaining, etc, up to 7 remaining.  This might take a lot of code, but may be
the best way to optimize all cases in combination with a deep pipelined loop.

@item
A computed jump into the middle of the loop, thus making the first iteration
handle the excess.  This should make times smoothly increase with size, which
is attractive, but setups for the jump and adjustments for pointers can be
tricky and could become quite difficult in combination with deep pipelining.
@end itemize

One way to write the setups and finishups for a pipelined unrolled loop is
simply to duplicate the loop at the start and the end, then delete
instructions at the start which have no valid antecedents, and delete
instructions at the end whose results are unwanted.  Sizes not a multiple of
the unrolling can then be handled as desired.


@node Internals, Contributors, Algorithms, Top
@chapter Internals

@strong{This chapter is provided only for informational purposes and the
various internals described here may change in future GMP releases.
Applications expecting to be compatible with future releases should use only
the documented interfaces described in previous chapters.}

@menu
* Integer Internals::           
* Rational Internals::          
* Float Internals::             
* Raw Output Internals::        
* C++ Interface Internals::     
@end menu

@node Integer Internals, Rational Internals, Internals, Internals
@section Integer Internals

@code{mpz_t} variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated.  The fields are as follows.

@table @asis
@item @code{_mp_size}
The number of limbs, or the negative of that when representing a negative
integer.  Zero is represented by @code{_mp_size} set to zero, in which case
the @code{_mp_d} data is unused.

@item @code{_mp_d}
A pointer to an array of limbs which is the magnitude.  These are stored
``little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the
least significant limb and @code{_mp_d[ABS(_mp_size)-1]} is the most
significant.  Whenever @code{_mp_size} is non-zero, the most significant limb
is non-zero.

Currently there's always at least one limb allocated, so for instance
@code{mpz_set_ui} never needs to reallocate, and @code{mpz_get_ui} can fetch
@code{_mp_d[0]} unconditionally (though its value is then only wanted if
@code{_mp_size} is non-zero).

@item @code{_mp_alloc}
@code{_mp_alloc} is the number of limbs currently allocated at @code{_mp_d},
and naturally @code{_mp_alloc >= ABS(_mp_size)}.  When an @code{mpz} routine
is about to (or might be about to) increase @code{_mp_size}, it checks
@code{_mp_alloc} to see whether there's enough space, and reallocates if not.
@code{MPZ_REALLOC} is generally used for this.
@end table

The various bitwise logical functions like @code{mpz_and} behave as if
negative values were twos complement.  But sign and magnitude is always used
internally, and necessary adjustments are made during the calculations.
Sometimes this isn't pretty, but sign and magnitude are best for other
routines.

Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these
have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory
allocation functions.  Care is taken to ensure that these are big enough that
no reallocation is necessary (since it would have unpredictable consequences).


@node Rational Internals, Float Internals, Integer Internals, Internals
@section Rational Internals

@code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and
denominator (@pxref{Integer Internals}).

The canonical form adopted is denominator positive (and non-zero), no common
factors between numerator and denominator, and zero uniquely represented as
0/1.

It's believed that casting out common factors at each stage of a calculation
is best in general.  A GCD is an @ma{O(N^2)} operation so it's better to do a
few small ones immediately than to delay and have to do a big one later.
Knowing the numerator and denominator have no common factors can be used for
example in @code{mpq_mul} to make only two cross GCDs necessary, not four.

This general approach to common factors is badly sub-optimal in the presence
of simple factorizations or little prospect for cancellation, but GMP has no
way to know when this will occur.  As per @ref{Efficiency}, that's left to
applications.  The @code{mpq_t} framework might still suit, with
@code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and
denominator, or of course @code{mpz_t} variables can be used directly.


@node Float Internals, Raw Output Internals, Rational Internals, Internals
@section Float Internals

Efficient calculation is the primary aim of GMP floats and the use of whole
limbs and simple rounding facilitates this.

@code{mpf_t} floats have a variable precision mantissa and a single machine
word signed exponent.  The mantissa is represented using sign and magnitude.

@c FIXME: The arrow heads don't join to the lines exactly.
@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
\GMPdisplay{%
\vbox{%
  \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb}
  \vskip 0.7ex
  \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
  \hbox {
    \hbox to 3\GMPboxwidth {%
      \setbox 0 = \hbox{@code{\_mp\_exp}}%
      \dimen0=3\GMPboxwidth
      \advance\dimen0 by -\wd0
      \divide\dimen0 by 2
      \advance\dimen0 by -1em
      \setbox1 = \hbox{$\rightarrow$}%
      \dimen1=\dimen0
      \advance\dimen1 by -\wd1
      \GMPcentreline{\dimen0}%
      \hfil
      \box0%
      \hfil
      \GMPcentreline{\dimen1{}}%
      \box1}
    \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}}
  \vskip 0.5ex
  \vbox {%
    \hrule
    \hbox{%
      \vrule height 2ex depth 1ex
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule}
    \hrule
  }
  \hbox {%
    \hbox to 0.8 pt {}
    \hbox to 3\GMPboxwidth {%
      \hfil $\cdot$} \hbox {$\leftarrow$ radix point\hfil}}
  \hbox to 5\GMPboxwidth{%
    \setbox 0 = \hbox{@code{\_mp\_size}}%
    \dimen0 = 5\GMPboxwidth
    \advance\dimen0 by -\wd0
    \divide\dimen0 by 2
    \advance\dimen0 by -1em
    \dimen1 = \dimen0
    \setbox1 = \hbox{$\leftarrow$}%
    \setbox2 = \hbox{$\rightarrow$}%
    \advance\dimen0 by -\wd1
    \advance\dimen1 by -\wd2
    \hbox to 0.3 em {}%
    \box1
    \GMPcentreline{\dimen0}%
    \hfil
    \box0
    \hfil
    \GMPcentreline{\dimen1}%
    \box2}
}}
@end tex
@ifnottex
@example
   most                   least
significant            significant
   limb                   limb

                            _mp_d
 |---- _mp_exp --->           |
  _____ _____ _____ _____ _____
 |_____|_____|_____|_____|_____|
                   . <------------ radix point

  <-------- _mp_size --------->
@sp 1
@end example
@end ifnottex

@noindent
The fields are as follows.

@table @asis
@item @code{_mp_size}
The number of limbs currently in use, or the negative of that when
representing a negative value.  Zero is represented by @code{_mp_size} and
@code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is
unused.  (In the future @code{_mp_exp} might be undefined when representing
zero.)

@item @code{_mp_prec}
The precision of the mantissa, in limbs.  In any calculation the aim is to
produce @code{_mp_prec} limbs of result (the most significant being non-zero).

@item @code{_mp_d}
A pointer to the array of limbs which is the absolute value of the mantissa.
These are stored ``little endian'' as per the @code{mpn} functions, so
@code{_mp_d[0]} is the least significant limb and
@code{_mp_d[ABS(_mp_size)-1]} the most significant.

The most significant limb is always non-zero, but there are no other
restrictions on its value, in particular the highest 1 bit can be anywhere
within the limb.

@code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being
for convenience (see below).  There are no reallocations during a calculation,
only in a change of precision with @code{mpf_set_prec}.

@item @code{_mp_exp}
The exponent, in limbs, determining the location of the implied radix point.
Zero means the radix point is just above the most significant limb.  Positive
values mean a radix point offset towards the lower limbs and hence a value
@ma{@ge{} 1}, as for example in the diagram above.  Negative exponents mean a
radix point further above the highest limb.

Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data
are treated as zero.
@end table

@sp 1
@noindent
The following various points should be noted.

@table @asis
@item Low Zeros
The least significant limbs @code{_mp_d[0]} etc can be zero, though such low
zeros can always be ignored.  Routines likely to produce low zeros check and
avoid them to save time in subsequent calculations, but for most routines
they're quite unlikely and aren't checked.

@item Mantissa Size Range
The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if
the value can be represented in less.  This means low precision values or
small integers stored in a high precision @code{mpf_t} can still be operated
on efficiently.

@code{_mp_size} can also be greater than @code{_mp_prec}.  Firstly a value is
allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d},
and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves
@code{_mp_size} unchanged and so the size can be arbitrarily bigger than
@code{_mp_prec}.

@item Rounding
All rounding is done on limb boundaries.  Calculating @code{_mp_prec} limbs
with the high non-zero will ensure the application requested minimum precision
is obtained.

The use of simple ``trunc'' rounding towards zero is efficient, since there's
no need to examine extra limbs and increment or decrement.

@item Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic operations
like @code{mpf_add} and @code{mpf_mul}.  When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.

Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.

@item Use of @code{_mp_prec+1} Limbs
The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just
@code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its
operation.  @code{mpf_add} for instance will do an @code{mpn_add} of
@code{_mp_prec} limbs.  If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
@code{_mp_size} becomes @code{_mp_prec+1}.

Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not
needed for the intended precision, only the @code{_mp_prec} high limbs.  But
zeroing it out or moving the rest down is unnecessary.  Subsequent routines
reading the value will simply take the high limbs they need, and this will be
@code{_mp_prec} if their target has that same precision.  This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.

Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs
if available.  This ensures that a variable which has @code{_mp_size} equal to
@code{_mp_prec+1} will get its full exact value copied.  Strictly speaking
this is unnecessary since only @code{_mp_prec} limbs are needed for the
application's requested precision, but it's considered that an @code{mpf_set}
from one variable into another of the same precision ought to produce an exact
copy.

@item Application Precisions
@code{__GMPF_BITS_TO_PREC} converts an application requested precision to an
@code{_mp_prec}.  The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of @code{_mp_d} is only
non-zero and therefore might contain only one bit.

@code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra
limb from @code{_mp_prec} before converting to bits.  The net effect of
reading back with @code{mpf_get_prec} is simply the precision rounded up to a
multiple of @code{mp_bits_per_limb}.

Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to @code{_mp_d}.  For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
@code{_mp_prec} of 9.  @code{_mp_d} then gets 10 limbs allocated.  Reading
back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and
multiply by 32, giving 256 bits.

Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice
multiple of the limb size.
@end table


@node Raw Output Internals, C++ Interface Internals, Float Internals, Internals
@section Raw Output Internals

@noindent
@code{mpz_out_raw} uses the following format.

@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
\GMPdisplay{%
\vbox{%
  \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
  \vbox {%
    \hrule
    \hbox{%
      \vrule height 2.5ex depth 1.5ex
      \hbox to \GMPboxwidth {\hfil size\hfil}%
      \vrule
      \hbox to 3\GMPboxwidth {\hfil data bytes\hfil}%
      \vrule}
    \hrule}
}}
@end tex
@ifnottex
@example
+------+------------------------+
| size |       data bytes       |
+------+------------------------+
@end example
@end ifnottex

The size is 4 bytes written most significant byte first, being the number of
subsequent data bytes, or the twos complement negative of that when a negative
integer is represented.  The data bytes are the absolute value of the integer,
written most significant byte first.

The most significant data byte is always non-zero, so the output is the same
on all systems, irrespective of limb size.

In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
of the limb size.  @code{mpz_inp_raw} will still accept this, for
compatibility.

The use of ``big endian'' for both the size and data fields is deliberate, it
makes the data easy to read in a hex dump of a file.  Unfortunately it also
means that the limb data must be reversed when reading or writing, so neither
a big endian nor little endian system can just read and write @code{_mp_d}.


@node C++ Interface Internals,  , Raw Output Internals, Internals
@section C++ Interface Internals

A system of expression templates is used to ensure something like @code{a=b+c}
turns into a simple call to @code{mpz_add} etc.  For @code{mpf_class} and
@code{mpfr_class} the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
@code{f=w*x+y*z}.  These are important features which a naive implementation
cannot provide.

A simplified description of the scheme follows.  The true scheme is
complicated by the fact that expressions have different return types.  For
detailed information, refer to the source code.

To perform an operation, say, addition, we first define a ``function object''
evaluating it,

@example
struct __gmp_binary_plus
@{
  static void eval(mpf_t f, mpf_t g, mpf_t h) @{ mpf_add(f, g, h); @}
@};
@end example

@noindent
And an ``additive expression'' object,

@example
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
@{
  return __gmp_expr
    <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
@}
@end example

The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<...>>} is used to
encapsulate any possible kind of expression into a single template type.  In
fact even @code{mpf_class} etc are @code{typedef} specializations of
@code{__gmp_expr}.

Next we define assignment of @code{__gmp_expr} to @code{mpf_class}.

@example
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
@{
  expr.eval(this->get_mpf_t(), this->precision());
  return *this;
@}

template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, unsigned long int precision)
@{
  Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
@}
@end example

where @code{expr.val1} and @code{expr.val2} are references to the expression's
operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the
@code{__gmp_expr}).

This way, the expression is actually evaluated only at the time of assignment,
when the required precision (that of @code{f}) is known.  Furthermore the
target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly
with @code{f} as the output argument.

Compound expressions are handled by defining operators taking subexpressions
as their arguments, like this:

@example
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
@{
  return __gmp_expr
    <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
    (expr1, expr2);
@}
@end example

And the corresponding specializations of @code{__gmp_expr::eval}:

@example
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, unsigned long int precision)
@{
  // declare two temporaries
  mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
  Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
@}
@end example

The expression is thus recursively evaluated to any level of complexity and
all subexpressions are evaluated to the precision of @code{f}.


@node Contributors, References, Internals, Top
@comment  node-name,  next,  previous,  up
@appendix Contributors
@cindex Contributors

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

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

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

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

John Amanatides of York University in Canada contributed the function
@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 GMP to use Cygnus' configure.
He has also made valuable suggestions and tested numerous intermediary
releases.

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

Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
@code{mpz_legendre}.

Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
@file{mpn/m68k/rshift.S} (now in @file{.asm} form).

The development of floating point functions of GNU MP 2, were supported in part
by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
System SOlving).

GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the
IDA Center for Computing Sciences, USA.

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

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

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

Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the
REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square
root.  The ECMNET project Paul is organizing was a driving force behind many
of the optimizations in GMP 3.

Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.

Kent Boortz made the Macintosh port.

Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros,
parameter tuning, speed measuring, the configure system, function inlining,
divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number
functions, printf and scanf functions, perl interface, demo expression parser,
the algorithms chapter in the manual, gmpasm-mode.el, and various
miscellaneous improvements elsewhere.

Steve Root helped write the optimized alpha 21264 assembly code.

Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++
istream input routines.

GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde.
Torbjorn's work was partially funded by the IDA Center for Computing Sciences,
USA.

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


@node References, GNU Free Documentation License, Contributors, Top
@comment  node-name,  next,  previous,  up
@appendix References
@cindex References

@c  FIXME: In tex, the @uref's are unhyphenated, which is good for clarity,
@c  but being long words they upset paragraph formatting (the preceding line
@c  can get badly stretched).  Would like an conditional @* style line break
@c  if the uref is too long to fit on the last line of the paragraph, but it's
@c  not clear how to do that.  For now explicit @texlinebreak{}s are used on
@c  paragraphs that come out bad.

@section Books

@itemize @bullet
@item
Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate
Texts in Mathematics number 138, Springer-Verlag, 1993.
@texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen}

@item
Donald E. Knuth, ``The Art of Computer Programming'', volume 2,
``Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998.
@texlinebreak{} @uref{http://www-cs-faculty.stanford.edu/~knuth/taocp.html}

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

@item
Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of
Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}

@item
Richard M. Stallman, ``Using and Porting GCC'', Free Software Foundation, 1999,
available online @uref{http://www.gnu.org/software/gcc/onlinedocs/}, and in
the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/}
@end itemize

@section Papers

@itemize @bullet
@item
Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'',
Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{}
@uref{http://www.mpi-sb.mpg.de/~ziegler/TechRep.ps.gz}

@item
Torbjorn Granlund and Peter L. Montgomery, ``Division by Invariant Integers
using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June
1994.  Also available @uref{ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz}
(and .psl.gz).

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

@item
Tudor Jebelean,
``An algorithm for exact division'',
Journal of Symbolic Computation,
volume 15, 1993, pp. 169-180.
Research report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz}

@item
Tudor Jebelean, ``Exact Division with Karatsuba Complexity - Extended
Abstract'', RISC-Linz technical report 96-31, @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz}

@item
Tudor Jebelean, ``Practical Integer Division with Karatsuba Complexity'',
ISSAC 97, pp. 339-341.  Technical report available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz}

@item
Tudor Jebelean, ``A Generalization of the Binary GCD Algorithm'', ISSAC 93,
pp. 111-116.  Technical report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz}

@item
Tudor Jebelean, ``A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD
of Long Integers'', Journal of Symbolic Computation, volume 19, 1995,
pp. 145-157.  Technical report version also available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz}

@item
Werner Krandick and Tudor Jebelean, ``Bidirectional Exact Integer Division'',
Journal of Symbolic Computation, volume 21, 1996, pp. 441-455.  Early
technical report version also available
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz}

@item
R. Moenck and A. Borodin, ``Fast Modular Transforms via Division'',
Proceedings of the 13th Annual IEEE Symposium on Switching and Automata
Theory, October 1972, pp. 90-96.  Reprinted as ``Fast Modular Transforms'',
Journal of Computer and System Sciences, volume 8, number 3, June 1974,
pp. 366-386.

@item
Arnold Sch@"onhage and Volker Strassen, ``Schnelle Multiplikation grosser
Zahlen'', Computing 7, 1971, pp. 281-292.

@item
Kenneth Weber, ``The accelerated integer GCD algorithm'',
ACM Transactions on Mathematical Software,
volume 21, number 1, March 1995, pp. 111-122.

@item
Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805,
November 1999, @uref{http://www.inria.fr/RRRT/RR-3805.html}

@item
Paul Zimmermann, ``A Proof of GMP Fast Division and Square Root
Implementations'', @texlinebreak{}
@uref{http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz}

@item
Dan Zuras, ``On Squaring and Multiplying Large Integers'', ARITH-11: IEEE
Symposium on Computer Arithmetic, 1993, pp. 260 to 271.  Reprinted as ``More
on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers,
volume 43, number 8, August 1994, pp. 899-908.
@end itemize


@node GNU Free Documentation License, Concept Index, References, Top
@appendix GNU Free Documentation License
@cindex GNU Free Documentation License
@include fdl.texi


@node Concept Index, Function Index, GNU Free Documentation License, Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

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

@bye

@c Local variables:
@c fill-column: 78
@c compile-command: "make gmp.info"
@c End: