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gmp.texi：源码内容

For C++, add @samp{--enable-cxx CXXFLAGS=-g}.
@item Checker
@cindex Checker
@cindex GCC Checker
The GCC checker (@uref{http://savannah.nongnu.org/projects/checker/}) can be
used with GMP@. It contains a stub library which means GMP applications
compiled with checker can use a normal GMP build.
A build of GMP with checking within GMP itself can be made. This will run
very very slowly. On GNU/Linux for example,
@cindex @command{checkergcc}
@example
./configure --host=none-pc-linux-gnu CC=checkergcc
@end example
@samp{--host=none} must be used, since the GMP assembly code doesn't support
the checking scheme. The GMP C++ features cannot be used, since current
versions of checker (0.9.9.1) don't yet support the standard C++ library.
@item Valgrind
@cindex Valgrind
The valgrind program (@uref{http://valgrind.org/}) is a memory
checker for x86s. It translates and emulates machine instructions to do
strong checks for uninitialized data (at the level of individual bits), memory
accesses through bad pointers, and memory leaks.
Recent versions of Valgrind are getting support for MMX and SSE/SSE2
instructions, for past versions GMP will need to be configured not to use
those, ie.@: for an x86 without them (for instance plain @samp{i486}).
@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
@cindex Execution profiling
@cindex @code{--enable-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. The profiling choices
for a GMP build are as follows.
@table @asis
@item @samp{--disable-profiling}
The default is to add nothing special for profiling.
It should be possible to just compile the mainline of a program with @code{-p}
and use @command{prof} to get a profile consisting of timer-based sampling of
the program counter. Most of the GMP assembly code has the necessary symbol
information.
This approach has the advantage of minimizing interference with normal program
operation, but on most systems the resolution of the sampling is quite low (10
milliseconds for instance), requiring long runs to get accurate information.
@item @samp{--enable-profiling=prof}
@cindex @code{prof}
Build with support for the system @command{prof}, which means @samp{-p} added
to the @samp{CFLAGS}.
This provides call counting in addition to program counter sampling, which
allows the most frequently called routines to be identified, and an average
time spent in each routine to be determined.
The x86 assembly code has support for this option, but on other processors
the assembly routines will be as if compiled without @samp{-p} and therefore
won't appear in the call counts.
On some systems, such as GNU/Linux, @samp{-p} in fact means @samp{-pg} and in
this case @samp{--enable-profiling=gprof} described below should be used
instead.
@item @samp{--enable-profiling=gprof}
@cindex @code{gprof}
Build with support for @command{gprof}, which means @samp{-pg} added to the
@samp{CFLAGS}.
This provides call graph construction in addition to call counting and program
counter sampling, which makes it possible to count calls coming from different
locations. For example the number of calls to @code{mpn_mul} from
@code{mpz_mul} versus the number from @code{mpf_mul}. The program counter
sampling is still flat though, so only a total time in @code{mpn_mul} would be
accumulated, not a separate amount for each call site.
The x86 assembly code has support for this option, but on other processors
the assembly routines will be as if compiled without @samp{-pg} and therefore
not be included in the call counts.
On x86 and m68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are
incompatible, so the latter is omitted from the default flags in that case,
which might result in poorer code generation.
Incidentally, it should be possible to use the @command{gprof} program with a
plain @samp{--enable-profiling=prof} build. But in that case only the
@samp{gprof -p} flat profile and call counts can be expected to be valid, not
the @samp{gprof -q} call graph.
@item @samp{--enable-profiling=instrument}
@cindex @code{-finstrument-functions}
@cindex @code{instrument-functions}
Build with the GCC option @samp{-finstrument-functions} added to the
@samp{CFLAGS} (@pxref{Code Gen Options,, Options for Code Generation, gcc,
Using the GNU Compiler Collection (GCC)}).
This inserts special instrumenting calls at the start and end of each
function, allowing exact timing and full call graph construction.
This instrumenting is not normally a standard system feature and will require
support from an external library, such as
@cindex FunctionCheck
@cindex fnccheck
@display
@uref{http://sourceforge.net/projects/fnccheck/}
@end display
This should be included in @samp{LIBS} during the GMP configure so that test
programs will link. For example,
@example
./configure --enable-profiling=instrument LIBS=-lfc
@end example
On a GNU system the C library provides dummy instrumenting functions, so
programs compiled with this option will link. In this case it's only
necessary to ensure the correct library is added when linking an application.
The x86 assembly code supports this option, but on other processors the
assembly routines will be as if compiled without
@samp{-finstrument-functions} meaning time spent in them will effectively be
attributed to their caller.
@end table
@node Autoconf, Emacs, Profiling, GMP Basics
@section Autoconf
@cindex Autoconf
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,
@cindex @code{AC_CHECK_LIB}
@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://gmplib.org/])])
@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://gmplib.org/])])
@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.
In general it's recommended 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
@node Emacs, , Autoconf, GMP Basics
@section Emacs
@cindex Emacs
@cindex @code{info-lookup-symbol}
@key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation
on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup,
emacs, The Emacs Editor}).
The GMP manual can be included in such lookups by putting the following in
your @file{.emacs},
@c This isn't pretty, but there doesn't seem to be a better way (in emacs
@c 21.2 at least). info-lookup->mode-value could be used for the "assoc"s,
@c but that function isn't documented, whereas info-lookup-alist is.
@c
@example
(eval-after-load "info-look"
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
(setcar (nthcdr 3 mode-value)
(cons '("(gmp)Function Index" nil "^ -.* " "\>")
(nth 3 mode-value)))))
@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://gmplib.org/} 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 compressed 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{gmp-bugs@@gmplib.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::
* Integer Import and Export::
* Miscellaneous Integer Functions::
* Integer Special 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{x})
Initialize @var{x}, and set its value to 0.
@end deftypefun
@deftypefun void mpz_inits (mpz_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpz_t} variables, and set their
values to 0.
@end deftypefun
@deftypefun void mpz_init2 (mpz_t @var{x}, mp_bitcnt_t @var{n})
Initialize @var{x}, with space for @var{n}-bit numbers, and set its value to 0.
Calling this function instead of @code{mpz_init} or @code{mpz_inits} is never
necessary; reallocation is handled automatically by GMP when needed.
@var{n} is only the initial space, @var{x} 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{x})
Free the space occupied by @var{x}. Call this function for all @code{mpz_t}
variables when you are done with them.
@end deftypefun
@deftypefun void mpz_clears (mpz_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpz_t} variables.
@end deftypefun
@deftypefun void mpz_realloc2 (mpz_t @var{x}, mp_bitcnt_t @var{n})
Change the space allocated for @var{x} to @var{n} bits. The value in @var{x}
is preserved if it fits, or is set to 0 if not.
Calling this function is never necessary; reallocation is handled automatically
by GMP when needed. But 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
@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 @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
@code{0B} for binary, @code{0} for octal, or decimal otherwise.
For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value. For bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.
This function returns 0 if the entire string is a valid number in base
@var{base}. Otherwise it returns @minus{}1.
@c
@c It turns out that it is not entirely true that this function ignores
@c white-space. It does ignore it between digits, but not after a minus sign
@c or within or after ``0x''. Some thought was given to disallowing all
@c whitespace, but that would be an incompatible change, whitespace has been
@c documented as ignored ever since GMP 1.
@c
@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 Integer assignment functions
@cindex Assignment functions
@cindex Integer initialization functions
@cindex Initialization 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{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 value of @var{op} as an @code{unsigned long}.
If @var{op} is too big to fit an @code{unsigned long} then just the least
significant bits that do fit are returned. The sign of @var{op} is ignored,
only the absolute value is used.
@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 big 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}, truncating if necessary (ie.@: rounding
towards zero).
If the exponent from the conversion is too big, the result is system
dependent. An infinity is returned where available. A hardware overflow trap
may or may not occur.
@end deftypefun
@deftypefun double mpz_get_d_2exp (signed long int *@var{exp}, mpz_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
towards zero), and returning the exponent separately.
The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the
exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} *
2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the
return is @math{0.0} and 0 is stored to @code{*@var{exp}}.
@cindex @code{frexp}
This is similar to the standard C @code{frexp} function (@pxref{Normalization
Functions,,, libc, The GNU C Library Reference Manual}).
@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 argument
may vary from 2 to 62 or from @minus{}2 to @minus{}36.
For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.
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
@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 @math{@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})
@deftypefunx void mpz_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, mpz_t @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 @math{@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 @math{@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 @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun
@deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, mp_bitcnt_t @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{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @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{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @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{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @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 @math{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.
For the @code{2exp} variants the divisor is @m{2^b,2^@var{b}}. These
functions are implemented as right shifts and bit masks, but of course they
round 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}, mp_bitcnt_t @var{b})
@cindex Divisibility functions
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}}.
@var{n} is divisible by @var{d} if there exists an integer @var{q} satisfying
@math{@var{n} = @var{q}@GMPmultiply{}@var{d}}. Unlike the other division
functions, @math{@var{d}=0} is accepted and following the rule it can be seen
that only 0 is considered divisible by 0.
@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}, mp_bitcnt_t @var{b})
@cindex Divisibility functions
@cindex Congruence functions
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}}.
@var{n} is congruent to @var{c} mod @var{d} if there exists an integer @var{q}
satisfying @math{@var{n} = @var{c} + @var{q}@GMPmultiply{}@var{d}}. Unlike
the other division functions, @math{@var{d}=0} is accepted and following the
rule it can be seen that @var{n} and @var{c} are considered congruent mod 0
only when exactly equal.
@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 @math{@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_powm_sec (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
Set @var{rop} to @m{base^{exp} bmod mod, (@var{base} raised to @var{exp})
modulo @var{mod}}.
It is required that @math{@var{exp} > 0} and that @var{mod} is odd.
This function is designed to take the same time and have the same cache access
patterns for any two same-size arguments, assuming that function arguments are
placed at the same position and that the machine state is identical upon
function entry. This function is intended for cryptographic purposes, where
resilience to side-channel attacks is desired.
@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
@math{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{lfloorroot 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_rootrem (mpz_t @var{root}, mpz_t @var{rem}, mpz_t @var{u}, unsigned long int @var{n})
Set @var{root} to @m{lfloorroot n of {u}rfloor@C{},} the truncated
integer part of the @var{n}th root of @var{u}. Set @var{rem} to the
remainder, @m{(@var{u} - @var{root}^n),
@var{u}@minus{}@var{root}**@var{n}}.
@end deftypefun
@deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to @m{lfloorsqrt{@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{lfloorsqrt{@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})
@cindex Perfect power functions
@cindex Root testing functions
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})
@cindex Perfect square functions
@cindex Root testing functions
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
@cindex Probable 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})
@cindex Next prime function
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
@cindex GCD 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
@cindex GCD extended
Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
addition set @var{s} and @var{t} to coefficients satisfying
@math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}.
The value in @var{g} is always positive, even if one or both of @var{a} and
@var{b} are negative. The values in @var{s} and @var{t} are chosen such that
@math{@GMPabs{@var{s}} @le{} @GMPabs{@var{b}}} and @math{@GMPabs{@var{t}}
@le{} @GMPabs{@var{a}}}.
If @var{t} is @code{NULL} then that value 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
@cindex LCM 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
@cindex Inverse modulo 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 @math{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 bright),
(@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})
@cindex Legendre symbol functions
Calculate the Legendre symbol @m{left(a over pright),
(@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 bright),
(@var{a}/@var{b})} with the Kronecker extension @m{left(a over
2right) = left(2 over aright), (a/2)=(2/a)} when @math{a} odd, or
@m{left(a over 2right) = 0, (a/2)=0} when @math{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 {mp_bitcnt_t} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f})
@cindex Remove factor functions
@cindex Factor removal functions
Remove all occurrences of the factor @var{f} from @var{op} and store the
result in @var{rop}. The return value is how many such occurrences were
removed.
@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 @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
@math{@var{op1} < @var{op2}}.
@code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate their
arguments more than once. @code{mpz_cmp_d} can be called with an infinity,
but results are undefined for a NaN.
@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 @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
@math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
@math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.
@code{mpz_cmpabs_d} can be called with an infinity, but results are undefined
for a NaN.
@end deftypefn
@deftypefn Macro int mpz_sgn (mpz_t @var{op})
@cindex Sign tests
@cindex Integer sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@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 logical 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} bitwise-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} bitwise 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} bitwise 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 {mp_bitcnt_t} mpz_popcount (mpz_t @var{op})
If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is the
number of 1 bits in the binary representation. If @math{@var{op}<0}, the
number of 1s is infinite, and the return value is the largest possible
@code{mp_bitcnt_t}.
@end deftypefun
@deftypefun {mp_bitcnt_t} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<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
@math{@ge{}0} and the other @math{<0} then the number of bits different is
infinite, and the return value is the largest possible @code{mp_bitcnt_t}.
@end deftypefun
@deftypefun {mp_bitcnt_t} mpz_scan0 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
@deftypefunx {mp_bitcnt_t} mpz_scan1 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
@cindex Bit scanning functions
@cindex Scan bit functions
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 the largest possible @code{mp_bitcnt_t} is
returned. This will happen in @code{mpz_scan0} past the end of a negative
number, or @code{mpz_scan1} past the end of a nonnegative number.
@end deftypefun
@deftypefun void mpz_setbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Set bit @var{bit_index} in @var{rop}.
@end deftypefun
@deftypefun void mpz_clrbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Clear bit @var{bit_index} in @var{rop}.
@end deftypefun
@deftypefun void mpz_combit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Complement bit @var{bit_index} in @var{rop}.
@end deftypefun
@deftypefun int mpz_tstbit (mpz_t @var{op}, mp_bitcnt_t @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 argument may vary from 2 to 62 or from @minus{}2 to
@minus{}36.
For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.
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 @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
@code{0B} for binary, @code{0} for octal, or decimal otherwise.
For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value. For bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.
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, Integer Import and Export, 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}, mp_bitcnt_t @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 @math{@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}, mp_bitcnt_t @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
@node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions
@section Integer Import and Export
@code{mpz_t} variables can be converted to and from arbitrary words of binary
data with the following functions.
@deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op})
@cindex Integer import
@cindex Import
Set @var{rop} from an array of word data at @var{op}.
The parameters specify the format of the data. @var{count} many words are
read, each @var{size} bytes. @var{order} can be 1 for most significant word
first or -1 for least significant first. Within each word @var{endian} can be
1 for most significant byte first, -1 for least significant first, or 0 for
the native endianness of the host CPU@. The most significant @var{nails} bits
of each word are skipped, this can be 0 to use the full words.
There is no sign taken from the data, @var{rop} will simply be a positive
integer. An application can handle any sign itself, and apply it for instance
with @code{mpz_neg}.
There are no data alignment restrictions on @var{op}, any address is allowed.
Here's an example converting an array of @code{unsigned long} data, most
significant element first, and host byte order within each value.
@example
unsigned long a[20];
/* Initialize @var{z} and @var{a} */
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
@end example
This example assumes the full @code{sizeof} bytes are used for data in the
given type, which is usually true, and certainly true for @code{unsigned long}
everywhere we know of. However on Cray vector systems it may be noted that
@code{short} and @code{int} are always stored in 8 bytes (and with
@code{sizeof} indicating that) but use only 32 or 46 bits. The @var{nails}
feature can account for this, by passing for instance
@code{8*sizeof(int)-INT_BIT}.
@end deftypefun
@deftypefun {void *} mpz_export (void *@var{rop}, size_t *@var{countp}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, mpz_t @var{op})
@cindex Integer export
@cindex Export
Fill @var{rop} with word data from @var{op}.
The parameters specify the format of the data produced. Each word will be
@var{size} bytes and @var{order} can be 1 for most significant word first or
-1 for least significant first. Within each word @var{endian} can be 1 for
most significant byte first, -1 for least significant first, or 0 for the
native endianness of the host CPU@. The most significant @var{nails} bits of
each word are unused and set to zero, this can be 0 to produce full words.
The number of words produced is written to @code{*@var{countp}}, or
@var{countp} can be @code{NULL} to discard the count. @var{rop} must have
enough space for the data, or if @var{rop} is @code{NULL} then a result array
of the necessary size is allocated using the current GMP allocation function
(@pxref{Custom Allocation}). In either case the return value is the
destination used, either @var{rop} or the allocated block.
If @var{op} is non-zero then the most significant word produced will be
non-zero. If @var{op} is zero then the count returned will be zero and
nothing written to @var{rop}. If @var{rop} is @code{NULL} in this case, no
block is allocated, just @code{NULL} is returned.
The sign of @var{op} is ignored, just the absolute value is exported. An
application can use @code{mpz_sgn} to get the sign and handle it as desired.
(@pxref{Integer Comparisons})
There are no data alignment restrictions on @var{rop}, any address is allowed.
When an application is allocating space itself the required size can be
determined with a calculation like the following. Since @code{mpz_sizeinbase}
always returns at least 1, @code{count} here will be at least one, which
avoids any portability problems with @code{malloc(0)}, though if @code{z} is
zero no space at all is actually needed (or written).
@example
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);
@end example
@end deftypefun
@need 2000
@node Miscellaneous Integer Functions, Integer Special Functions, Integer Import and Export, 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_sizeinbase (mpz_t @var{op}, int @var{base})
@cindex Size in digits
@cindex Digits in an integer
Return the size of @var{op} measured in number of digits in the given
@var{base}. @var{base} can vary from 2 to 62. The sign of @var{op} is
ignored, just the absolute value is used. The result will be either exact or
1 too big. If @var{base} is a power of 2, the result is always exact. If
@var{op} is zero the return value is always 1.
This function can be used to determine the space required when 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.
@cindex Most significant bit
It will be noted that @code{mpz_sizeinbase(@var{op},2)} can be used to locate
the most significant 1 bit in @var{op}, counting from 1. (Unlike the bitwise
functions which start from 0, @xref{Integer Logic and Bit Fiddling,, Logical
and Bit Manipulation Functions}.)
@end deftypefun
@node Integer Special Functions, , Miscellaneous Integer Functions, Integer Functions
@section Special Functions
@cindex Special integer functions
@cindex Integer special functions
The functions in this section are for various special purposes. Most
applications will not need them.
@deftypefun void mpz_array_init (mpz_t @var{integer_array}, mp_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} is allocated to each of the @var{array_size} integers in
@var{integer_array}. There is no way to free the storage allocated by this
function. Don't call @code{mpz_clear}!
The @var{integer_array} parameter is the first @code{mpz_t} in the array. For
example,
@example
mpz_t arr[20000];
mpz_array_init (arr[0], 20000, 512);
@end example
@c In case anyone's wondering, yes this parameter style is a bit anomalous,
@c it'd probably be nicer if it was "arr" instead of "arr[0]". Obviously the
@c two differ only in the declaration, not the pointer value, but changing is
@c not possible since it'd provoke warnings or errors in existing sources.
This function is only intended for programs that create a large number
of integers and need to reduce memory usage by avoiding the overheads of
allocating and reallocating lots of small blocks. In normal programs this
function is not recommended.
The space allocated to each integer by this function will not be automatically
increased, unlike the normal @code{mpz_init}, so an application must ensure it
is sufficient for any value stored. The following space requirements apply to
various routines,
@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}.
@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 its size in limbs.
@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
@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
@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 Rational assignment functions
@cindex Assignment functions
@cindex Rational initialization functions
@cindex Initialization functions
@deftypefun void mpq_init (mpq_t @var{x})
Initialize @var{x} 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_inits (mpq_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpq_t} variables, and set their
values to 0/1.
@end deftypefun
@deftypefun void mpq_clear (mpq_t @var{x})
Free the space occupied by @var{x}. Make sure to call this function for all
@code{mpq_t} variables when you are done with them.
@end deftypefun
@deftypefun void mpq_clears (mpq_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpq_t} variables.
@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 62, or
if @var{base} is 0 then the leading characters are used: @code{0x} or @code{0X} for hex,
@code{0b} or @code{0B} for binary,
@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}, truncating if necessary (ie.@: rounding
towards zero).
If the exponent from the conversion is too big or too small to fit a
@code{double} then the result is system dependent. For too big an infinity is
returned when available. For too small @math{0.0} is normally returned.
Hardware overflow, underflow and denorm traps may or may not occur.
@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}. There is no rounding, this conversion
is exact.
@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 @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
@end deftypefun
@deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, mp_bitcnt_t @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}, mp_bitcnt_t @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 @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@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
@math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
@var{num2}/@var{den2}}, and a negative value if @math{@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 @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@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 @math{16/11}, whereas @samp{0x10/0x11} is
@math{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
@math{2^@W{-68719476768}} to @math{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.
The mantissa in stored in binary, as might be imagined from the fact
precisions are expressed in bits. One consequence of this is that decimal
fractions like @math{0.1} cannot be represented exactly. The same is true of
plain IEEE @code{double} floats. This makes both highly unsuitable for
calculations involving money or other values that should be exact decimal
fractions. (Suitably scaled integers, or perhaps rationals, are better
choices.)
@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 (mp_bitcnt_t @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 {mp_bitcnt_t} mpf_get_default_prec (void)
Return the 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}, mp_bitcnt_t @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_inits (mpf_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpf_t} variables, and set their
values to 0. The precision of the initialized variables is undefined unless a
default precision has already been established by a call to
@code{mpf_set_default_prec}.
@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
@deftypefun void mpf_clears (mpf_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpf_t} variables.
@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 {mp_bitcnt_t} 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}, mp_bitcnt_t @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}, mp_bitcnt_t @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 62, or @minus{}62 to
@minus{}2. Negative values are used to specify that the exponent is in
decimal.
For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value; for bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.
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 Float assignment functions
@cindex Assignment functions
@cindex Float initialization functions
@cindex Initialization 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}, truncating if necessary (ie.@: rounding
towards zero).
If the exponent in @var{op} is too big or too small to fit a @code{double}
then the result is system dependent. For too big an infinity is returned when
available. For too small @math{0.0} is normally returned. Hardware overflow,
underflow and denorm traps may or may not occur.
@end deftypefun
@deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, mpf_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (ie.@: rounding
towards zero), and with an exponent returned separately.
The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the
exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} *
2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the
return is @math{0.0} and 0 is stored to @code{*@var{exp}}.
@cindex @code{frexp}
This is similar to the standard C @code{frexp} function (@pxref{Normalization
Functions,,, libc, The GNU C Library Reference Manual}).
@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}. The base argument
may vary from 2 to 62 or from @minus{}2 to @minus{}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.
For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.
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
@math{@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 @math{@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 @math{@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}, mp_bitcnt_t @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}, mp_bitcnt_t @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 @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.
@code{mpf_cmp_d} can be called with an infinity, but results are undefined for
a NaN.
@end deftypefun
@deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, mp_bitcnt_t op3)
Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
equal, zero otherwise. I.e., test if @var{op1} and @var{op2} are approximately
equal.
Caution 1: All version of GMP up to version 4.2.4 compared just whole limbs,
meaning sometimes more than @var{op3} bits, sometimes fewer.
Caution 2: This function will consider XXX11...111 and XX100...000 different,
even if ... is replaced by a semi-infinite number of bits. Such numbers are
really just one ulp off, and should be considered equal.
@end deftypefun
@deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
Compute the relative difference between @var{op1} and @var{op2} and store the
result in @var{rop}. This is @math{@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 @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@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 62 or from @minus{}2 to @minus{}36. An exponent is
then printed, separated by an @samp{e}, or if the 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}.
For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.
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})
@cindex Rounding functions
@cindex Float rounding functions
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}, mp_bitcnt_t @var{nbits})
@cindex Random number functions
@cindex Float random number functions
Generate a uniformly distributed random float in @var{rop}, such that @math{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} (in limbs). This function is
useful for testing functions and algorithms, since these 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
It should be noted that the @code{mpn} functions make no attempt to identify
high or low zero limbs on their operands, or other special forms. On random
data such cases will be unlikely and it'd be wasteful for every function to
check every time. An application knowing something about its data can take
steps to trim or perhaps split its calculations.
@c
@c For reference, within gmp mpz_t operands never have high zero limbs, and
@c we rate low zero limbs as unlikely too (or something an application should
@c handle). This is a prime motivation for not stripping zero limbs in say
@c mpn_mul_n etc.
@c
@c Other applications doing variable-length calculations will quite likely do
@c something similar to mpz. And even if not then it's highly likely zero
@c limb stripping can be done at just a few judicious points, which will be
@c more efficient than having lots of mpn functions checking every time.
@sp 1
@noindent
In the notation used below, 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_neg (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Perform the negation of @{@var{sp}, @var{n}@}, and write the result to
@{@var{rp}, @var{n}@}. Return carry-out.
@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. No overlap is permitted between the
destination and either source.
If the two input operands are the same, use @code{mpn_sqr}.
@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
(@var{s1n}+@var{s2n})-limb 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
product's most significant limb is zero. No overlap is permitted between the
destination and either source.
This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun
@deftypefun void mpn_sqr (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Compute the square of @{@var{s1p}, @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 result's
most significant limb is zero. No overlap is permitted between the
destination and the source.
@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 @math{@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, plus
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 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, except that @var{np} might equal
@var{rp}. The dividend size @var{nn} must be greater than or equal to divisor
size @var{dn}. The most significant limb of the divisor must be non-zero. The
@var{qxn} operand must be zero.
@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