This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.4.
Copyright 1991, 1993-2016 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in GNU Free Documentation License.
• Copying: | MPFR Copying Conditions (LGPL). | |
• Introduction to MPFR: | Brief introduction to GNU MPFR. | |
• Installing MPFR: | How to configure and compile the MPFR library. | |
• Reporting Bugs: | How to usefully report bugs. | |
• MPFR Basics: | What every MPFR user should now. | |
• MPFR Interface: | MPFR functions and macros. | |
• API Compatibility: | API compatibility with previous MPFR versions. | |
• Contributors: | ||
• References: | ||
• GNU Free Documentation License: | ||
• Concept Index: | ||
• Function and Type Index: |
Next: Introduction to MPFR, Previous: Top, Up: Top [Index]
The GNU MPFR library (or MPFR for short) is free; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.
Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the GNU MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LESSER.
Next: Installing MPFR, Previous: Copying, Up: Top [Index]
MPFR is a portable library written in C for arbitrary precision arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are:
mp_bits_per_limb
(64 on most current processors);
In particular, with a precision of 53 bits, MPFR is able to
exactly reproduce all computations with double-precision machine
floating-point numbers (e.g., double
type in C, with a C
implementation that rigorously follows Annex F of the ISO C99 standard
and FP_CONTRACT
pragma set to OFF
) on the four arithmetic
operations and the square root, except the default exponent range is much
wider and subnormal numbers are not implemented (but can be emulated).
This version of MPFR is released under the GNU Lesser General Public License, version 3 or any later version. It is permitted to link MPFR to most non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided.
Everyone should read MPFR Basics. If you need to install the library yourself, you need to read Installing MPFR, too. To use the library you will need to refer to MPFR Interface.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
Next: Reporting Bugs, Previous: Introduction to MPFR, Up: Top [Index]
The MPFR library is already installed on some GNU/Linux distributions,
but the development files necessary to the compilation such as
mpfr.h are not always present. To check that MPFR is fully
installed on your computer, you can check the presence of the file
mpfr.h in /usr/include, or try to compile a small program
having #include <mpfr.h>
(since mpfr.h may be installed
somewhere else). For instance, you can try to compile:
#include <stdio.h> #include <mpfr.h> int main (void) { printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n", mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL); return 0; }
with
cc -o version version.c -lmpfr -lgmp
and if you get errors whose first line looks like
version.c:2:19: error: mpfr.h: No such file or directory
then MPFR is probably not installed. Running this program will give you the MPFR version.
If MPFR is not installed on your computer, or if you want to install a different version, please follow the steps below.
Here are the steps needed to install the library on Unix systems (more details are provided in the INSTALL file):
Then, in the MPFR build directory, type the following commands.
This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default /usr/local), threading support, and so on. See the INSTALL file and/or the output of ‘./configure --help’ for more information, in particular if you get error messages.
This will compile MPFR, and create a library archive file libmpfr.a. On most platforms, a dynamic library will be produced too.
This will make sure that MPFR was built correctly. If any test fails, information about this failure can be found in the tests/test-suite.log file. If you want the contents of this file to be automatically output in case of failure, you can set the ‘VERBOSE’ environment variable to 1 before running ‘make check’, for instance by typing:
‘VERBOSE=1 make check’
In case of failure, you may want to check whether the problem is already known. If not, please report this failure to the MPFR mailing-list ‘mpfr@inria.fr’. For details, See Reporting Bugs.
This will copy the files mpfr.h and mpf2mpfr.h to the directory /usr/local/include, the library files (libmpfr.a and possibly others) to the directory /usr/local/lib, the file mpfr.info to the directory /usr/local/share/info, and some other documentation files to the directory /usr/local/share/doc/mpfr (or if you passed the ‘--prefix’ option to configure, using the prefix directory given as argument to ‘--prefix’ instead of /usr/local).
There are some other useful make targets:
Create or update an info version of the manual, in mpfr.info.
This file is already provided in the MPFR archives.
Create a PDF version of the manual, in mpfr.pdf.
Create a DVI version of the manual, in mpfr.dvi.
Create a Postscript version of the manual, in mpfr.ps.
Create a HTML version of the manual, in several pages in the directory doc/mpfr.html; if you want only one output HTML file, then type ‘makeinfo --html --no-split mpfr.texi’ from the ‘doc’ directory instead.
Delete all object files and archive files, but not the configuration files.
Delete all generated files not included in the distribution.
Delete all files copied by ‘make install’.
In case of problem, please read the INSTALL file carefully before reporting a bug, in particular section “In case of problem”. Some problems are due to bad configuration on the user side (not specific to MPFR). Problems are also mentioned in the FAQ http://www.mpfr.org/faq.html.
Please report problems to the MPFR mailing-list ‘mpfr@inria.fr’. See Reporting Bugs. Some bug fixes are available on the MPFR 3.1.4 web page http://www.mpfr.org/mpfr-3.1.4/.
The latest version of MPFR is available from ftp://ftp.gnu.org/gnu/mpfr/ or http://www.mpfr.org/.
Next: MPFR Basics, Previous: Installing MPFR, Up: Top [Index]
If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.1.4 web page http://www.mpfr.org/mpfr-3.1.4/ and the FAQ http://www.mpfr.org/faq.html: perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: https://sympa.inria.fr/sympa/arc/mpfr. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find.
There are a few things you should think about when you put your bug report together.
You have to send us a test case that makes it possible for us to reproduce the bug, i.e., a small self-content program, using no other library than MPFR. Include instructions on how to run the test case.
You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way.
Please include compiler version information in your bug report. This can be extracted using ‘cc -V’ on some machines, or, if you’re using GCC, ‘gcc -v’. Also, include the output from ‘uname -a’ and the MPFR version (the GMP version may be useful too). If you get a failure while running ‘make’ or ‘make check’, please include the config.log file in your bug report, and in case of test failure, the tests/test-suite.log file too.
If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we will not do anything about it (aside of chiding you to send better bug reports).
Send your bug report to the MPFR mailing-list ‘mpfr@inria.fr’.
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.
Next: MPFR Interface, Previous: Reporting Bugs, Up: Top [Index]
• Headers and Libraries: | ||
• Nomenclature and Types: | ||
• MPFR Variable Conventions: | ||
• Rounding Modes: | ||
• Floating-Point Values on Special Numbers: | ||
• Exceptions: | ||
• Memory Handling: |
Next: Nomenclature and Types, Previous: MPFR Basics, Up: MPFR Basics [Index]
All declarations needed to use MPFR are collected in the include file mpfr.h. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library:
#include <mpfr.h>
Note however that prototypes for MPFR functions with FILE *
parameters
are provided only if <stdio.h>
is included too (before mpfr.h):
#include <stdio.h> #include <mpfr.h>
Likewise <stdarg.h>
(or <varargs.h>
) is required for prototypes
with va_list
parameters, such as mpfr_vprintf
.
And for any functions using intmax_t
, you must include
<stdint.h>
or <inttypes.h>
before mpfr.h, to
allow mpfr.h to define prototypes for these functions. Moreover,
users of C++ compilers under some platforms may need to define
MPFR_USE_INTMAX_T
(and should do it for portability) before
mpfr.h has been included; of course, it is possible to do that
on the command line, e.g., with -DMPFR_USE_INTMAX_T
.
Note: If mpfr.h and/or gmp.h (used by mpfr.h)
are included several times (possibly from another header file),
<stdio.h>
and/or <stdarg.h>
(or <varargs.h>
)
should be included before the first inclusion of
mpfr.h or gmp.h. Alternatively, you can define
MPFR_USE_FILE
(for MPFR I/O functions) and/or
MPFR_USE_VA_LIST
(for MPFR functions with va_list
parameters) anywhere before the last inclusion of mpfr.h.
As a consequence, if your file is a public header that includes
mpfr.h, you need to use the latter method.
When calling a MPFR macro, it is not allowed to have previously defined
a macro with the same name as some keywords (currently do
,
while
and sizeof
).
You can avoid the use of MPFR macros encapsulating functions by defining
the MPFR_USE_NO_MACRO
macro before mpfr.h is included. In
general this should not be necessary, but this can be useful when debugging
user code: with some macros, the compiler may emit spurious warnings with
some warning options, and macros can prevent some prototype checking.
All programs using MPFR must link against both libmpfr and libgmp libraries. On a typical Unix-like system this can be done with ‘-lmpfr -lgmp’ (in that order), for example:
gcc myprogram.c -lmpfr -lgmp
MPFR is built using Libtool and an application can use that to link if desired, see GNU Libtool.
If MPFR has been installed to a non-standard location, then it may be necessary to set up environment variables such as ‘C_INCLUDE_PATH’ and ‘LIBRARY_PATH’, or use ‘-I’ and ‘-L’ compiler options, in order to point to the right directories. For a shared library, it may also be necessary to set up some sort of run-time library path (e.g., ‘LD_LIBRARY_PATH’) on some systems. Please read the INSTALL file for additional information.
Next: MPFR Variable Conventions, Previous: Headers and Libraries, Up: MPFR Basics [Index]
A floating-point number, or float for short, is an arbitrary
precision significand (also called mantissa) with a limited precision
exponent. The C data type
for such objects is mpfr_t
(internally defined as a one-element
array of a structure, and mpfr_ptr
is the C data type representing
a pointer to this structure). A floating-point number can have
three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN
represents an uninitialized object, the result of an invalid operation
(like 0 divided by 0), or a value that cannot be determined (like
+Infinity minus +Infinity). Moreover, like in the IEEE 754 standard,
zero is signed, i.e., there are both +0 and -0; the behavior
is the same as in the IEEE 754 standard and it is generalized to
the other functions supported by MPFR. Unless documented otherwise,
the sign bit of a NaN is unspecified.
The precision is the number of bits used to represent the significand
of a floating-point number;
the corresponding C data type is mpfr_prec_t
.
The precision can be any integer between MPFR_PREC_MIN
and
MPFR_PREC_MAX
. In the current implementation, MPFR_PREC_MIN
is equal to 2.
Warning! MPFR needs to increase the precision internally, in order to
provide accurate results (and in particular, correct rounding). Do not
attempt to set the precision to any value near MPFR_PREC_MAX
,
otherwise MPFR will abort due to an assertion failure. Moreover, you
may reach some memory limit on your platform, in which case the program
may abort, crash or have undefined behavior (depending on your C
implementation).
The rounding mode specifies the way to round the result of a
floating-point operation, in case the exact result can not be represented
exactly in the destination significand;
the corresponding C data type is mpfr_rnd_t
.
Next: Rounding Modes, Previous: Nomenclature and Types, Up: MPFR Basics [Index]
Before you can assign to an MPFR variable, you need to initialize it by calling one of the special initialization functions. When you’re done with a variable, you need to clear it out, using one of the functions for that purpose. A variable should only be initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited. You do not need to be concerned about allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floating-point variable will have the same allocated space during all its life.
As a general rule, all MPFR functions expect output arguments before input
arguments. This notation is based on an analogy with the assignment operator.
MPFR allows you to use the same variable for both input and output in the same
expression. For example, the main function for floating-point multiplication,
mpfr_mul
, can be used like this: mpfr_mul (x, x, x, rnd)
.
This
computes the square of x with rounding mode rnd
and puts the result back in x.
Next: Floating-Point Values on Special Numbers, Previous: MPFR Variable Conventions, Up: MPFR Basics [Index]
The following five rounding modes are supported:
MPFR_RNDN
: round to nearest (roundTiesToEven in IEEE 754-2008),
MPFR_RNDZ
: round toward zero (roundTowardZero in IEEE 754-2008),
MPFR_RNDU
: round toward plus infinity (roundTowardPositive in IEEE 754-2008),
MPFR_RNDD
: round toward minus infinity (roundTowardNegative in IEEE 754-2008),
MPFR_RNDA
: round away from zero.
The ‘round to nearest’ mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the drift phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2).
Most MPFR functions take as first argument the destination variable, as
second and following arguments the input variables, as last argument a
rounding mode, and have a return value of type int
, called the
ternary value. The value stored in the destination variable is
correctly rounded, i.e., MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this variable.
The input variables are regarded as exact (in particular, their precision
does not affect the result).
As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding).
Unless documented otherwise, functions returning an int
return
a ternary value.
If the ternary value is zero, it means that the value stored in the
destination variable is the exact result of the corresponding mathematical
function. If the ternary value is positive (resp. negative), it means
the value stored in the destination variable is greater (resp. lower)
than the exact result. For example with the MPFR_RNDU
rounding mode,
the ternary value is usually positive, except when the result is exact, in
which case it is zero. In the case of an infinite result, it is considered
as inexact when it was obtained by overflow, and exact otherwise. A NaN
result (Not-a-Number) always corresponds to an exact return value.
The opposite of a returned ternary value is guaranteed to be representable
in an int
.
Unless documented otherwise, functions returning as result the value 1
(or any other value specified in this manual)
for special cases (like acos(0)
) yield an overflow or
an underflow if that value is not representable in the current exponent range.
Next: Exceptions, Previous: Rounding Modes, Up: MPFR Basics [Index]
This section specifies the floating-point values (of type mpfr_t
)
returned by MPFR functions (where by “returned” we mean here the modified
value of the destination object, which should not be mixed with the ternary
return value of type int
of those functions).
For functions returning several values (like
mpfr_sin_cos
), the rules apply to each result separately.
Functions can have one or several input arguments. An input point is a mapping from these input arguments to the set of the MPFR numbers. When none of its components are NaN, an input point can also be seen as a tuple in the extended real numbers (the set of the real numbers with both infinities).
When the input point is in the domain of the mathematical function, the result is rounded as described in Section “Rounding Modes” (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the description of the MPFR function (MPFR Interface).
When the input point is not in the domain of the mathematical function
but is in its closure in the extended real numbers and the function can
be extended by continuity, the result is the obtained limit.
Examples: mpfr_hypot
on (+Inf,0) gives +Inf. But mpfr_pow
cannot be defined on (1,+Inf) using this rule, as one can find
sequences (x_n,y_n) such that
x_n goes to 1, y_n goes to +Inf
and x_n to the y_n goes to any
positive value when n goes to the infinity.
When the input point is in the closure of the domain of the mathematical
function and an input argument is +0 (resp. -0), one considers
the limit when the corresponding argument approaches 0 from above
(resp. below), if possible. If the limit is not defined (e.g.,
mpfr_sqrt
and mpfr_log
on -0), the behavior is
specified in the description of the MPFR function, but must be consistent
with the rule from the above paragraph (e.g., mpfr_log
on ±0
gives -Inf).
When the result is equal to 0, its sign is determined by considering the
limit as if the input point were not in the domain: If one approaches 0
from above (resp. below), the result is +0 (resp. -0);
for example, mpfr_sin
on -0 gives -0 and
mpfr_acos
on 1 gives +0 (in all rounding modes).
In the other cases, the sign is specified in the description of the MPFR
function; for example mpfr_max
on -0 and +0 gives +0.
When the input point is not in the closure of the domain of the function,
the result is NaN. Example: mpfr_sqrt
on -17 gives NaN.
When an input argument is NaN, the result is NaN, possibly except when
a partial function is constant on the finite floating-point numbers;
such a case is always explicitly specified in MPFR Interface.
Example: mpfr_hypot
on (NaN,0) gives NaN, but mpfr_hypot
on (NaN,+Inf) gives +Inf (as specified in Special Functions),
since for any finite or infinite input x, mpfr_hypot
on
(x,+Inf) gives +Inf.
Next: Memory Handling, Previous: Floating-Point Values on Special Numbers, Up: MPFR Basics [Index]
MPFR supports 6 exception types:
Note: This is not the single possible definition of the underflow. MPFR chooses to consider the underflow after rounding. The underflow before rounding can also be defined. For instance, consider a function that has the exact result 7 multiplied by two to the power e-4, where e is the smallest exponent (for a significand between 1/2 and 1), with a 2-bit target precision and rounding toward plus infinity. The exact result has the exponent e-1. With the underflow before rounding, such a function call would yield an underflow, as e-1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to e, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR.
mpfr_cmp
, or a
conversion to an integer cannot be represented in the target type).
MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in Exception Related Functions.
Differences with the ISO C99 standard:
Previous: Exceptions, Up: MPFR Basics [Index]
MPFR functions may create caches, e.g., when computing constants such
as Pi, either because the user has called a function like
mpfr_const_pi
directly or because such a function was called
internally by the MPFR library itself to compute some other function.
At any time, the user can free the various caches with
mpfr_free_cache
. It is strongly advised to do that before
terminating a thread, or before exiting when using tools like
‘valgrind’ (to avoid memory leaks being reported).
MPFR internal data such as flags, the exponent range, the default precision and rounding mode, and caches (i.e., data that are not accessed via parameters) are either global (if MPFR has not been compiled as thread safe) or per-thread (thread local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier).
Next: API Compatibility, Previous: MPFR Basics, Up: Top [Index]
The floating-point functions expect arguments of type mpfr_t
.
The MPFR floating-point functions have an interface that is similar to the
GNU MP
functions. The function prefix for floating-point operations is mpfr_
.
The user has to specify the precision of each variable. A computation that assigns a variable will take place with the precision of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average).
The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with “infinite accuracy”), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system.
MPFR does not keep track of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision.
The value of the standard C macro errno
may be set to non-zero by
any MPFR function or macro, whether or not there is an error.
Next: Assignment Functions, Previous: MPFR Interface, Up: MPFR Interface [Index]
An mpfr_t
object must be initialized before storing the first value in
it. The functions mpfr_init
and mpfr_init2
are used for that
purpose.
Initialize x, set its precision to be exactly prec bits and its value to NaN. (Warning: the corresponding MPF function initializes to zero instead.)
Normally, a variable should be initialized once only or at
least be cleared, using mpfr_clear
, between initializations.
To change the precision of a variable which has already been initialized,
use mpfr_set_prec
.
The precision prec must be an integer between MPFR_PREC_MIN
and
MPFR_PREC_MAX
(otherwise the behavior is undefined).
Initialize all the mpfr_t
variables of the given variable
argument va_list
, set their precision to be exactly
prec bits and their value to NaN.
See mpfr_init2
for more details.
The va_list
is assumed to be composed only of type mpfr_t
(or equivalently mpfr_ptr
).
It begins from x, and ends when it encounters a null pointer (whose
type must also be mpfr_ptr
).
Free the space occupied by the significand of
x. Make sure to call this function for all
mpfr_t
variables when you are done with them.
Free the space occupied by all the mpfr_t
variables of the given
va_list
. See mpfr_clear
for more details.
The va_list
is assumed to be composed only of type mpfr_t
(or equivalently mpfr_ptr
).
It begins from x, and ends when it encounters a null pointer (whose
type must also be mpfr_ptr
).
Here is an example of how to use multiple initialization functions
(since NULL
is not necessarily defined in this context, we use
(mpfr_ptr) 0
instead, but (mpfr_ptr) NULL
is also correct).
{ mpfr_t x, y, z, t; mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); … mpfr_clears (x, y, z, t, (mpfr_ptr) 0); }
Initialize x, set its precision to the default precision,
and set its value to NaN.
The default precision can be changed by a call to mpfr_set_default_prec
.
Warning! In a given program, some other libraries might change the default
precision and not restore it. Thus it is safer to use mpfr_init2
.
Initialize all the mpfr_t
variables of the given va_list
,
set their precision to the default precision and their value to NaN.
See mpfr_init
for more details.
The va_list
is assumed to be composed only of type mpfr_t
(or equivalently mpfr_ptr
).
It begins from x, and ends when it encounters a null pointer (whose
type must also be mpfr_ptr
).
Warning! In a given program, some other libraries might change the default
precision and not restore it. Thus it is safer to use mpfr_inits2
.
This macro declares name as an automatic variable of type mpfr_t
,
initializes it and sets its precision to be exactly prec bits
and its value to NaN. name must be a valid identifier.
You must use this macro in the declaration section.
This macro is much faster than using mpfr_init2
but has some
drawbacks:
mpfr_clear
with variables
created with this macro (the storage is allocated at the point of declaration
and deallocated when the brace-level is exited).
MPFR_USE_EXTENSION
macro to avoid warnings
due to the MPFR_DECL_INIT
implementation.
Set the default precision to be exactly prec bits, where
prec can be any integer between MPFR_PREC_MIN
and
MPFR_PREC_MAX
.
The
precision of a variable means the number of bits used to store its significand.
All
subsequent calls to mpfr_init
or mpfr_inits
will use this precision, but previously
initialized variables are unaffected.
The default precision is set to 53 bits initially.
Note: when MPFR is built with the --enable-thread-safe
configure option,
the default precision is local to each thread. See Memory Handling, for
more information.
Return the current default MPFR precision in bits.
See the documentation of mpfr_set_default_prec
.
Here is an example on how to initialize floating-point variables:
{ mpfr_t x, y; mpfr_init (x); /* use default precision */ mpfr_init2 (y, 256); /* precision exactly 256 bits */ … /* When the program is about to exit, do ... */ mpfr_clear (x); mpfr_clear (y); mpfr_free_cache (); /* free the cache for constants like pi */ }
The following 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.
Reset the precision of x to be exactly prec bits,
and set its value to NaN.
The previous value stored in x is lost. It is equivalent to
a call to mpfr_clear(x)
followed by a call to
mpfr_init2(x, prec)
, but more efficient as no allocation is done in
case the current allocated space for the significand of x is enough.
The precision prec can be any integer between MPFR_PREC_MIN
and
MPFR_PREC_MAX
.
In case you want to keep the previous value stored in x,
use mpfr_prec_round
instead.
Warning! You must not use this function if x was initialized
with MPFR_DECL_INIT
or with mpfr_custom_init_set
(see Custom Interface).
Return the precision of x, i.e., the number of bits used to store its significand.
Next: Combined Initialization and Assignment Functions, Previous: Initialization Functions, Up: MPFR Interface [Index]
These functions assign new values to already initialized floats (see Initialization Functions).
Set the value of rop from op, rounded
toward the given direction rnd.
Note that the input 0 is converted to +0 by mpfr_set_ui
,
mpfr_set_si
, mpfr_set_uj
, mpfr_set_sj
,
mpfr_set_z
, mpfr_set_q
and
mpfr_set_f
, regardless of the rounding mode.
If the system does not support the IEEE 754 standard,
mpfr_set_flt
, mpfr_set_d
, mpfr_set_ld
and
mpfr_set_decimal64
might not preserve the signed zeros.
The mpfr_set_decimal64
function is built only with the configure
option ‘--enable-decimal-float’, which also requires
‘--with-gmp-build’, and when the compiler or
system provides the ‘_Decimal64’ data type
(recent versions of GCC support this data type);
to use mpfr_set_decimal64
, one should define the macro
MPFR_WANT_DECIMAL_FLOATS
before including mpfr.h.
mpfr_set_q
might fail if the numerator (or the
denominator) can not be represented as a mpfr_t
.
Note: If you want to store a floating-point constant to a mpfr_t
,
you should use mpfr_set_str
(or one of the MPFR constant functions,
such as mpfr_const_pi
for Pi) instead of
mpfr_set_flt
, mpfr_set_d
,
mpfr_set_ld
or mpfr_set_decimal64
.
Otherwise the floating-point constant will be first
converted into a reduced-precision (e.g., 53-bit) binary
(or decimal, for mpfr_set_decimal64
) number before
MPFR can work with it.
Set the value of rop from op multiplied by two to the power e, rounded toward the given direction rnd. Note that the input 0 is converted to +0.
Set rop to the value of the string s in base base,
rounded in the direction rnd.
See the documentation of mpfr_strtofr
for a detailed description
of the valid string formats.
Contrary to mpfr_strtofr
, mpfr_set_str
requires the
whole string to represent a valid floating-point number.
The meaning of the return value differs from other MPFR functions:
it is 0 if the entire string up to the final null character
is a valid number in base base; otherwise it is -1, and
rop may have changed (users interested in the ternary value
should use mpfr_strtofr
instead).
Note: it is preferable to use mpfr_strtofr
if one wants to distinguish
between an infinite rop value coming from an infinite s or from
an overflow.
Read a floating-point number from a string nptr in base base,
rounded in the direction rnd; base must be either 0 (to
detect the base, as described below) or a number from 2 to 62 (otherwise
the behavior is undefined). If nptr starts with valid data, the
result is stored in rop and *endptr
points to the
character just after the valid data (if endptr is not a null pointer);
otherwise rop is set to zero (for consistency with strtod
)
and the value of nptr is stored
in the location referenced by endptr (if endptr is not a null
pointer). The usual ternary value is returned.
Parsing follows the standard C strtod
function with some extensions.
After optional leading whitespace, one has a subject sequence consisting of an
optional sign (+
or -
), and either numeric data or special
data. The subject sequence is defined as the longest initial subsequence of
the input string, starting with the first non-whitespace character, that is of
the expected form.
The form of numeric data is a non-empty sequence of significand digits with an
optional decimal point, and an optional exponent consisting of an exponent
prefix followed by an optional sign and a non-empty sequence of decimal
digits. A significand digit is either a decimal digit or a Latin letter (62
possible characters), with A
= 10, B
= 11, …, Z
=
35; case is ignored in bases less or equal to 36, in bases larger than 36,
a
= 36, b
= 37, …, z
= 61.
The value of a
significand digit must be strictly less than the base. The decimal point can
be either the one defined by the current locale or the period (the first one
is accepted for consistency with the C standard and the practice, the second
one is accepted to allow the programmer to provide MPFR numbers from strings
in a way that does not depend on the current locale).
The exponent prefix can be e
or E
for bases up to 10, or
@
in any base; it indicates a multiplication by a power of the
base. In bases 2 and 16, the exponent prefix can also be p
or P
,
in which case the exponent, called binary exponent, indicates a
multiplication by a power of 2 instead of the base (there is a difference
only for base 16); in base 16 for example 1p2
represents 4 whereas
1@2
represents 256. The value of an exponent is always written in
base 10.
If the argument base is 0, then the base is automatically detected
as follows. If the significand starts with 0b
or 0B
, base 2
is assumed. If the significand starts with 0x
or 0X
, base 16
is assumed. Otherwise base 10 is assumed.
Note: The exponent (if present)
must contain at least a digit. Otherwise the possible
exponent prefix and sign are not part of the number (which ends with the
significand). Similarly, if 0b
, 0B
, 0x
or 0X
is not followed by a binary/hexadecimal digit, then the subject sequence
stops at the character 0
, thus 0 is read.
Special data (for infinities and NaN) can be @inf@
or
@nan@(n-char-sequence-opt)
, and if base <= 16,
it can also be infinity
, inf
, nan
or
nan(n-char-sequence-opt)
, all case insensitive.
A n-char-sequence-opt
is a possibly empty string containing only digits,
Latin letters and the underscore (0, 1, 2, …, 9, a, b, …, z,
A, B, …, Z, _). Note: one has an optional sign for all data, even
NaN.
For example, -@nAn@(This_Is_Not_17)
is a valid representation for NaN
in base 17.
Set the variable x to NaN (Not-a-Number), infinity or zero respectively.
In mpfr_set_inf
or mpfr_set_zero
, x is set to plus
infinity or plus zero iff sign is nonnegative;
in mpfr_set_nan
, the sign bit of the result is unspecified.
Swap the structures pointed to by x and y. In particular,
the values are exchanged without rounding (this may be different from
three mpfr_set
calls using a third auxiliary variable).
Warning! Since the precisions are exchanged, this will affect future
assignments. Moreover, since the significand pointers are also exchanged,
you must not use this function if the allocation method used for x
and/or y does not permit it. This is the case when x and/or
y were declared and initialized with MPFR_DECL_INIT
, and
possibly with mpfr_custom_init_set
(see Custom Interface).
Next: Conversion Functions, Previous: Assignment Functions, Up: MPFR Interface [Index]
Initialize rop and set its value from op, rounded in the direction
rnd.
The precision of rop will be taken from the active default precision,
as set by mpfr_set_default_prec
.
Initialize x and set its value from
the string s in base base,
rounded in the direction rnd.
See mpfr_set_str
.
Next: Basic Arithmetic Functions, Previous: Combined Initialization and Assignment Functions, Up: MPFR Interface [Index]
Convert op to a float
(respectively double
,
long double
or _Decimal64
), using the rounding mode rnd.
If op is NaN, some fixed NaN (either quiet or signaling) or the result
of 0.0/0.0 is returned. If op is ±Inf, an infinity of the same
sign or the result of ±1.0/0.0 is returned. If op is zero, these
functions return a zero, trying to preserve its sign, if possible.
The mpfr_get_decimal64
function is built only under some conditions:
see the documentation of mpfr_set_decimal64
.
Convert op to a long
, an unsigned long
,
an intmax_t
or an uintmax_t
(respectively) after rounding
it with respect to rnd.
If op is NaN, 0 is returned and the erange flag is set.
If op is too big for the return type, the function returns the maximum
or the minimum of the corresponding C type, depending on the direction
of the overflow; the erange flag is set too.
See also mpfr_fits_slong_p
, mpfr_fits_ulong_p
,
mpfr_fits_intmax_p
and mpfr_fits_uintmax_p
.
Return d and set exp (formally, the value pointed to by exp) such that 0.5<=abs(d)<1 and d times 2 raised to exp equals op rounded to double (resp. long double) precision, using the given rounding mode. If op is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and exp is set to 0. If op is NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, and exp is undefined.
Set exp (formally, the value pointed to by exp) and y such that 0.5<=abs(y)<1 and y times 2 raised to exp equals x rounded to the precision of y, using the given rounding mode. If x is zero, then y is set to a zero of the same sign and exp is set to 0. If x is NaN or an infinity, then y is set to the same value and exp is undefined.
Put the scaled significand of op (regarded as an integer, with the
precision of op) into rop, and return the exponent exp
(which may be outside the current exponent range) such that op
exactly equals
rop times 2 raised to the power exp.
If op is zero, the minimal exponent emin
is returned.
If op is NaN or an infinity, the erange flag is set, rop
is set to 0, and the the minimal exponent emin
is returned.
The returned exponent may be less than the minimal exponent emin
of MPFR numbers in the current exponent range; in case the exponent is
not representable in the mpfr_exp_t
type, the erange flag
is set and the minimal value of the mpfr_exp_t
type is returned.
Convert op to a mpz_t
, after rounding it with respect to
rnd. If op is NaN or an infinity, the erange flag is
set, rop is set to 0, and 0 is returned.
Convert op to a mpf_t
, after rounding it with respect to
rnd.
The erange flag is set if op is NaN or an infinity, which
do not exist in MPF. If op is NaN, then rop is undefined.
If op is +Inf (resp. -Inf), then rop is set to
the maximum (resp. minimum) value in the precision of the MPF number;
if a future MPF version supports infinities, this behavior will be
considered incorrect and will change (portable programs should assume
that rop is set either to this finite number or to an infinite
number).
Note that since MPFR currently has the same exponent type as MPF (but
not with the same radix), the range of values is much larger in MPF
than in MPFR, so that an overflow or underflow is not possible.
Convert op to a string of digits in base b, with rounding in
the direction rnd, where n is either zero (see below) or the
number of significant digits output in the string; in the latter case,
n must be greater or equal to 2. The base may vary from 2 to 62;
otherwise the function does nothing and immediately returns a null pointer.
If the input number is an ordinary number, the exponent is written through
the pointer expptr (for input 0, the current minimal exponent is
written); the type mpfr_exp_t
is large enough to hold the exponent
in all cases.
The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at expptr. If rnd is to nearest, and op is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of op. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal.
If n is zero, the number of digits of the significand is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of op. More precisely, in most cases, the chosen precision of str is the minimal precision m depending only on p = PREC(op) and b that satisfies the above property, i.e., m = 1 + ceil(p*log(2)/log(b)), with p replaced by p-1 if b is a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is when p equals 186564318007 for bases 7 and 49).
If str is a null pointer, space for the significand is allocated using
the current allocation function and a pointer to the string is returned
(unless the base is invalid).
To free the returned string, you must use mpfr_free_str
.
If str is not a null pointer, it should point to a block of storage
large enough for the significand, i.e., at least max(n + 2, 7)
.
The extra two bytes are for a possible minus sign, and for the terminating null
character, and the value 7 accounts for -@Inf@
plus the terminating null character. The pointer to the string str
is returned (unless the base is invalid).
Note: The NaN and inexact flags are currently not set when need be; this will be fixed in future versions. Programmers should currently assume that whether the flags are set by this function is unspecified.
Free a string allocated by mpfr_get_str
using the current unallocation
function.
The block is assumed to be strlen(str)+1
bytes.
For more information about how it is done:
see Section “Custom Allocation” in GNU MP.
Return non-zero if op would fit in the respective C data type,
respectively unsigned long
, long
, unsigned int
,
int
, unsigned short
, short
, uintmax_t
,
intmax_t
, when rounded to an integer in the direction rnd.
Next: Comparison Functions, Previous: Conversion Functions, Up: MPFR Interface [Index]
Set rop to op1 + op2 rounded in the direction
rnd. The IEEE-754 rules are used, in particular for signed zeros.
But for types having no signed zeros, 0 is considered unsigned
(i.e., (+0) + 0 = (+0) and (-0) + 0 = (-0)).
The mpfr_add_d
function assumes that the radix of the double
type
is a power of 2, with a precision at most that declared by the C implementation
(macro IEEE_DBL_MANT_DIG
, and if not defined 53 bits).
Set rop to op1 - op2 rounded in the direction
rnd. The IEEE-754 rules are used, in particular for signed zeros.
But for types having no signed zeros, 0 is considered unsigned
(i.e., (+0) - 0 = (+0), (-0) - 0 = (-0),
0 - (+0) = (-0) and 0 - (-0) = (+0)).
The same restrictions than for mpfr_add_d
apply to mpfr_d_sub
and mpfr_sub_d
.
Set rop to op1 times op2 rounded in the
direction rnd.
When a result is zero, its sign is the product of the signs of the operands
(for types having no signed zeros, 0 is considered positive).
The same restrictions than for mpfr_add_d
apply to mpfr_mul_d
.
Set rop to the square of op rounded in the direction rnd.
Set rop to op1/op2 rounded in the direction rnd.
When a result is zero, its sign is the product of the signs of the operands
(for types having no signed zeros, 0 is considered positive).
The same restrictions than for mpfr_add_d
apply to mpfr_d_div
and mpfr_div_d
.
Set rop to the square root of op rounded in the direction rnd. Set rop to -0 if op is -0, to be consistent with the IEEE 754 standard. Set rop to NaN if op is negative.
Set rop to the reciprocal square root of op rounded in the direction rnd. Set rop to +Inf if op is ±0, +0 if op is +Inf, and NaN if op is negative. Warning! Therefore the result on -0 is different from the one of the rSqrt function recommended by the IEEE 754-2008 standard (Section 9.2.1), which is -Inf instead of +Inf.
Set rop to the cubic root (resp. the kth root) of op rounded in the direction rnd. For k odd (resp. even) and op negative (including -Inf), set rop to a negative number (resp. NaN). The kth root of -0 is defined to be -0, whatever the parity of k.
Set rop to op1 raised to op2,
rounded in the direction rnd.
Special values are handled as described in the ISO C99 and IEEE 754-2008
standards for the pow
function:
pow(±0, y)
returns plus or minus infinity for y a negative odd integer.
pow(±0, y)
returns plus infinity for y negative and not an odd integer.
pow(±0, y)
returns plus or minus zero for y a positive odd integer.
pow(±0, y)
returns plus zero for y positive and not an odd integer.
pow(-1, ±Inf)
returns 1.
pow(+1, y)
returns 1 for any y, even a NaN.
pow(x, ±0)
returns 1 for any x, even a NaN.
pow(x, y)
returns NaN for finite negative x and finite non-integer y.
pow(x, -Inf)
returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1.
pow(x, +Inf)
returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1.
pow(-Inf, y)
returns minus zero for y a negative odd integer.
pow(-Inf, y)
returns plus zero for y negative and not an odd integer.
pow(-Inf, y)
returns minus infinity for y a positive odd integer.
pow(-Inf, y)
returns plus infinity for y positive and not an odd integer.
pow(+Inf, y)
returns plus zero for y negative, and plus infinity for y positive.
Set rop to -op and the absolute value of op respectively, rounded in the direction rnd. Just changes or adjusts the sign if rop and op are the same variable, otherwise a rounding might occur if the precision of rop is less than that of op.
Set rop to the positive difference of op1 and op2, i.e., op1 - op2 rounded in the direction rnd if op1 > op2, +0 if op1 <= op2, and NaN if op1 or op2 is NaN.
Set rop to op1 times 2 raised to op2 rounded in the direction rnd. Just increases the exponent by op2 when rop and op1 are identical.
Set rop to op1 divided by 2 raised to op2 rounded in the direction rnd. Just decreases the exponent by op2 when rop and op1 are identical.
Next: Special Functions, Previous: Basic Arithmetic Functions, Up: MPFR Interface [Index]
Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. Both op1 and op2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the erange flag and return zero.
Note: These functions may be useful to distinguish the three possible cases.
If you need to distinguish two cases only, it is recommended to use the
predicate functions (e.g., mpfr_equal_p
for the equality) described
below; they behave like the IEEE 754 comparisons, in particular when one
or both arguments are NaN. But only floating-point numbers can be compared
(you may need to do a conversion first).
Compare op1 and op2 multiplied by two to the power e. Similar as above.
Compare |op1| and |op2|. Return a positive value if |op1| > |op2|, zero if |op1| = |op2|, and a negative value if |op1| < |op2|. If one of the operands is NaN, set the erange flag and return zero.
Return non-zero if op is respectively NaN, an infinity, an ordinary number (i.e., neither NaN nor an infinity), zero, or a regular number (i.e., neither NaN, nor an infinity nor zero). Return zero otherwise.
Return a positive value if op > 0, zero if op = 0,
and a negative value if op < 0.
If the operand is NaN, set the erange flag and return zero.
This is equivalent to mpfr_cmp_ui (op, 0)
, but more efficient.
Return non-zero if op1 > op2, op1 >= op2, op1 < op2, op1 <= op2, op1 = op2 respectively, and zero otherwise. Those functions return zero whenever op1 and/or op2 is NaN.
Return non-zero if op1 < op2 or op1 > op2 (i.e., neither op1, nor op2 is NaN, and op1 <> op2), zero otherwise (i.e., op1 and/or op2 is NaN, or op1 = op2).
Return non-zero if op1 or op2 is a NaN (i.e., they cannot be compared), zero otherwise.
Next: Input and Output Functions, Previous: Comparison Functions, Up: MPFR Interface [Index]
All those functions, except explicitly stated (for example
mpfr_sin_cos
), return a ternary value, i.e., zero for an
exact return value, a positive value for a return value larger than the
exact result, and a negative value otherwise.
Important note: in some domains, computing special functions (either with correct or incorrect rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument.
Set rop to the natural logarithm of op, log2(op) or log10(op), respectively, rounded in the direction rnd. Set rop to +0 if op is 1 (in all rounding modes), for consistency with the ISO C99 and IEEE 754-2008 standards. Set rop to -Inf if op is ±0 (i.e., the sign of the zero has no influence on the result).
Set rop to the exponential of op, to 2 power of op or to 10 power of op, respectively, rounded in the direction rnd.
Set rop to the cosine of op, sine of op, tangent of op, rounded in the direction rnd.
Set simultaneously sop to the sine of op and cop to the cosine of op, rounded in the direction rnd with the corresponding precisions of sop and cop, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if sop is exact, s=1 if sop is larger than the sine of op, s=2 if sop is smaller than the sine of op, and similarly for c and the cosine of op.
Set rop to the secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set rop to the arc-cosine, arc-sine or arc-tangent of op,
rounded in the direction rnd.
Note that since acos(-1)
returns the floating-point number closest to
Pi according to the given rounding mode, this number might not be
in the output range 0 <= rop < \pi
of the arc-cosine function;
still, the result lies in the image of the output range
by the rounding function.
The same holds for asin(-1)
, asin(1)
, atan(-Inf)
,
atan(+Inf)
or for atan(op)
with large op and
small precision of rop.
Set rop to the arc-tangent2 of y and x,
rounded in the direction rnd:
if x > 0
, atan2(y, x) = atan (y/x)
;
if x < 0
, atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))
,
thus a number from -Pi to Pi.
As for atan
, in case the exact mathematical result is +Pi or
-Pi,
its rounded result might be outside the function output range.
atan2(y, 0)
does not raise any floating-point exception.
Special values are handled as described in the ISO C99 and IEEE 754-2008
standards for the atan2
function:
atan2(+0, -0)
returns +Pi.
atan2(-0, -0)
returns -Pi.
atan2(+0, +0)
returns +0.
atan2(-0, +0)
returns -0.
atan2(+0, x)
returns +Pi for x < 0.
atan2(-0, x)
returns -Pi for x < 0.
atan2(+0, x)
returns +0 for x > 0.
atan2(-0, x)
returns -0 for x > 0.
atan2(y, 0)
returns -Pi/2 for y < 0.
atan2(y, 0)
returns +Pi/2 for y > 0.
atan2(+Inf, -Inf)
returns +3*Pi/4.
atan2(-Inf, -Inf)
returns -3*Pi/4.
atan2(+Inf, +Inf)
returns +Pi/4.
atan2(-Inf, +Inf)
returns -Pi/4.
atan2(+Inf, x)
returns +Pi/2 for finite x.
atan2(-Inf, x)
returns -Pi/2 for finite x.
atan2(y, -Inf)
returns +Pi for finite y > 0.
atan2(y, -Inf)
returns -Pi for finite y < 0.
atan2(y, +Inf)
returns +0 for finite y > 0.
atan2(y, +Inf)
returns -0 for finite y < 0.
Set rop to the hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.
Set simultaneously sop to the hyperbolic sine of op and
cop to the hyperbolic cosine of op,
rounded in the direction rnd with the corresponding precision of
sop and cop, which must be different variables.
Return 0 iff both results are exact (see mpfr_sin_cos
for a more
detailed description of the return value).
Set rop to the hyperbolic secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set rop to the inverse hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.
Set rop to the factorial of op, rounded in the direction rnd.
Set rop to the logarithm of one plus op, rounded in the direction rnd.
Set rop to the exponential of op followed by a subtraction by one, rounded in the direction rnd.
Set rop to the exponential integral of op, rounded in the direction rnd. For positive op, the exponential integral is the sum of Euler’s constant, of the logarithm of op, and of the sum for k from 1 to infinity of op to the power k, divided by k and factorial(k). For negative op, rop is set to NaN (this definition for negative argument follows formula 5.1.2 from the Handbook of Mathematical Functions from Abramowitz and Stegun, a future version might use another definition).
Set rop to real part of the dilogarithm of op, rounded in the direction rnd. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to op.
Set rop to the value of the Gamma function on op, rounded in the direction rnd. When op is a negative integer, rop is set to NaN.
Set rop to the value of the logarithm of the Gamma function on op,
rounded in the direction rnd.
When op is 1 or 2, set rop to +0 (in all rounding modes).
When op is an infinity or a nonpositive integer, set rop to +Inf,
following the general rules on special values.
When -2k-1 < op < -2k,
k being a nonnegative integer, set rop to NaN.
See also mpfr_lgamma
.
Set rop to the value of the logarithm of the absolute value of the Gamma function on op, rounded in the direction rnd. The sign (1 or -1) of Gamma(op) is returned in the object pointed to by signp. When op is 1 or 2, set rop to +0 (in all rounding modes). When op is an infinity or a nonpositive integer, set rop to +Inf. When op is NaN, -Inf or a negative integer, *signp is undefined, and when op is ±0, *signp is the sign of the zero.
Set rop to the value of the Digamma (sometimes also called Psi) function on op, rounded in the direction rnd. When op is a negative integer, set rop to NaN.
Set rop to the value of the Riemann Zeta function on op, rounded in the direction rnd.
Set rop to the value of the error function on op (resp. the complementary error function on op) rounded in the direction rnd.
Set rop to the value of the first kind Bessel function of order 0, (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN, rop is always set to NaN. When op is plus or minus Infinity, rop is set to +0. When op is zero, and n is not zero, rop is set to +0 or -0 depending on the parity and sign of n, and the sign of op.
Set rop to the value of the second kind Bessel function of order 0 (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN or negative, rop is always set to NaN. When op is +Inf, rop is set to +0. When op is zero, rop is set to +Inf or -Inf depending on the parity and sign of n.
Set rop to (op1 times op2) + op3 (resp. (op1 times op2) - op3) rounded in the direction rnd. Concerning special values (signed zeros, infinities, NaN), these functions behave like a multiplication followed by a separate addition or subtraction. That is, the fused operation matters only for rounding.
Set rop to the arithmetic-geometric mean of op1 and op2, rounded in the direction rnd. The arithmetic-geometric mean is the common limit of the sequences u_n and v_n, where u_0=op1, v_0=op2, u_(n+1) is the arithmetic mean of u_n and v_n, and v_(n+1) is the geometric mean of u_n and v_n. If any operand is negative, set rop to NaN.
Set rop to the Euclidean norm of x and y, i.e., the square root of the sum of the squares of x and y, rounded in the direction rnd. Special values are handled as described in the ISO C99 (Section F.9.4.3) and IEEE 754-2008 (Section 9.2.1) standards: If x or y is an infinity, then +Inf is returned in rop, even if the other number is NaN.
Set rop to the value of the Airy function Ai on x, rounded in the direction rnd. When x is NaN, rop is always set to NaN. When x is +Inf or -Inf, rop is +0. The current implementation is not intended to be used with large arguments. It works with abs(x) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version.
Set rop to the logarithm of 2, the value of Pi,
of Euler’s constant 0.577…, of Catalan’s constant 0.915…,
respectively, rounded in the direction
rnd. These functions cache the computed values to avoid other
calculations if a lower or equal precision is requested. To free these caches,
use mpfr_free_cache
.
Free various caches used by MPFR internally, in particular the
caches used by the functions computing constants (mpfr_const_log2
,
mpfr_const_pi
,
mpfr_const_euler
and mpfr_const_catalan
).
You should call this function before terminating a thread, even if you did
not call these functions directly (they could have been called internally).
Set rop to the sum of all elements of tab, whose size is n,
rounded in the direction rnd. Warning: for efficiency reasons,
tab is an array of pointers
to mpfr_t
, not an array of mpfr_t
.
If the returned int
value is zero, rop is guaranteed to be the
exact sum; otherwise rop might be smaller than, equal to, or larger than
the exact sum (in accordance to the rounding mode).
However, mpfr_sum
does guarantee the result is correctly rounded.
Next: Formatted Output Functions, Previous: Special Functions, Up: MPFR Interface [Index]
This section describes functions that perform input from an input/output
stream, and functions that output to an input/output stream.
Passing a null pointer for a stream
to any of these functions will make
them read from stdin
and write to stdout
, respectively.
When using any of these functions, you must include the <stdio.h>
standard header before mpfr.h, to allow mpfr.h to define
prototypes for these functions.
Output op on stream stream, as a string of digits in
base base, rounded in the direction rnd.
The base may vary from 2 to 62. Print n significant digits exactly,
or if n is 0, enough digits so that op can be read back
exactly (see mpfr_get_str
).
In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form ‘eNNN’, are printed. If base is greater than 10, ‘@’ will be used instead of ‘e’ as exponent delimiter.
Return the number of characters written, or if an error occurred, return 0.
Input a string in base base from stream stream, rounded in the direction rnd, and put the read float in rop.
This function reads a word (defined as a sequence of characters between
whitespace) and parses it using mpfr_set_str
.
See the documentation of mpfr_strtofr
for a detailed description
of the valid string formats.
Return the number of bytes read, or if an error occurred, return 0.
Next: Integer Related Functions, Previous: Input and Output Functions, Up: MPFR Interface [Index]
The class of mpfr_printf
functions provides formatted output in a
similar manner as the standard C printf
. These functions are defined
only if your system supports ISO C variadic functions and the corresponding
argument access macros.
When using any of these functions, you must include the <stdio.h>
standard header before mpfr.h, to allow mpfr.h to define
prototypes for these functions.
The format specification accepted by mpfr_printf
is an extension of the
printf
one. The conversion specification is of the form:
% [flags] [width] [.[precision]] [type] [rounding] conv
‘flags’, ‘width’, and ‘precision’ have the same meaning as for
the standard printf
(in particular, notice that the ‘precision’ is
related to the number of digits displayed in the base chosen by ‘conv’
and not related to the internal precision of the mpfr_t
variable).
mpfr_printf
accepts the same ‘type’ specifiers as GMP (except the
non-standard and deprecated ‘q’, use ‘ll’ instead), namely the
length modifiers defined in the C standard:
‘h’ short
‘hh’ char
‘j’ intmax_t
oruintmax_t
‘l’ long
orwchar_t
‘ll’ long long
‘L’ long double
‘t’ ptrdiff_t
‘z’ size_t
and the ‘type’ specifiers defined in GMP plus ‘R’ and ‘P’ specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of ‘conv’ specifier to use after the ‘type’ specifier):
‘F’ mpf_t
, float conversions‘Q’ mpq_t
, integer conversions‘M’ mp_limb_t
, integer conversions‘N’ mp_limb_t
array, integer conversions‘Z’ mpz_t
, integer conversions‘P’ mpfr_prec_t
, integer conversions‘R’ mpfr_t
, float conversions
The ‘type’ specifiers have the same restrictions as those
mentioned in the GMP documentation:
see Section “Formatted Output Strings” in GNU MP.
In particular, the ‘type’ specifiers (except ‘R’ and ‘P’) are
supported only if they are supported by gmp_printf
in your GMP build;
this implies that the standard specifiers, such as ‘t’, must also
be supported by your C library if you want to use them.
The ‘rounding’ field is specific to mpfr_t
arguments and should
not be used with other types.
With conversion specification not involving ‘P’ and ‘R’ types,
mpfr_printf
behaves exactly as gmp_printf
.
The ‘P’ type specifies that a following ‘d’, ‘i’,
‘o’, ‘u’, ‘x’, or ‘X’ conversion specifier applies
to a mpfr_prec_t
argument.
It is needed because the mpfr_prec_t
type does not necessarily
correspond to an int
or any fixed standard type.
The ‘precision’ field specifies the minimum number of digits to
appear. The default ‘precision’ is 1.
For example:
mpfr_t x; mpfr_prec_t p; mpfr_init (x); … p = mpfr_get_prec (x); mpfr_printf ("variable x with %Pu bits", p);
The ‘R’ type specifies that a following ‘a’, ‘A’, ‘b’,
‘e’, ‘E’, ‘f’, ‘F’, ‘g’, ‘G’, or ‘n’
conversion specifier applies to a mpfr_t
argument.
The ‘R’ type can be followed by a ‘rounding’ specifier denoted by
one of the following characters:
‘U’ round toward plus infinity ‘D’ round toward minus infinity ‘Y’ round away from zero ‘Z’ round toward zero ‘N’ round to nearest (with ties to even) ‘*’ rounding mode indicated by the mpfr_rnd_t
argument just before the correspondingmpfr_t
variable.
The default rounding mode is rounding to nearest. The following three examples are equivalent:
mpfr_t x; mpfr_init (x); … mpfr_printf ("%.128Rf", x); mpfr_printf ("%.128RNf", x); mpfr_printf ("%.128R*f", MPFR_RNDN, x);
Note that the rounding away from zero mode is specified with ‘Y’ because ISO C reserves the ‘A’ specifier for hexadecimal output (see below).
The output ‘conv’ specifiers allowed with mpfr_t
parameter are:
‘a’ ‘A’ hex float, C99 style ‘b’ binary output ‘e’ ‘E’ scientific format float ‘f’ ‘F’ fixed point float ‘g’ ‘G’ fixed or scientific float
The conversion specifier ‘b’ which displays the argument in binary is
specific to mpfr_t
arguments and should not be used with other types.
Other conversion specifiers have the same meaning as for a double
argument.
In case of non-decimal output, only the significand is written in the
specified base, the exponent is always displayed in decimal.
Special values are always displayed as nan
, -inf
, and inf
for ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers and
NAN
, -INF
, and INF
for ‘A’, ‘E’, ‘F’, and
‘G’ specifiers.
If the ‘precision’ field is not empty, the mpfr_t
number is
rounded to the given precision in the direction specified by the rounding
mode.
If the precision is zero with rounding to nearest mode and one of the
following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’,
‘E’, tie case is rounded to even when it lies between two consecutive
values at the
wanted precision which have the same exponent, otherwise, it is rounded away
from zero.
For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the
format specification "%.0RNe"
.
This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses
the ‘e’ (resp. ‘E’) style.
If the precision is set to a value greater than the maximum value for an
int
, it will be silently reduced down to INT_MAX
.
If the ‘precision’ field is empty (as in %Re
or %.RE
) with
‘conv’ specifier ‘e’ and ‘E’, the number is displayed with
enough digits so that it can be read back exactly, assuming that the input and
output variables have the same precision and that the input and output
rounding modes are both rounding to nearest (as for mpfr_get_str
).
The default precision for an empty ‘precision’ field with ‘conv’
specifiers ‘f’, ‘F’, ‘g’, and ‘G’ is 6.
For all the following functions, if the number of characters which ought to be
written appears to exceed the maximum limit for an int
, nothing is
written in the stream (resp. to stdout
, to buf, to str),
the function returns -1, sets the erange flag, and (in
POSIX system only) errno
is set to EOVERFLOW
.
Print to the stream stream the optional arguments under the control of the template string template. Return the number of characters written or a negative value if an error occurred.
Print to stdout
the optional arguments under the control of the
template string template.
Return the number of characters written or a negative value if an error
occurred.
Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. No overlap is permitted between buf and the other arguments. Return the number of characters written in the array buf not counting the terminating null character or a negative value if an error occurred.
Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. If n is zero, nothing is written and buf may be a null pointer, otherwise, the n-1 first characters are written in buf and the n-th is a null character. Return the number of characters that would have been written had n be sufficiently large, not counting the terminating null character, or a negative value if an error occurred.
Write their output as a null terminated string in a block of memory allocated
using the current allocation function. A pointer to the block is stored in
str. The block of memory must be freed using mpfr_free_str
.
The return value is the number of characters written in the string, excluding
the null-terminator, or a negative value if an error occurred.
Next: Rounding Related Functions, Previous: Formatted Output Functions, Up: MPFR Interface [Index]
Set rop to op rounded to an integer.
mpfr_rint
rounds to the nearest representable integer in the
given direction rnd, mpfr_ceil
rounds
to the next higher or equal representable integer, mpfr_floor
to
the next lower or equal representable integer, mpfr_round
to the
nearest representable integer, rounding halfway cases away from zero
(as in the roundTiesToAway mode of IEEE 754-2008),
and mpfr_trunc
to the next representable integer toward zero.
The returned value is zero when the result is exact, positive when it is greater than the original value of op, and negative when it is smaller. More precisely, the returned value is 0 when op is an integer representable in rop, 1 or -1 when op is an integer that is not representable in rop, 2 or -2 when op is not an integer.
When op is NaN, the NaN flag is set as usual. In the other cases,
the inexact flag is set when rop differs from op, following
the ISO C99 rule for the rint
function. If you want the behavior to
be more like IEEE 754 / ISO TS 18661-1, i.e., the usual behavior where the
round-to-integer function is regarded as any other mathematical function,
you should use one the mpfr_rint_*
functions instead (however it is
not possible to round to nearest with the even rounding rule yet).
Note that mpfr_round
is different from mpfr_rint
called with
the rounding to nearest mode (where halfway cases are rounded to an even
integer or significand). Note also that no double rounding is performed; for
instance, 10.5 (1010.1 in binary) is rounded by mpfr_rint
with
rounding to nearest to 12 (1100
in binary) in 2-bit precision, because the two enclosing numbers representable
on two bits are 8 and 12, and the closest is 12.
(If one first rounded to an integer, one would round 10.5 to 10 with
even rounding, and then 10 would be rounded to 8 again with even rounding.)
Set rop to op rounded to an integer.
mpfr_rint_ceil
rounds to the next higher or equal integer,
mpfr_rint_floor
to the next lower or equal integer,
mpfr_rint_round
to the nearest integer, rounding halfway cases away
from zero, and mpfr_rint_trunc
to the next integer toward zero.
If the result is not representable, it is rounded in the direction rnd.
The returned value is the ternary value associated with the considered
round-to-integer function (regarded in the same way as any other
mathematical function).
Contrary to mpfr_rint
, those functions do perform a double rounding:
first op is rounded to the nearest integer in the direction given by
the function name, then this nearest integer (if not representable) is
rounded in the given direction rnd. Thus these round-to-integer
functions behave more like the other mathematical functions, i.e., the
returned result is the correct rounding of the exact result of the function
in the real numbers.
For example, mpfr_rint_round
with rounding to nearest and a precision
of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is
rounded to 8 by the round-even rule, despite the fact that 6 is also
representable on two bits, and is closer to 6.5 than 8.
Set rop to the fractional part of op, having the same sign as
op, rounded in the direction rnd (unlike in mpfr_rint
,
rnd affects only how the exact fractional part is rounded, not how
the fractional part is generated).
Set simultaneously iop to the integral part of op and fop to
the fractional part of op, rounded in the direction rnd with the
corresponding precision of iop and fop (equivalent to
mpfr_trunc(iop, op, rnd)
and
mpfr_frac(fop, op, rnd)
). The variables iop and
fop must be different. Return 0 iff both results are exact (see
mpfr_sin_cos
for a more detailed description of the return value).
Set r to the value of x - ny, rounded
according to the direction rnd, where n is the integer quotient
of x divided by y, defined as follows: n is rounded
toward zero for mpfr_fmod
, and to the nearest integer (ties rounded
to even) for mpfr_remainder
and mpfr_remquo
.
Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If x is infinite or y is zero, r is NaN. If y is infinite and x is finite, r is x rounded to the precision of r. If r is zero, it has the sign of x. The return value is the ternary value corresponding to r.
Additionally, mpfr_remquo
stores
the low significant bits from the quotient n in *q
(more precisely the number of bits in a long
minus one),
with the sign of x divided by y
(except if those low bits are all zero, in which case zero is returned).
Note that x may be so large in magnitude relative to y that an
exact representation of the quotient is not practical.
The mpfr_remainder
and mpfr_remquo
functions are useful for
additive argument reduction.
Return non-zero iff op is an integer.
Next: Miscellaneous Functions, Previous: Integer Related Functions, Up: MPFR Interface [Index]
Set the default rounding mode to rnd. The default rounding mode is to nearest initially.
Get the default rounding mode.
Round x according to rnd with precision prec, which
must be an integer between MPFR_PREC_MIN
and MPFR_PREC_MAX
(otherwise the behavior is undefined).
If prec is greater or equal to the precision of x, then new
space is allocated for the significand, and it is filled with zeros.
Otherwise, the significand is rounded to precision prec with the given
direction. In both cases, the precision of x is changed to prec.
Here is an example of how to use mpfr_prec_round
to implement
Newton’s algorithm to compute the inverse of a, assuming x is
already an approximation to n bits:
mpfr_set_prec (t, 2 * n); mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */
Warning! You must not use this function if x was initialized
with MPFR_DECL_INIT
or with mpfr_custom_init_set
(see Custom Interface).
Assuming b is an approximation of an unknown number x in the direction rnd1 with error at most two to the power E(b)-err where E(b) is the exponent of b, return a non-zero value if one is able to round correctly x to precision prec with the direction rnd2, and 0 otherwise (including for NaN and Inf). This function does not modify its arguments.
If rnd1 is MPFR_RNDN
, then the sign of the error is
unknown, but its absolute value is the same, so that the possible range
is twice as large as with a directed rounding for rnd1.
Note: if one wants to also determine the correct ternary value when rounding b to precision prec with rounding mode rnd, a useful trick is the following:
if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ...
Indeed, if rnd is MPFR_RNDN
, this will check if one can
round to prec+1 bits with a directed rounding:
if so, one can surely round to nearest to prec bits,
and in addition one can determine the correct ternary value, which would not
be the case when b is near from a value exactly representable on
prec bits.
Return the minimal number of bits required to store the significand of
x, and 0 for special values, including 0. (Warning: the returned
value can be less than MPFR_PREC_MIN
.)
The function name is subject to change.
Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode rnd, or a null pointer if rnd is an invalid rounding mode.
Next: Exception Related Functions, Previous: Rounding Related Functions, Up: MPFR Interface [Index]
If x or y is NaN, set x to NaN. If x and y are equal, x is unchanged. Otherwise, if x is different from y, replace x by the next floating-point number (with the precision of x and the current exponent range) in the direction of y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated.
Equivalent to mpfr_nexttoward
where y is plus infinity
(resp. minus infinity).
Set rop to the minimum (resp. maximum) of op1 and op2. If op1 and op2 are both NaN, then rop is set to NaN. If op1 or op2 is NaN, then rop is set to the numeric value. If op1 and op2 are zeros of different signs, then rop is set to -0 (resp. +0).
Generate a uniformly distributed random float in the interval 0 <= rop < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if e denotes the exponent after normalization, then the least -e significant bits of the significand are always 0).
Return 0, unless the exponent is not in the current exponent range, in
which case rop is set to NaN and a non-zero value is returned (this
should never happen in practice, except in very specific cases). The
second argument is a gmp_randstate_t
structure which should be
created using the GMP gmp_randinit
function (see the GMP manual).
Note: for a given version of MPFR, the returned value of rop and the new value of state (which controls further random values) do not depend on the machine word size.
Generate a uniformly distributed random float. The floating-point number rop can be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the direction rnd.
The second argument is a gmp_randstate_t
structure which should be
created using the GMP gmp_randinit
function (see the GMP manual).
Note: the note for mpfr_urandomb
holds too. In addition, the exponent
range and the rounding mode might have a side effect on the next random state.
Generate two random floats according to a standard normal gaussian distribution. If rop2 is a null pointer, then only one value is generated and stored in rop1.
The floating-point number rop1 (and rop2) can be seen as if a random real number were generated according to the standard normal gaussian distribution and then rounded in the direction rnd.
The third argument is a gmp_randstate_t
structure, which should be
created using the GMP gmp_randinit
function (see the GMP manual).
The combination of the ternary values is returned like with
mpfr_sin_cos
. If rop2 is a null pointer, the second ternary
value is assumed to be 0 (note that the encoding of the only ternary value
is not the same as the usual encoding for functions that return only one
result). Otherwise the ternary value of a random number is always non-zero.
Note: the note for mpfr_urandomb
holds too. In addition, the exponent
range and the rounding mode might have a side effect on the next random state.
Return the exponent of x, assuming that x is a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined.
Set the exponent of x if e is in the current exponent range, and return 0 (even if x is not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1).
Return a non-zero value iff op has its sign bit set (i.e., if it is negative, -0, or a NaN whose representation has its sign bit set).
Set the value of rop from op, rounded toward the given direction rnd, then set (resp. clear) its sign bit if s is non-zero (resp. zero), even when op is a NaN.
Set the value of rop from op1, rounded toward the given
direction rnd, then set its sign bit to that of op2 (even
when op1 or op2 is a NaN). This function is equivalent to
mpfr_setsign (rop, op1, mpfr_signbit (op2), rnd)
.
Return the MPFR version, as a null-terminated string.
MPFR_VERSION
is the version of MPFR as a preprocessing constant.
MPFR_VERSION_MAJOR
, MPFR_VERSION_MINOR
and
MPFR_VERSION_PATCHLEVEL
are respectively the major, minor and patch
level of MPFR version, as preprocessing constants.
MPFR_VERSION_STRING
is the version (with an optional suffix, used
in development and pre-release versions) as a string constant, which can
be compared to the result of mpfr_get_version
to check at run time
the header file and library used match:
if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Warning: header and library do not match\n");
Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system).
Create an integer in the same format as used by MPFR_VERSION
from the
given major, minor and patchlevel.
Here is an example of how to check the MPFR version at compile time:
#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0))) # error "Wrong MPFR version." #endif
Return a null-terminated string containing the ids of the patches applied to the MPFR library (contents of the PATCHES file), separated by spaces. Note: If the program has been compiled with an older MPFR version and is dynamically linked with a new MPFR library version, the identifiers of the patches applied to the old (compile-time) MPFR version are not available (however this information should not have much interest in general).
Return a non-zero value if MPFR was compiled as thread safe using
compiler-level Thread Local Storage (that is, MPFR was built with the
--enable-thread-safe
configure option, see INSTALL
file), return
zero otherwise.
Return a non-zero value if MPFR was compiled with decimal float support (that
is, MPFR was built with the --enable-decimal-float
configure option),
return zero otherwise.
Return a non-zero value if MPFR was compiled with GMP internals
(that is, MPFR was built with either --with-gmp-build
or
--enable-gmp-internals
configure option), return zero otherwise.
Return a string saying which thresholds file has been used at compile time. This file is normally selected from the processor type.
Next: Compatibility with MPF, Previous: Miscellaneous Functions, Up: MPFR Interface [Index]
Return the (current) smallest and largest exponents allowed for a floating-point variable. The smallest positive value of a floating-point variable is one half times 2 raised to the smallest exponent and the largest value has the form (1 - epsilon) times 2 raised to the largest exponent, where epsilon depends on the precision of the considered variable.
Set the smallest and largest exponents allowed for a floating-point variable.
Return a non-zero value when exp is not in the range accepted by the
implementation (in that case the smallest or largest exponent is not changed),
and zero otherwise.
If the user changes the exponent range, it is her/his responsibility to check
that all current floating-point variables are in the new allowed range
(for example using mpfr_check_range
), otherwise the subsequent
behavior will be undefined, in the sense of the ISO C standard.
Return the minimum and maximum of the exponents
allowed for mpfr_set_emin
and mpfr_set_emax
respectively.
These values are implementation dependent, thus a program using
mpfr_set_emax(mpfr_get_emax_max())
or mpfr_set_emin(mpfr_get_emin_min())
may not be portable.
This function assumes that x is the correctly-rounded value of some
real value y in the direction rnd and some extended exponent
range, and that t is the corresponding ternary value.
For example, one performed t = mpfr_log (x, u, rnd)
, and y is the
exact logarithm of u.
Thus t is negative if x is smaller than y,
positive if x is larger than y, and zero if x equals y.
This function modifies x if needed
to be in the current range of acceptable values: It
generates an underflow or an overflow if the exponent of x is
outside the current allowed range; the value of t may be used
to avoid a double rounding. This function returns zero if the new value of
x equals the exact one y, a positive value if that new value
is larger than y, and a negative value if it is smaller than y.
Note that unlike most functions,
the new result x is compared to the (unknown) exact one y,
not the input value x, i.e., the ternary value is propagated.
Note: If x is an infinity and t is different from zero (i.e.,
if the rounded result is an inexact infinity), then the overflow flag is
set. This is useful because mpfr_check_range
is typically called
(at least in MPFR functions) after restoring the flags that could have
been set due to internal computations.
This function rounds x emulating subnormal number arithmetic:
if x is outside the subnormal exponent range, it just propagates the
ternary value t; otherwise, it rounds x to precision
EXP(x)-emin+1
according to rounding mode rnd and previous
ternary value t, avoiding double rounding problems.
More precisely in the subnormal domain, denoting by e the value of
emin
, x is rounded in fixed-point
arithmetic to an integer multiple of two to the power
e-1; as a consequence, 1.5 multiplied by two to the power e-1 when t is zero
is rounded to two to the power e with rounding to nearest.
PREC(x)
is not modified by this function.
rnd and t must be the rounding mode
and the returned ternary value used when computing x
(as in mpfr_check_range
).
The subnormal exponent range is from emin
to emin+PREC(x)-1
.
If the result cannot be represented in the current exponent range
(due to a too small emax
), the behavior is undefined.
Note that unlike most functions, the result is compared to the exact one,
not the input value x, i.e., the ternary value is propagated.
As usual, if the returned ternary value is non zero, the inexact flag is set. Moreover, if a second rounding occurred (because the input x was in the subnormal range), the underflow flag is set.
This is an example of how to emulate binary double IEEE 754 arithmetic (binary64 in IEEE 754-2008) using MPFR:
{ mpfr_t xa, xb; int i; volatile double a, b; mpfr_set_default_prec (53); mpfr_set_emin (-1073); mpfr_set_emax (1024); mpfr_init (xa); mpfr_init (xb); b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN); a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN); a /= b; i = mpfr_div (xa, xa, xb, MPFR_RNDN); i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */ mpfr_clear (xa); mpfr_clear (xb); }
Warning: this emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware.
Clear the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.
Set the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.
Clear all global flags (underflow, overflow, divide-by-zero, invalid, inexact, erange).
Return the corresponding (underflow, overflow, divide-by-zero, invalid, inexact, erange) flag, which is non-zero iff the flag is set.
Next: Custom Interface, Previous: Exception Related Functions, Up: MPFR Interface [Index]
A header file mpf2mpfr.h is included in the distribution of MPFR for
compatibility with the GNU MP class MPF.
By inserting the following two lines after the #include <gmp.h>
line,
#include <mpfr.h> #include <mpf2mpfr.h>
any program written for
MPF can be compiled directly with MPFR without any changes
(except the gmp_printf
functions will not work for arguments of type
mpfr_t
).
All operations are then performed with the default MPFR rounding mode,
which can be reset with mpfr_set_default_rounding_mode
.
Warning: the mpf_init
and mpf_init2
functions initialize
to zero, whereas the corresponding MPFR functions initialize to NaN:
this is useful to detect uninitialized values, but is slightly incompatible
with MPF.
Reset the precision of x to be exactly prec bits.
The only difference with mpfr_set_prec
is that prec is assumed to
be small enough so that the significand fits into the current allocated memory
space for x. Otherwise the behavior is undefined.
Return non-zero if op1 and op2 are both non-zero ordinary numbers with the same exponent and the same first op3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of op3 larger than 1.
Compute the relative difference between op1 and op2 and store the result in rop. This function does not guarantee the correct rounding on the relative difference; it just computes |op1-op2|/op1, using the precision of rop and the rounding mode rnd for all operations.
These functions are identical to mpfr_mul_2ui
and mpfr_div_2ui
respectively.
These functions are only kept for compatibility with MPF, one should
prefer mpfr_mul_2ui
and mpfr_div_2ui
otherwise.
Next: Internals, Previous: Compatibility with MPF, Up: MPFR Interface [Index]
Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface.
The following interface allows one to use MPFR in two ways:
mpfr_t
on the stack.
mpfr_t
each time it is needed.
Nothing has to be done to destroy the floating-point numbers except garbaging the used memory: all the memory management (allocating, destroying, garbaging) is left to the application.
Each function in this interface is also implemented as a macro for
efficiency reasons: for example mpfr_custom_init (s, p)
uses the macro, while (mpfr_custom_init) (s, p)
uses the function.
Note 1: MPFR functions may still initialize temporary floating-point numbers
using mpfr_init
and similar functions. See Custom Allocation (GNU MP).
Note 2: MPFR functions may use the cached functions (mpfr_const_pi
for
example), even if they are not explicitly called. You have to call
mpfr_free_cache
each time you garbage the memory iff mpfr_init
,
through GMP Custom Allocation, allocates its memory on the application stack.
Return the needed size in bytes to store the significand of a floating-point number of precision prec.
Initialize a significand of precision prec, where
significand must be an area of mpfr_custom_get_size (prec)
bytes
at least and be suitably aligned for an array of mp_limb_t
(GMP type,
see Internals).
Perform a dummy initialization of a mpfr_t
and set it to:
ABS(kind) == MPFR_NAN_KIND
, x is set to NaN;
ABS(kind) == MPFR_INF_KIND
, x is set to the infinity
of sign sign(kind)
;
ABS(kind) == MPFR_ZERO_KIND
, x is set to the zero of
sign sign(kind)
;
ABS(kind) == MPFR_REGULAR_KIND
, x is set to a regular
number: x = sign(kind)*significand*2^exp
.
In all cases, it uses significand directly for further computing
involving x. It will not allocate anything.
A floating-point number initialized with this function cannot be resized using
mpfr_set_prec
or mpfr_prec_round
,
or cleared using mpfr_clear
!
The significand must have been initialized with mpfr_custom_init
using the same precision prec.
Return the current kind of a mpfr_t
as created by
mpfr_custom_init_set
.
The behavior of this function for any mpfr_t
not initialized
with mpfr_custom_init_set
is undefined.
Return a pointer to the significand used by a mpfr_t
initialized with
mpfr_custom_init_set
.
The behavior of this function for any mpfr_t
not initialized
with mpfr_custom_init_set
is undefined.
Return the exponent of x, assuming that x is a non-zero ordinary
number. The return value for NaN, Infinity or zero is unspecified but does not
produce any trap.
The behavior of this function for any mpfr_t
not initialized
with mpfr_custom_init_set
is undefined.
Inform MPFR that the significand of x has moved due to a garbage collect
and update its new position to new_position
.
However the application has to move the significand and the mpfr_t
itself.
The behavior of this function for any mpfr_t
not initialized
with mpfr_custom_init_set
is undefined.
Previous: Custom Interface, Up: MPFR Interface [Index]
A limb means the part of a multi-precision number that fits in a single
word. Usually a limb contains
32 or 64 bits. The C data type for a limb is mp_limb_t
.
The mpfr_t
type is internally defined as a one-element
array of a structure, and mpfr_ptr
is the C data type representing
a pointer to this structure.
The mpfr_t
type consists of four fields:
_mpfr_prec
field is used to store the precision of
the variable (in bits); this is not less than MPFR_PREC_MIN
.
_mpfr_sign
field is used to store the sign of the variable.
_mpfr_exp
field stores the exponent.
An exponent of 0 means a radix point just above the most significant
limb. Non-zero values n are a multiplier 2^n relative to that
point.
A NaN, an infinity and a zero are indicated by special values of the exponent
field.
_mpfr_d
field is a pointer to the limbs, least
significant limbs stored first.
The number of limbs in use is controlled by _mpfr_prec
, namely
ceil(_mpfr_prec
/mp_bits_per_limb
).
Non-singular (i.e., different from NaN, Infinity or zero)
values always have the most significant bit of the most
significant limb set to 1. When the precision does not correspond to a
whole number of limbs, the excess bits at the low end of the data are zeros.
Next: Contributors, Previous: MPFR Interface, Up: Top [Index]
The goal of this section is to describe some API changes that occurred from one version of MPFR to another, and how to write code that can be compiled and run with older MPFR versions. The minimum MPFR version that is considered here is 2.2.0 (released on 20 September 2005).
API changes can only occur between major or minor versions. Thus the patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior.
As a general rule, a program written for some MPFR version should work with later versions, possibly except at a new major version, where some features (described as obsolete for some time) can be removed. In such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the various changes below (they are minimal, and most software should be unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (see Reporting Bugs).
However, a program written for the current MPFR version (as documented by this manual) may not necessarily work with previous versions of MPFR. This section should help developers to write portable code.
Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being classified like here), together with other changes.
• Type and Macro Changes: | ||
• Added Functions: | ||
• Changed Functions: | ||
• Removed Functions: | ||
• Other Changes: |
Next: Added Functions, Previous: API Compatibility, Up: API Compatibility [Index]
The official type for exponent values changed from mp_exp_t
to
mpfr_exp_t
in MPFR 3.0. The type mp_exp_t
will remain
available as it comes from GMP (with a different meaning). These types
are currently the same (mpfr_exp_t
is defined as mp_exp_t
with typedef
), so that programs can still use mp_exp_t
;
but this may change in the future.
Alternatively, using the following code after including mpfr.h
will work with official MPFR versions, as mpfr_exp_t
was never
defined in MPFR 2.x:
#if MPFR_VERSION_MAJOR < 3 typedef mp_exp_t mpfr_exp_t; #endif
The official types for precision values and for rounding modes
respectively changed from mp_prec_t
and mp_rnd_t
to mpfr_prec_t
and mpfr_rnd_t
in MPFR 3.0. This
change was actually done a long time ago in MPFR, at least since
MPFR 2.2.0, with the following code in mpfr.h:
#ifndef mp_rnd_t # define mp_rnd_t mpfr_rnd_t #endif #ifndef mp_prec_t # define mp_prec_t mpfr_prec_t #endif
This means that it is safe to use the new official types
mpfr_prec_t
and mpfr_rnd_t
in your programs.
The types mp_prec_t
and mp_rnd_t
(defined
in MPFR only) may be removed in the future, as the prefix
mp_
is reserved by GMP.
The precision type mpfr_prec_t
(mp_prec_t
) was unsigned
before MPFR 3.0; it is now signed. MPFR_PREC_MAX
has not changed,
though. Indeed the MPFR code requires that MPFR_PREC_MAX
be
representable in the exponent type, which may have the same size as
mpfr_prec_t
but has always been signed.
The consequence is that valid code that does not assume anything about
the signedness of mpfr_prec_t
should work with past and new MPFR
versions.
This change was useful as the use of unsigned types tends to convert
signed values to unsigned ones in expressions due to the usual arithmetic
conversions, which can yield incorrect results if a negative value is
converted in such a way.
Warning! A program assuming (intentionally or not) that
mpfr_prec_t
is signed may be affected by this problem when
it is built and run against MPFR 2.x.
The rounding modes GMP_RNDx
were renamed to MPFR_RNDx
in MPFR 3.0. However the old names GMP_RNDx
have been kept for
compatibility (this might change in future versions), using:
#define GMP_RNDN MPFR_RNDN #define GMP_RNDZ MPFR_RNDZ #define GMP_RNDU MPFR_RNDU #define GMP_RNDD MPFR_RNDD
The rounding mode “round away from zero” (MPFR_RNDA
) was added in
MPFR 3.0 (however no rounding mode GMP_RNDA
exists).
Next: Changed Functions, Previous: Type and Macro Changes, Up: API Compatibility [Index]
We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version.
mpfr_add_d
in MPFR 2.4.
mpfr_ai
in MPFR 3.0 (incomplete, experimental).
mpfr_asprintf
in MPFR 2.4.
mpfr_buildopt_decimal_p
and mpfr_buildopt_tls_p
in MPFR 3.0.
mpfr_buildopt_gmpinternals_p
and mpfr_buildopt_tune_case
in MPFR 3.1.
mpfr_clear_divby0
in MPFR 3.1 (new divide-by-zero exception).
mpfr_copysign
in MPFR 2.3.
Note: MPFR 2.2 had a mpfr_copysign
function that was available,
but not documented,
and with a slight difference in the semantics (when
the second input operand is a NaN).
mpfr_custom_get_significand
in MPFR 3.0.
This function was named mpfr_custom_get_mantissa
in previous
versions; mpfr_custom_get_mantissa
is still available via a
macro in mpfr.h:
#define mpfr_custom_get_mantissa mpfr_custom_get_significand
Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
use mpfr_custom_get_mantissa
.
mpfr_d_div
and mpfr_d_sub
in MPFR 2.4.
mpfr_digamma
in MPFR 3.0.
mpfr_divby0_p
in MPFR 3.1 (new divide-by-zero exception).
mpfr_div_d
in MPFR 2.4.
mpfr_fmod
in MPFR 2.4.
mpfr_fms
in MPFR 2.3.
mpfr_fprintf
in MPFR 2.4.
mpfr_frexp
in MPFR 3.1.
mpfr_get_flt
in MPFR 3.0.
mpfr_get_patches
in MPFR 2.3.
mpfr_get_z_2exp
in MPFR 3.0.
This function was named mpfr_get_z_exp
in previous versions;
mpfr_get_z_exp
is still available via a macro in mpfr.h:
#define mpfr_get_z_exp mpfr_get_z_2exp
Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
use mpfr_get_z_exp
.
mpfr_grandom
in MPFR 3.1.
mpfr_j0
, mpfr_j1
and mpfr_jn
in MPFR 2.3.
mpfr_lgamma
in MPFR 2.3.
mpfr_li2
in MPFR 2.4.
mpfr_min_prec
in MPFR 3.0.
mpfr_modf
in MPFR 2.4.
mpfr_mul_d
in MPFR 2.4.
mpfr_printf
in MPFR 2.4.
mpfr_rec_sqrt
in MPFR 2.4.
mpfr_regular_p
in MPFR 3.0.
mpfr_remainder
and mpfr_remquo
in MPFR 2.3.
mpfr_set_divby0
in MPFR 3.1 (new divide-by-zero exception).
mpfr_set_flt
in MPFR 3.0.
mpfr_set_z_2exp
in MPFR 3.0.
mpfr_set_zero
in MPFR 3.0.
mpfr_setsign
in MPFR 2.3.
mpfr_signbit
in MPFR 2.3.
mpfr_sinh_cosh
in MPFR 2.4.
mpfr_snprintf
and mpfr_sprintf
in MPFR 2.4.
mpfr_sub_d
in MPFR 2.4.
mpfr_urandom
in MPFR 3.0.
mpfr_vasprintf
, mpfr_vfprintf
, mpfr_vprintf
,
mpfr_vsprintf
and mpfr_vsnprintf
in MPFR 2.4.
mpfr_y0
, mpfr_y1
and mpfr_yn
in MPFR 2.3.
mpfr_z_sub
in MPFR 3.1.
Next: Removed Functions, Previous: Added Functions, Up: API Compatibility [Index]
The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below.
mpfr_check_range
changed in MPFR 2.3.2 and MPFR 2.4.
If the value is an inexact infinity, the overflow flag is now set
(in case it was lost), while it was previously left unchanged.
This is really what is expected in practice (and what the MPFR code
was expecting), so that the previous behavior was regarded as a bug.
Hence the change in MPFR 2.3.2.
mpfr_get_f
changed in MPFR 3.0.
This function was returning zero, except for NaN and Inf, which do not
exist in MPF. The erange flag is now set in these cases,
and mpfr_get_f
now returns the usual ternary value.
mpfr_get_si
, mpfr_get_sj
, mpfr_get_ui
and mpfr_get_uj
changed in MPFR 3.0.
In previous MPFR versions, the cases where the erange flag
is set were unspecified.
mpfr_get_z
changed in MPFR 3.0.
The return type was void
; it is now int
, and the usual
ternary value is returned. Thus programs that need to work with both
MPFR 2.x and 3.x must not use the return value. Even in this case,
C code using mpfr_get_z
as the second or third term of
a conditional operator may also be affected. For instance, the
following is correct with MPFR 3.0, but not with MPFR 2.x:
bool ? mpfr_get_z(...) : mpfr_add(...);
On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0:
bool ? mpfr_get_z(...) : (void) mpfr_add(...);
Portable code should cast mpfr_get_z(...)
to void
to
use the type void
for both terms of the conditional operator,
as in:
bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);
Alternatively, if ... else
can be used instead of the
conditional operator.
Moreover the cases where the erange flag is set were unspecified in MPFR 2.x.
mpfr_get_z_exp
changed in MPFR 3.0.
In previous MPFR versions, the cases where the erange flag
is set were unspecified.
Note: this function has been renamed to mpfr_get_z_2exp
in MPFR 3.0, but mpfr_get_z_exp
is still available for
compatibility reasons.
mpfr_strtofr
changed in MPFR 2.3.1 and MPFR 2.4.
This was actually a bug fix since the code and the documentation did
not match. But both were changed in order to have a more consistent
and useful behavior. The main changes in the code are as follows.
The binary exponent is now accepted even without the 0b
or
0x
prefix. Data corresponding to NaN can now have an optional
sign (such data were previously invalid).
mpfr_strtofr
changed in MPFR 3.0.
This function now accepts bases from 37 to 62 (no changes for the other
bases). Note: if an unsupported base is provided to this function,
the behavior is undefined; more precisely, in MPFR 2.3.1 and later,
providing an unsupported base yields an assertion failure (this
behavior may change in the future).
mpfr_subnormalize
changed in MPFR 3.1.
This was actually regarded as a bug fix. The mpfr_subnormalize
implementation up to MPFR 3.0.0 did not change the flags. In particular,
it did not follow the generic rule concerning the inexact flag (and no
special behavior was specified). The case of the underflow flag was more
a lack of specification.
mpfr_urandom
and mpfr_urandomb
changed in MPFR 3.1.
Their behavior no longer depends on the platform (assuming this is also true
for GMP’s random generator, which is not the case between GMP 4.1 and 4.2 if
gmp_randinit_default
is used). As a consequence, the returned values
can be different between MPFR 3.1 and previous MPFR versions.
Note: as the reproducibility of these functions was not specified
before MPFR 3.1, the MPFR 3.1 behavior is not regarded as
backward incompatible with previous versions.
Next: Other Changes, Previous: Changed Functions, Up: API Compatibility [Index]
Functions mpfr_random
and mpfr_random2
have been
removed in MPFR 3.0 (this only affects old code built against
MPFR 3.0 or later).
(The function mpfr_random
had been deprecated since at least MPFR 2.2.0,
and mpfr_random2
since MPFR 2.4.0.)
Previous: Removed Functions, Up: API Compatibility [Index]
For users of a C++ compiler, the way how the availability of intmax_t
is detected has changed in MPFR 3.0.
In MPFR 2.x, if a macro INTMAX_C
or UINTMAX_C
was defined
(e.g. when the __STDC_CONSTANT_MACROS
macro had been defined
before <stdint.h>
or <inttypes.h>
has been included),
intmax_t
was assumed to be defined.
However this was not always the case (more precisely, intmax_t
can be defined only in the namespace std
, as with Boost), so
that compilations could fail.
Thus the check for INTMAX_C
or UINTMAX_C
is now disabled for
C++ compilers, with the following consequences:
intmax_t
may no longer
be compiled against MPFR 3.0: a #define MPFR_USE_INTMAX_T
may be
necessary before mpfr.h is included.
intmax_t
and uintmax_t
in the global
namespace, though this is not clean.
The divide-by-zero exception is new in MPFR 3.1. However it should not introduce incompatible changes for programs that strictly follow the MPFR API since the exception can only be seen via new functions.
As of MPFR 3.1, the mpfr.h header can be included several times, while still supporting optional functions (see Headers and Libraries).
Next: References, Previous: API Compatibility, Up: Top [Index]
The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Philippe Théveny and Paul Zimmermann.
Sylvie Boldo from ENS-Lyon, France,
contributed the functions mpfr_agm
and mpfr_log
.
Sylvain Chevillard contributed the mpfr_ai
function.
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base-2 exponential, and the factorial function.
Alain Delplanque contributed the new version of the mpfr_get_str
function.
Mathieu Dutour contributed the functions mpfr_acos
, mpfr_asin
and mpfr_atan
, and a previous version of mpfr_gamma
.
Laurent Fousse contributed the mpfr_sum
function.
Emmanuel Jeandel, from ENS-Lyon too,
contributed the generic hypergeometric code,
as well as the internal function mpfr_exp3
,
a first implementation of the sine and cosine,
and improved versions of
mpfr_const_log2
and mpfr_const_pi
.
Ludovic Meunier helped in the design of the mpfr_erf
code.
Jean-Luc Rémy contributed the mpfr_zeta
code.
Fabrice Rouillier contributed the mpfr_xxx_z
and mpfr_xxx_q
functions, and helped to the Microsoft Windows porting.
Damien Stehlé contributed the mpfr_get_ld_2exp
function.
We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004.
The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao, Caramel and Caramba project-teams at LORIA and of the Arénaire and AriC project-teams at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly supported by the ERC grant ANTICS of Andreas Enge.
Next: GNU Free Documentation License, Previous: Contributors, Up: Top [Index]
Next: Concept Index, Previous: References, Up: Top [Index]
Copyright © 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.
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This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.
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The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.
The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.
A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.
Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.
The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work’s title, preceding the beginning of the body of the text.
A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.
The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.
You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.
You may also lend copies, under the same conditions stated above, and you may publicly display copies.
If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document’s license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.
If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.
If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.
It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.
You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:
If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles.
You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.
You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.
You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.
The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”
You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.
A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.
If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.
You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.
To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:
Copyright (C) year your name. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''.
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.” line with this:
with the Invariant Sections being list their titles, with the Front-Cover Texts being list, and with the Back-Cover Texts being list.
If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.
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