This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 4.0.2.
Copyright 1991, 1993-2019 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. | |
| • MPFR and the IEEE 754 Standard: | ||
| • 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), possibly
except in faithful rounding.
It does not depend either on the machine rounding mode or rounding precision;
In particular, with a precision of 53 bits and in any of the four standard
rounding modes, 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 https://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 4.0.2 web page https://www.mpfr.org/mpfr-4.0.2/.
The latest version of MPFR is available from https://ftp.gnu.org/gnu/mpfr/ or https://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 4.0.2 web page https://www.mpfr.org/mpfr-4.0.2/ and the FAQ https://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 are 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]
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, under some platforms (in particular with C++ compilers),
users 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.
Alternatively, it is possible to use ‘pkg-config’ (a file ‘mpfr.pc’ is provided as of MPFR 4.0):
cc myprogram.c $(pkg-config --cflags --libs mpfr)
Note that the ‘MPFR_’ and ‘mpfr_’ prefixes are reserved for MPFR. As a general rule, in order to avoid clashes, software using MPFR (directly or indirectly) and system headers/libraries should not define macros and symbols using these prefixes.
Next: MPFR Variable Conventions, Previous: Headers and Libraries, Up: MPFR Basics [Index]
A floating-point number, or float for short, is an object representing a radix-2 floating-point number consisting of a sign, an arbitrary-precision normalized significand (also called mantissa), and an exponent (an integer in some given range); these are called regular numbers. Like in the IEEE 754 standard, a floating-point number can also have three kinds of special values: a signed zero, a signed infinity, and Not-a-Number (NaN). NaN can represent the default value of a floating-point object and the result of some operations for which no other results would make sense, such as 0 divided by 0 or +Infinity minus +Infinity; unless documented otherwise, the sign bit of a NaN is unspecified. Note that contrary to IEEE 754, MPFR has a single kind of NaN and does not have subnormals. Other than that, the behavior is very similar to IEEE 754, but there may be some differences.
The C data type for such objects is mpfr_t, internally defined
as a one-element array of a structure (so that when passed as an
argument to a function, it is the pointer that is actually passed),
and mpfr_ptr is the C data type representing a pointer to this
structure.
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 1.
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).
An exponent is a component of a regular floating-point number.
Its C data type is mpfr_exp_t. Valid exponents are restricted
to a subset of this type, and the exponent range can be changed globally
as described in Exception Related Functions. Special values do not
have an exponent.
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;
the corresponding C data type is mpfr_rnd_t.
MPFR has a global (or per-thread) flag for each supported exception and provides operations on flags (Exceptions). This C data type is used to represent a group of flags (or a mask).
Next: Rounding Modes, Previous: Nomenclature and Types, Up: MPFR Basics [Index]
Before you can assign to a MPFR variable, you need to initialize it by calling one of the special initialization functions. When you are 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 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.
MPFR_RNDF: faithful rounding. This feature is currently
experimental. Specific support for this rounding mode has been added
to some functions, such as the basic operations (addition, subtraction,
multiplication, square, division, square root) or when explicitly
documented. It might also work with other functions, as it is possible that
they do not need modification in their code; even though a correct behavior
is not guaranteed yet (corrections were done when failures occurred in the
test suite, but almost nothing has been checked manually), failures should
be regarded as bugs and reported, so that they can be fixed.
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).
The MPFR_RNDF mode works as follows: the computed value is either
that corresponding to MPFR_RNDD or that corresponding to
MPFR_RNDU.
In particular when those values are identical,
i.e., when the result of the corresponding operation is exactly
representable, that exact result is returned.
Thus, the computed result can take at most two possible values, and
in absence of underflow/overflow, the corresponding error is strictly
less than one ulp (unit in the last place) of that result and of the
exact result.
For MPFR_RNDF, the ternary value (defined below) and the inexact flag
(defined later, as with the other flags) are unspecified, the divide-by-zero
flag is as with other roundings, and the underflow and overflow flags match
what would be obtained in the case the computed value is the same as with
MPFR_RNDD or MPFR_RNDU.
The results may not be reproducible.
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 defines a global (or per-thread) flag for each supported exception. A macro evaluating to a power of two is associated with each flag and exception, in order to be able to specify a group of flags (or a mask) by OR’ing such macros.
Flags can be cleared (lowered), set (raised), and tested by functions described in Exception Related Functions.
The supported exceptions are listed below. The macro associated with each exception is in parentheses.
MPFR_FLAGS_UNDERFLOW):
An underflow occurs when the exact result of a function is a non-zero
real number and the result obtained after the rounding, assuming an
unbounded exponent range (for the rounding), has an exponent smaller
than the minimum value of the current exponent range. (In the round-to-nearest
mode, the halfway case is rounded toward zero.)
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_FLAGS_OVERFLOW):
An overflow occurs when the exact result of a function is a non-zero
real number and the result obtained after the rounding, assuming an
unbounded exponent range (for the rounding), has an exponent larger
than the maximum value of the current exponent range. In the round-to-nearest
mode, the result is infinite.
Note: unlike the underflow case, there is only one possible definition of
overflow here.
MPFR_FLAGS_DIVBY0):
An exact infinite result is obtained from finite inputs.
MPFR_FLAGS_NAN):
A NaN exception occurs when the result of a function is NaN.
MPFR_FLAGS_INEXACT):
An inexact exception occurs when the result of a function cannot be
represented exactly and must be rounded.
MPFR_FLAGS_ERANGE):
A range exception occurs when a function that does not return a MPFR
number (such as comparisons and conversions to an integer) has an
invalid result (e.g., an argument is NaN in mpfr_cmp, or a
conversion to an integer cannot be represented in the target type).
Moreover, the group consisting of all the flags is represented by
the MPFR_FLAGS_ALL macro (if new flags are added in future
MPFR versions, they will be added to this macro too).
Differences with the ISO C99 standard:
Next: Getting the Best Efficiency Out of MPFR, 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.
When more precision is needed, the value is automatically recomputed;
a minimum of 10% increase of the precision is guaranteed to avoid too
many recomputations.
MPFR functions may also create thread-local pools for internal use
to avoid the cost of memory allocation. The pools can be freed with
mpfr_free_pool (but with a default MPFR build, they should not
take much memory, as the allocation size is limited).
At any time, the user can free various caches and pools with
mpfr_free_cache and mpfr_free_cache2. It is strongly advised
to free thread-local caches before terminating a thread, and all caches
before exiting when using tools like ‘valgrind’ (to avoid memory leaks
being reported).
MPFR allocates its memory either on the stack (for temporary memory only)
or with the same allocator as the one configured for GMP:
see Section “Custom Allocation” in GNU MP.
This means that the application must make sure that data allocated with the
current allocator will not be reallocated or freed with a new allocator.
So, in practice, if an application needs to change the allocator with
mp_set_memory_functions, it should first free all data allocated
with the current allocator: for its own data, with mpfr_clear,
etc.; for the caches and pools, with mpfr_mp_memory_cleanup in
all threads where MPFR is potentially used. This function is currently
equivalent to mpfr_free_cache, but mpfr_mp_memory_cleanup
is the recommended way in case the allocation method changes in the future
(for instance, one may choose to allocate the caches for floating-point
constants with malloc to avoid freeing them if the allocator
changes). Developers should also be aware that MPFR may also be used
indirectly by libraries, so that libraries based on MPFR should provide
a clean-up function calling mpfr_mp_memory_cleanup and/or warn
their users about this issue.
Note: For multithreaded applications, the allocator must be valid in all threads where MPFR may be used; data allocated in one thread may be reallocated and/or freed in some other thread.
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).
Writers of libraries using MPFR should be aware that the application and/or another library used by the application may also use MPFR, so that changing the exponent range, the default precision, or the default rounding mode may have an effect on this other use of MPFR since these data are not duplicated (unless they are in a different thread). Therefore any such value changed in a library function should be restored before the function returns (unless the purpose of the function is to do such a change). Writers of software using MPFR should also be careful when changing such a value if they use a library using MPFR (directly or indirectly), in order to make sure that such a change is compatible with the library.
Previous: Memory Handling, Up: MPFR Basics [Index]
Here are a few hints to get the best efficiency out of MPFR:
mpfr_swap instead of mpfr_set whenever possible.
This will avoid copying the significands;
a = a + b you don’t
need an auxiliary variable, you can directly write
mpfr_add (a, a, b, ...).
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 after
calling any MPFR function or macro, whether or not there is an error. Except
when documented, MPFR will not set errno, but functions called by the
MPFR code (libc functions, memory allocator, etc.) may do so.
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,
The mpfr_set_float128 function is built only with the configure
option ‘--enable-float128’, which requires the compiler or
system provides the ‘__float128’ data type
(GCC 4.3 or later supports this data type);
to use mpfr_set_float128, one should define the macro
MPFR_WANT_FLOAT128 before including mpfr.h.
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’, 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.
For mpfr_set, the sign of a NaN is propagated in order to mimic the
IEEE 754 copy operation. But contrary to IEEE 754, the NaN flag is
set as usual.
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_float128 and mpfr_get_decimal64 functions are built
only under some conditions: see the documentation of mpfr_set_float128
and mpfr_set_decimal64 respectively.
Convert op to a long, an unsigned long,
an intmax_t or an uintmax_t (respectively) after rounding
it to an integer 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.
When there is no such range error, if the return value differs from
op, i.e., if op is not an integer, the inexact flag is set.
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. Otherwise the return
value is zero when rop is equal to op (i.e., when op
is an integer), positive when it is greater than op, and negative
when it is smaller than op; moreover, if rop differs from
op, i.e., if op is not an integer, the inexact flag is set.
Convert op to a mpq_t.
If op is NaN or an infinity, the erange flag is
set and rop is set to 0. Otherwise the conversion is always exact.
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 abs(b), with rounding in the direction rnd, where n is either zero (see below) or the number of significant digits output in the string. The argument b may vary from 2 to 62 or from -2 to -36; otherwise the function does nothing and immediately returns a null pointer.
For b in the range 2 to 36, digits and lower-case letters are used;
for -2 to -36, digits and upper-case letters are used; for
37 to 62, digits, upper-case letters, and lower-case letters, in that
significance order, are used. Warning! This implies that for
b > 10, the successor of the digit 9 depends on b.
This choice has been done for compatibility with GMP’s mpf_get_str
function. Users who wish a more consistent behavior should write a simple
wrapper.
If the input is NaN, then the returned string is ‘@NaN@’ and the NaN flag is set. If the input is +Inf (resp. -Inf), then the returned string is ‘@Inf@’ (resp. ‘-@Inf@’).
If the input number is a finite 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 allocation function (see Memory Handling) 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. A safe block size (sufficient for any
value) is max(n + 2, 7) if n is not zero; if n is
zero, replace it by m+1, as discussed above. 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).
Like in usual functions, the inexact flag is set iff the result is inexact.
Free a string allocated by mpfr_get_str using the unallocation
function (see Memory Handling).
The block is assumed to be strlen(str)+1 bytes.
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.
For instance, with the MPFR_RNDU rounding mode on -0.5,
the result will be non-zero for all these functions.
For MPFR_RNDF, those functions return non-zero when it is guaranteed
that the corresponding conversion function (for example mpfr_get_ui
for mpfr_fits_ulong_p), when called with faithful rounding,
will always return a number that is representable in the corresponding type.
As a consequence, for MPFR_RNDF, mpfr_fits_ulong_p will return
non-zero for a non-negative number less or equal to ULONG_MAX.
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; but note that if
op1 is non-zero and op2 is zero, the result might change from
±Inf to NaN in future MPFR versions if there is an opposite decision
on the IEEE 754 side.
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 = 0, set rop to NaN. For k odd (resp. even) and op negative (including -Inf), set rop to a negative number (resp. NaN). If op is zero, set rop to zero with the sign obtained by the usual limit rules, i.e., the same sign as op if k is odd, and positive if k is even.
These functions agree with the rootn function of the IEEE 754-2008 standard (Section 9.2).
This function is the same as mpfr_rootn_ui except when op
is -0 and k is even: the result is -0 instead of +0
(the reason was to be consistent with mpfr_sqrt). Said otherwise,
if op is zero, set rop to op.
This function predates the IEEE 754-2008 standard and behaves differently from its rootn function. It is marked as deprecated and will be removed in a future release.
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.
Note: When 0 is of integer type, it is regarded as +0 by these functions.
We do not use the usual limit rules in this case, as these rules are not
used for pow.
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.
The sign rule also applies to NaN in order to mimic the IEEE 754
negate and abs operations, i.e., for mpfr_neg, the
sign is reversed, and for mpfr_abs, the sign is set to positive.
But contrary to IEEE 754, the NaN flag is set as usual.
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 (even more
with correct rounding) is expensive, even for small precision,
for example the trigonometric and Bessel functions for large argument.
For some functions, the memory usage might depend not only on the output
precision: it is the case of the mpfr_rootn_ui function where the
memory usage is also linear in the argument k,
and of the incomplete Gamma function (dependence on the precision of op).
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 logarithm of one plus op, rounded in the direction rnd. Set rop to -Inf if op is -1.
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 exponential of op followed by a subtraction by one, 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 exponential integral of op, rounded in the direction rnd. This is the sum of Euler’s constant, of the logarithm of the absolute value 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 positive op, it corresponds to the Ei function at op (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative op, to the opposite of the E1 function (sometimes called eint1) at -op (formula 5.1.1 from the same reference).
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, resp. the
incomplete Gamma function on op and op2,
rounded in the direction rnd.
(In the literature, mpfr_gamma_inc is called upper
incomplete Gamma function,
or sometimes complementary incomplete Gamma function.)
For mpfr_gamma (and mpfr_gamma_inc when op2 is zero),
when op is a negative integer, rop is set to NaN.
Note: the current implementation of mpfr_gamma_inc is slow for
large values of rop or op, in which case some internal overflow
might also occur.
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 Beta function at arguments op1 and op2. Note: the current code does not try to avoid internal overflow or underflow, and might use a huge internal precision in some cases.
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 (op1 times op2) + (op3 times op4) (resp. (op1 times op2) - (op3 times op4)) rounded in the direction rnd. In case the computation of op1 times op2 overflows or underflows (or that of op3 times op4), the result rop is computed as if the two intermediate products were computed with rounding toward zero.
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 and the other one is not zero, set rop to NaN. If any operand is zero and the other one is finite (resp. infinite), set rop to +0 (resp. 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 or mpfr_free_cache2.
Free all caches and pools used by MPFR internally (those local to the
current thread and those shared by all threads).
You should call this function before terminating a thread, even if you did
not call mpfr_const_* functions directly (they could have been called
internally).
Free various caches and pools used by MPFR internally, as specified by way, which is a set of flags:
MPFR_FREE_LOCAL_CACHE
is set;
MPFR_FREE_GLOBAL_CACHE
is set.
The other bits of way are currently ignored and are reserved for future use; they should be zero.
Note: mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)
is currently equivalent to mpfr_free_cache().
Free the pools used by MPFR internally.
Note: This function is automatically called after the thread-local caches
are freed (with mpfr_free_cache or mpfr_free_cache2).
This function should be called before calling mp_set_memory_functions.
See Memory Handling, for more information.
Zero is returned in case of success, non-zero in case of error.
Errors are currently not possible, but checking the return value
is recommended for future compatibility.
Set rop to the sum of all elements of tab, whose size is n,
correctly 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 n = 0, then the result is +0, and if n = 1, then the function
is equivalent to mpfr_set.
For the special exact cases, the result is the same as the one obtained
with a succession of additions (mpfr_add) in infinite precision.
In particular, if the result is an exact zero and n >= 1:
MPFR_RNDD rounding mode, where it is
-0.
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 a function that takes a FILE * argument, 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.
Export the number op to the stream stream in a floating-point interchange format. In particular one can export on a 32-bit computer and import on a 64-bit computer, or export on a little-endian computer and import on a big-endian computer. The precision of op and the sign bit of a NaN are stored too. Return 0 iff the export was successful.
Note: this function is experimental and its interface might change in future versions.
Import the number op from the stream stream in a floating-point
interchange format (see mpfr_fpif_export).
Note that the precision of op is set to the one read from the stream,
and the sign bit is always retrieved (even for NaN).
If the stored precision is zero or greater than MPFR_PREC_MAX, the
function fails (it returns non-zero) and op is unchanged. If the
function fails for another reason, op is set to NaN and it is
unspecified whether the precision of op has changed to the one
read from the file.
Return 0 iff the import was successful.
Note: this function is experimental and its interface might change in future versions.
Output op on stdout in some unspecified format, then a newline
character. This function is mainly for debugging purpose. Thus invalid data
may be supported. Everything that is not specified may change without
breaking the ABI and may depend on the environment.
The current output format is the following: a minus sign if the sign bit is set (even for NaN); ‘@NaN@’, ‘@Inf@’ or ‘0’ if the argument is NaN, an infinity or zero, respectively; otherwise the remaining of the output is as follows: ‘0.’ then the p bits of the binary significand, where p is the precision of the number; if the trailing bits are not all zeros (which must not occur with valid data), they are output enclosed by square brackets; the character ‘E’ followed by the exponent written in base 10; in case of invalid data or out-of-range exponent, this function outputs three exclamation marks (‘!!!’), followed by flags, followed by three exclamation marks (‘!!!’) again. These flags are: ‘N’ if the most significant bit of the significand is 0 (i.e., the number is not normalized); ‘T’ if there are non-zero trailing bits; ‘U’ if this is a UBF number (internal use only); ‘<’ if the exponent is less than the current minimum exponent; ‘>’ if the exponent is greater than the current maximum exponent.
Next: Integer and Remainder 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), but
note that for ‘Re’, the default precision is not the same as the one for
‘e’.
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_toruintmax_t‘l’ longorwchar_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_tarray, 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_targument just before the correspondingmpfr_tvariable.
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-point 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 that ought to be
written exceeds the maximum limit INT_MAX 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 errno
is set to EOVERFLOW if the EOVERFLOW macro is defined (such as
on POSIX systems). Note, however, that errno might be changed to
another value by some internal library call if another error occurs there
(currently, this would come from the unallocation function).
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 been 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 allocation function (see Memory Handling). 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, in which case
the contents of str are undefined.
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, and the other five functions behave in a
similar way with some fixed rounding mode:
mpfr_ceil: to the next higher or equal representable integer
(like mpfr_rint with MPFR_RNDU);
mpfr_floor to the next lower or equal representable integer
(like mpfr_rint with MPFR_RNDD);
mpfr_round to the nearest representable integer,
rounding halfway cases away from zero
(as in the roundTiesToAway mode of IEEE 754-2008);
mpfr_roundeven to the nearest representable integer,
rounding halfway cases with the even-rounding rule
(like mpfr_rint with MPFR_RNDN);
mpfr_trunc to the next representable integer toward zero
(like mpfr_rint with MPFR_RNDZ).
When op is a zero or an infinity, set rop to the same value (with the same sign).
The return 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 return 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.
Note 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: 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;
mpfr_rint_roundeven: to the nearest integer,
rounding halfway cases to the nearest even integer;
mpfr_rint_trunc to the next integer toward zero.
If the result is not representable, it is rounded in the direction rnd. When op is a zero or an infinity, set rop to the same value (with the same sign). The return 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).
When op is an integer or an infinity, set rop to zero with
the same sign as op.
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 mpfr_fmodquo,
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_fmodquo and mpfr_remquo store
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 and Remainder 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 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on b is bounded by two to the power k ulps, and b has precision prec, you should give err=prec-k. This function does not modify its arguments.
If rnd1 is MPFR_RNDN or MPFR_RNDF,
the error is considered to be either
positive or negative, thus the possible range
is twice as large as with a directed rounding for rnd1 (with the
same value of err).
When rnd2 is MPFR_RNDF, let rnd3 be the opposite direction
if rnd1 is a directed rounding, and MPFR_RNDN
if rnd1 is MPFR_RNDN or MPFR_RNDF.
The returned value of mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec)
is non-zero iff after
the call mpfr_set (y, b, rnd3) with y of precision prec,
y is guaranteed to be a faithful rounding of x.
Note: The ternary value cannot be determined in general with this
function. However, if it is known that the exact value is not exactly
representable in precision prec, then one can use the following
trick to determine the (non-zero) ternary value in any rounding mode
rnd2 (note that MPFR_RNDZ below can be replaced by any
directed rounding mode):
if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ,
prec + (rnd2 == MPFR_RNDN)))
{
/* round the approximation 'b' to the result 'r' of 'prec' bits
with rounding mode 'rnd2' and get the ternary value 'inex' */
inex = mpfr_set (r, b, rnd2);
}
Indeed, if rnd2 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.
A detailed example is available in the examples subdirectory, file can_round.c.
Return the minimal number of bits required to store the significand of x, and 0 for special values, including 0.
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.
Given a function foo and one or more values op (which may be
a mpfr_t, a long, a double, etc.), put in rop
the round-to-nearest-away rounding of foo(op,...).
This rounding is defined in the same way as round-to-nearest-even,
except in case of tie, where the value away from zero is returned.
The function foo takes as input, from second to
penultimate argument(s), the argument list given after rop,
a rounding mode as final argument,
puts in its first argument the value foo(op,...) rounded
according to this rounding mode, and returns the corresponding ternary value
(which is expected to be correct, otherwise mpfr_round_nearest_away
will not work as desired).
Due to implementation constraints, this function must not be called when
the minimal exponent emin is the smallest possible one.
This macro has been made such that the compiler is able to detect
mismatch between the argument list op
and the function prototype of foo.
Multiple input arguments op are supported only with C99 compilers.
Otherwise, for C89 compilers, only one such argument is supported.
Note: this macro is experimental and its interface might change in future versions.
unsigned long ul; mpfr_t f, r; /* Code that inits and sets r, f, and ul, and if needed sets emin */ int i = mpfr_round_nearest_away (mpfr_add_ui, r, f, ul);
Next: Exception Related Functions, Previous: Rounding-Related Functions, Up: MPFR Interface [Index]
If x or y is NaN, set x to NaN; note that the NaN flag is set as usual. 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, overflow, or inexact exception is raised.
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. Moreover, the exact number
(the random value to be rounded) and the next random state do not depend on
the current exponent range and the rounding mode. However, they depend on
the target precision: from the same state of the random generator, if the
precision of the destination is changed, then the value may be completely
different (and the state of the random generator is different too).
Generate one (possibly two for mpfr_grandom) random floating-point
number according to a standard normal Gaussian distribution (with mean zero
and variance one). For mpfr_grandom, 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 gmp_randstate_t argument should be
created using the GMP gmp_randinit function (see the GMP manual).
For mpfr_grandom,
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.
Note: mpfr_nrandom is much more efficient than mpfr_grandom,
especially for large precision. Thus mpfr_grandom is marked as
deprecated and will be removed in a future release.
Generate one random floating-point number according to an exponential
distribution, with mean one.
Other characteristics are identical to mpfr_nrandom.
Return the exponent of x, assuming that x is a non-zero ordinary number and the significand is considered in [1/2,1). For this function, x is allowed to be outside of the current range of acceptable values. The behavior for NaN, infinity or zero is undefined.
Set the exponent of x to e if x is a non-zero ordinary number and e is in the current exponent range, and return 0; otherwise, return a non-zero value (x is not changed).
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 ‘__float128’ support (that is, MPFR was built with the ‘--enable-float128’ configure option), 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 non-zero value if MPFR was compiled so that all threads share
the same cache for one MPFR constant, like mpfr_const_pi or
mpfr_const_log2 (that is, MPFR was built with the
‘--enable-shared-cache’ configure option), return zero otherwise.
If the return value is non-zero, MPFR applications may need to be compiled
with the ‘-pthread’ option.
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.
For the subsequent operations, it is the user’s responsibility to check
that any floating-point value used as an input is in the new exponent range
(for example using mpfr_check_range). If a floating-point value
outside the new exponent range is used as an input, the default behavior
is undefined, in the sense of the ISO C standard; the behavior may also be
explicitly documented, such as for mpfr_check_range.
Note: Caches may still have values outside the current exponent range. This is not an issue as the user cannot use these caches directly via the API (MPFR extends the exponent range internally when need be).
If emin > emax and a floating-point value needs to
be produced as output, the behavior is undefined (mpfr_set_emin
and mpfr_set_emax do not check this condition as it might occur
between successive calls to these two functions).
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 of the emulated
floating-point system, this function 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 of MPFR
(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.
Warning! If you change emin (with mpfr_set_emin) just before
calling mpfr_subnormalize, you need to make sure that the value is
in the current exponent range of MPFR. But it is better to change
emin before any computation, if possible.
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);
}
Note that mpfr_set_emin and mpfr_set_emax are called early
enough in order to make sure that all computed values are in the current
exponent range.
Warning! This emulates a double IEEE 754 arithmetic with correct rounding
in the subnormal range, which may not be the case for your hardware.
Below is another example showing how to emulate fixed-point arithmetic in a specific case. Here we compute the sine of the integers 1 to 17 with a result in a fixed-point arithmetic rounded at 2 power -42 (using the fact that the result is at most 1 in absolute value):
{
mpfr_t x; int i, inex;
mpfr_set_emin (-41);
mpfr_init2 (x, 42);
for (i = 1; i <= 17; i++)
{
mpfr_set_ui (x, i, MPFR_RNDN);
inex = mpfr_sin (x, x, MPFR_RNDZ);
mpfr_subnormalize (x, inex, MPFR_RNDZ);
mpfr_dump (x);
}
mpfr_clear (x);
}
Clear (lower) the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.
Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid,
inexact, erange). Note: a group of flags can be cleared by using
mpfr_flags_clear.
Set (raise) the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.
Return the corresponding (underflow, overflow, divide-by-zero, invalid, inexact, erange) flag, which is non-zero iff the flag is set.
The mpfr_flags_ functions below that take an argument mask
can operate on any subset of the exception flags: a flag is part of this
subset (or group) if and only if the corresponding bit of the argument
mask is set. The MPFR_FLAGS_ macros will normally be used
to build this argument. See Exceptions.
Clear (lower) the group of flags specified by mask.
Set (raise) the group of flags specified by mask.
Return the flags specified by mask. To test whether any flag from
mask is set, compare the return value to 0. You can also test
individual flags by AND’ing the result with MPFR_FLAGS_ macros.
Example:
mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW|
MPFR_FLAGS_OVERFLOW)
…
if (t) /* underflow and/or overflow (unlikely) */
{
if (t & MPFR_FLAGS_UNDERFLOW) { /* handle underflow */ }
if (t & MPFR_FLAGS_OVERFLOW) { /* handle overflow */ }
}
Return all the flags. It is equivalent to
mpfr_flags_test(MPFR_FLAGS_ALL).
Restore the flags specified by mask to their state represented in flags.
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>
many programs written for MPF can be compiled directly against MPFR
without any changes.
All operations are then performed with the default MPFR rounding mode,
which can be reset with mpfr_set_default_rounding_mode.
Warning! There are some differences. In particular:
gmp_printf, etc.) will not work
for arguments of arbitrary-precision floating-point type (mpf_t, which
mpf2mpfr.h redefines as mpfr_t).
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:
MPFR_NAN_KIND, x is set to NaN;
MPFR_INF_KIND, x is set to the
infinity of the same sign as kind;
MPFR_ZERO_KIND, x is set to the
zero of the same sign as kind;
MPFR_REGULAR_KIND, x is set to
the regular number whose sign is the one of kind, and whose exponent
and significand are given by exp and significand.
In all cases, significand will be used directly for further computing
involving x. This function does 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 and the significand is considered in [1/2,1).
But if x is NaN, infinity or zero, contrary to mpfr_get_exp
(where the behavior is undefined), the return value is here an unspecified,
valid value of the mpfr_exp_t type.
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: MPFR and the IEEE 754 Standard, 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).
Faithful rounding (MPFR_RNDF) was added in MPFR 4.0, but currently,
it is partially supported.
The flags-related macros, whose name starts with MPFR_FLAGS_,
were added in MPFR 4.0 (for the new functions mpfr_flags_clear,
mpfr_flags_restore, mpfr_flags_set and mpfr_flags_test,
in particular).
Next: Changed Functions, Previous: Type and Macro Changes, Up: API Compatibility [Index]
We give here in alphabetical order the functions (and function-like macros) 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_beta in MPFR 4.0 (incomplete, experimental).
mpfr_buildopt_decimal_p in MPFR 3.0.
mpfr_buildopt_float128_p in MPFR 4.0.
mpfr_buildopt_gmpinternals_p in MPFR 3.1.
mpfr_buildopt_sharedcache_p in MPFR 4.0.
mpfr_buildopt_tls_p in MPFR 3.0.
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_erandom in MPFR 4.0.
mpfr_flags_clear, mpfr_flags_restore,
mpfr_flags_save, mpfr_flags_set and mpfr_flags_test
in MPFR 4.0.
mpfr_fmma and mpfr_fmms in MPFR 4.0.
mpfr_fmod in MPFR 2.4.
mpfr_fmodquo in MPFR 4.0.
mpfr_fms in MPFR 2.3.
mpfr_fpif_export and mpfr_fpif_import in MPFR 4.0.
mpfr_fprintf in MPFR 2.4.
mpfr_free_cache2 in MPFR 4.0.
mpfr_free_pool in MPFR 4.0.
mpfr_frexp in MPFR 3.1.
mpfr_gamma_inc in MPFR 4.0.
mpfr_get_float128 in MPFR 4.0 if configured with
‘--enable-float128’.
mpfr_get_flt in MPFR 3.0.
mpfr_get_patches in MPFR 2.3.
mpfr_get_q in MPFR 4.0.
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_log_ui in MPFR 4.0.
mpfr_min_prec in MPFR 3.0.
mpfr_modf in MPFR 2.4.
mpfr_mp_memory_cleanup in MPFR 4.0.
mpfr_mul_d in MPFR 2.4.
mpfr_nrandom in MPFR 4.0.
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_rint_roundeven and mpfr_roundeven in MPFR 4.0.
mpfr_round_nearest_away in MPFR 4.0.
mpfr_rootn_ui in MPFR 4.0.
mpfr_set_divby0 in MPFR 3.1 (new divide-by-zero exception).
mpfr_set_float128 in MPFR 4.0 if configured with
‘--enable-float128’.
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_abs, mpfr_neg and mpfr_set changed in MPFR 4.0.
In previous MPFR versions, the sign bit of a NaN was unspecified; however,
in practice, it was set as now specified except for mpfr_neg with
a reused argument: mpfr_neg(x,x,rnd).
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_eint changed in MPFR 4.0.
This function now returns the value of the E1/eint1 function for
negative argument (before MPFR 4.0, it was returning NaN).
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_str changed in MPFR 4.0.
This function now sets the NaN flag on NaN input (to follow the usual MPFR
rules on NaN and IEEE 754-2008 recommendations on string conversions from
Subclause 5.12.1) and sets the inexact flag when the conversion is inexact.
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_set_exp changed in MPFR 4.0.
Before MPFR 4.0, the exponent was set whatever the contents of the MPFR
object in argument. In practice, this could be useful as a low-level
function when the MPFR number was being constructed by setting the fields
of its internal structure, but the API does not provide a way to do this
except by using internals. Thus, for the API, this behavior was useless
and could quickly lead to undefined behavior due to the fact that the
generated value could have an invalid format if the MPFR object contained
a special value (NaN, infinity or zero).
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_sum changed in MPFR 4.0.
The mpfr_sum function has completely been rewritten for MPFR 4.0,
with an update of the specification: the sign of an exact zero result
is now specified, and the return value is now the usual ternary value.
The old mpfr_sum implementation could also take all the memory
and crash on inputs of very different magnitude.
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.
mpfr_urandom changed in MPFR 4.0.
The next random state no longer depends on the current exponent range and
the rounding mode. The exceptions due to the rounding of the random number
are now correctly generated, following the uniform distribution.
As a consequence, the returned values can be different between MPFR 4.0
and previous MPFR 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.)
Macros mpfr_add_one_ulp and mpfr_sub_one_ulp have been
removed in MPFR 4.0. They were no longer documented since MPFR 2.1.0
and were announced as deprecated since MPFR 3.1.0.
Function mpfr_grandom is marked as deprecated in MPFR 4.0.
It will be removed in a future release.
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).
The way memory is allocated by MPFR should be regarded as well-specified only as of MPFR 4.0.
Next: Contributors, Previous: API Compatibility, Up: Top [Index]
This section describes differences between MPFR and the IEEE 754 standard, and behaviors that are not specified yet in IEEE 754.
The MPFR numbers do not include subnormals. The reason is that subnormals
are less useful than in IEEE 754 as the default exponent range in MPFR is
large and they would have made the implementation more complex.
However, subnormals can be emulated using mpfr_subnormalize.
MPFR has a single NaN. The behavior is similar either to a signaling NaN or to a quiet NaN, depending on the context. For any function returning a NaN (either produced or propagated), the NaN flag is set, while in IEEE 754, some operations are quiet (even on a signaling NaN).
The mpfr_rec_sqrt function differs from IEEE 754 on -0,
where it gives +Inf (like for +0), following the usual limit rules,
instead of -Inf.
The mpfr_root function predates IEEE 754-2008 and behaves differently
from its rootn operation.
It is deprecated and mpfr_rootn_ui should be used instead.
Operations with an unsigned zero: For functions taking an argument of
integer or rational type, a zero of such a type is unsigned unlike the
floating-point zero (this includes the zero of type unsigned long,
which is a mathematical, exact zero, as opposed to a floating-point zero,
which may come from an underflow and whose sign would correspond to the
sign of the real non-zero value). Unless documented otherwise, this zero
is regarded as +0, as if it were first converted to a MPFR number with
mpfr_set_ui or mpfr_set_si (thus the result may not agree
with the usual limit rules applied to a mathematical zero). This is not
the case of addition and subtraction (mpfr_add_ui, etc.), but for
these functions, only the sign of a zero result would be affected, with
+0 and -0 considered equal.
Such operations are currently out of the scope of the IEEE 754 standard,
and at the time of specification in MPFR, the Floating-Point Working Group
in charge of the revision of IEEE 754 did not want to discuss issues with
non-floating-point types in general.
Note also that some obvious differences may come from the fact that in
MPFR, each variable has its own precision. For instance, a subtraction
of two numbers of the same sign may yield an overflow; idem for a call
to mpfr_set, mpfr_neg or mpfr_abs, if the destination
variable has a smaller precision.
Next: References, Previous: MPFR and the IEEE 754 Standard, 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 original version of the mpfr_sum
function (used up to MPFR 3.1).
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.
Charles Karney contributed the mpfr_nrandom and mpfr_erandom
functions.
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.
The MPFR-MPC workshop in January 2013 was partly supported by the ERC
grant ANTICS, the GDR IM and the Caramel project-team, during which
Mickaël Gastineau contributed the MPFRbench program, and
Fredrik Johansson a faster version of mpfr_const_euler.
Next: GNU Free Documentation License, Previous: Contributors, Up: Top [Index]
Next: Concept Index, Previous: References, Up: Top [Index]
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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 https://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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