This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.0.0.
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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.
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.
MPFR is a portable library written in C for arbitrary precision arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are:
mp_bits_per_limb (64 on most current processors);
In particular, with a precision of 53 bits, MPFR is able to
exactly reproduce all computations with double-precision machine
floating-point numbers (e.g., double type in C, with a C
implementation that rigorously follows Annex F of the ISO C99 standard
and FP_CONTRACT pragma set to OFF) on the four arithmetic
operations and the square root, except the default exponent range is much
wider and subnormal numbers are not implemented (but can be emulated).
This version of MPFR is released under the GNU Lesser General Public License, version 3 or any later version. It is permitted to link MPFR to most non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided.
Everyone should read MPFR Basics. If you need to install the library yourself, you need to read Installing MPFR, too. To use the library you will need to refer to MPFR Interface.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
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 MPFR was built correctly. If you get error messages, please report this to ‘mpfr@loria.fr’. (See Reporting Bugs, for information on what to include in useful bug reports.)
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 mpfr.html; if you want only one output HTML file, then type ‘makeinfo --html --no-split mpfr.texi’ instead.
Delete all object files and archive files, but not the configuration files.
Delete all generated files not included in the distribution.
Delete all files copied by ‘make install’.
In case of problem, please read the INSTALL file carefully before reporting a bug, in particular section “In case of problem”. Some problems are due to bad configuration on the user side (not specific to MPFR). Problems are also mentioned in the FAQ http://www.mpfr.org/faq.html.
Please report problems to ‘mpfr@loria.fr’. See Reporting Bugs. Some bug fixes are available on the MPFR 3.0.0 web page http://www.mpfr.org/mpfr-3.0.0/.
The latest version of MPFR is available from ftp://ftp.gnu.org/gnu/mpfr/ or http://www.mpfr.org/.
If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.0.0 web page http://www.mpfr.org/mpfr-3.0.0/ and the FAQ http://www.mpfr.org/faq.html: perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: https://sympa.inria.fr/sympa/arc/mpfr. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find.
There are a few things you should think about when you put your bug report together.
You have to send us a test case that makes it possible for us to reproduce the bug, i.e., a small self-content program, using no other library than MPFR. Include instructions on how to run the test case.
You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way.
Please include compiler version information in your bug report. This can be extracted using ‘cc -V’ on some machines, or, if you're using GCC, ‘gcc -v’. Also, include the output from ‘uname -a’ and the MPFR version (the GMP version may be useful too).
If 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: ‘mpfr@loria.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.
All declarations needed to use MPFR are collected in the include file mpfr.h. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library:
#include <mpfr.h>
Note however that prototypes for MPFR functions with FILE * parameters
are provided only if <stdio.h> is included too (before mpfr.h):
#include <stdio.h>
#include <mpfr.h>
Likewise <stdarg.h> (or <varargs.h>) is required for prototypes
with va_list parameters, such as mpfr_vprintf.
And for any functions using intmax_t, you must include
<stdint.h> or <inttypes.h> before mpfr.h, to
allow mpfr.h to define prototypes for these functions. Moreover,
users of C++ compilers under some platforms may need to define
MPFR_USE_INTMAX_T (and should do it for portability) before
mpfr.h has been included; of course, it is possible to do that
on the command line, e.g., with -DMPFR_USE_INTMAX_T.
Note: If mpfr.h and/or gmp.h (used by mpfr.h)
are included several times (possibly from another header file), the
aforementioned standard headers should be included before the
first inclusion of mpfr.h or gmp.h. For the time being,
this problem is not avoidable in MPFR without a change in GMP.
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.
A floating-point number, or float for short, is an arbitrary
precision significand (also called mantissa) with a limited precision
exponent. The C data type
for such objects is mpfr_t (internally defined as a one-element
array of a structure, and mpfr_ptr is the C data type representing
a pointer to this structure). A floating-point number can have
three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN
represents an uninitialized object, the result of an invalid operation
(like 0 divided by 0), or a value that cannot be determined (like
+Infinity minus +Infinity). Moreover, like in the IEEE 754 standard,
zero is signed, i.e., there are both +0 and −0; the behavior
is the same as in the IEEE 754 standard and it is generalized to
the other functions supported by MPFR. Unless documented otherwise,
the sign bit of a NaN is unspecified.
The precision is the number of bits used to represent the significand
of a floating-point number;
the corresponding C data type is mpfr_prec_t.
The precision can be any integer between MPFR_PREC_MIN and
MPFR_PREC_MAX. In the current implementation, MPFR_PREC_MIN
is equal to 2.
Warning! MPFR needs to increase the precision internally, in order to
provide accurate results (and in particular, correct rounding). Do not
attempt to set the precision to any value near MPFR_PREC_MAX,
otherwise MPFR will abort due to an assertion failure. Moreover, you
may reach some memory limit on your platform, in which case the program
may abort, crash or have undefined behavior (depending on your C
implementation).
The rounding mode specifies the way to round the result of a
floating-point operation, in case the exact result can not be represented
exactly in the destination significand;
the corresponding C data type is mpfr_rnd_t.
Before you can assign to an MPFR variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. A variable should only be initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited. You do not need to be concerned about allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floating-point variable will have the same allocated space during all its life.
As a general rule, all MPFR functions expect output arguments before input
arguments. This notation is based on an analogy with the assignment operator.
MPFR allows you to use the same variable for both input and output in the same
expression. For example, the main function for floating-point multiplication,
mpfr_mul, can be used like this: mpfr_mul (x, x, x, rnd).
This
computes the square of x with rounding mode rnd
and puts the result back in x.
The following five rounding modes are supported:
MPFR_RNDN: round to nearest (roundTiesToEven in IEEE 754-2008),
MPFR_RNDZ: round toward zero (roundTowardZero in IEEE 754-2008),
MPFR_RNDU: round toward plus infinity (roundTowardPositive in IEEE 754-2008),
MPFR_RNDD: round toward minus infinity (roundTowardNegative in IEEE 754-2008),
MPFR_RNDA: round away from zero (experimental).
The ‘round to nearest’ mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the drift phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2).
Most MPFR functions take as first argument the destination variable, as
second and following arguments the input variables, as last argument a
rounding mode, and have a return value of type int, called the
ternary value. The value stored in the destination variable is
correctly rounded, i.e., MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this variable.
The input variables are regarded as exact (in particular, their precision
does not affect the result).
As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding).
Unless documented otherwise, functions returning an int return
a ternary value.
If the ternary value is zero, it means that the value stored in the
destination variable is the exact result of the corresponding mathematical
function. If the ternary value is positive (resp. negative), it means
the value stored in the destination variable is greater (resp. lower)
than the exact result. For example with the MPFR_RNDU rounding mode,
the ternary value is usually positive, except when the result is exact, in
which case it is zero. In the case of an infinite result, it is considered
as inexact when it was obtained by overflow, and exact otherwise. A NaN
result (Not-a-Number) always corresponds to an exact return value.
The opposite of a returned ternary value is guaranteed to be representable
in an int.
Unless documented otherwise, functions returning as result the value 1
(or any other value specified in this manual)
for special cases (like acos(0)) yield an overflow or
an underflow if that value is not representable in the current exponent range.
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 the limit is not defined (e.g., mpfr_log on
−0), the behavior is specified in the description of the MPFR function.
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.
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 input x, mpfr_hypot on (x,+Inf)
gives +Inf.
MPFR supports 5 exception types:
Note: This is not the single possible definition of the underflow. MPFR chooses to consider the underflow after rounding. The underflow before rounding can also be defined. For instance, consider a function that has the exact result 7 multiplied by two to the power e−4, where e is the smallest exponent (for a significand between 1/2 and 1), with a 2-bit target precision and rounding toward plus infinity. The exact result has the exponent e−1. With the underflow before rounding, such a function call would yield an underflow, as e−1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to e, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR.
mpfr_cmp, or a
conversion to an integer cannot be represented in the target type).
MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in Exception Related Functions.
Differences with the ISO C99 standard:
MPFR functions may create caches, e.g., when computing constants such
as Pi, either because the user has called a function like
mpfr_const_pi directly or because such a function was called
internally by the MPFR library itself to compute some other function.
At any time, the user can free the various caches with
mpfr_free_cache. It is strongly advised to do that before
terminating a thread, or before exiting when using tools like
‘valgrind’ (to avoid memory leaks being reported).
MPFR internal data such as flags, the exponent range, the default precision and rounding mode, and caches (i.e., data that are not accessed via parameters) are either global (if MPFR has not been compiled as thread safe) or per-thread (thread local storage).
The floating-point functions expect arguments of type mpfr_t.
The MPFR floating-point functions have an interface that is similar to the
GNU MP
functions. The function prefix for floating-point operations is mpfr_.
The user has to specify the precision of each variable. A computation that assigns a variable will take place with the precision of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average).
The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with “infinite accuracy”), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system.
MPFR does not keep track of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision.
The value of the standard C macro errno may be set to non-zero by
any MPFR function or macro, whether or not there is an error.
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, usempfr_set_prec. The precision prec must be an integer betweenMPFR_PREC_MINandMPFR_PREC_MAX(otherwise the behavior is undefined).
Initialize all the
mpfr_tvariables of the given variable argumentva_list, set their precision to be exactly prec bits and their value to NaN. Seempfr_init2for more details. Theva_listis assumed to be composed only of typempfr_t(or equivalentlympfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also bempfr_ptr).
Free the space occupied by the significand of x. Make sure to call this function for all
mpfr_tvariables when you are done with them.
Free the space occupied by all the
mpfr_tvariables of the givenva_list. Seempfr_clearfor more details. Theva_listis assumed to be composed only of typempfr_t(or equivalentlympfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also bempfr_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_tvariables of the givenva_list, set their precision to the default precision and their value to NaN. Seempfr_initfor more details. Theva_listis assumed to be composed only of typempfr_t(or equivalentlympfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also bempfr_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 usingmpfr_init2but has some drawbacks:
- You must not call
mpfr_clearwith variables created with this macro (the storage is allocated at the point of declaration and deallocated when the brace-level is exited).- You cannot change their precision.
- You should not create variables with huge precision with this macro.
- Your compiler must support ‘Non-Constant Initializers’ (standard in C++ and ISO C99) and ‘Token Pasting’ (standard in ISO C89). If prec is not a constant expression, your compiler must support ‘variable-length automatic arrays’ (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with ‘-pedantic’, you may want to define the
MPFR_USE_EXTENSIONmacro to avoid warnings due to theMPFR_DECL_INITimplementation.
Set the default precision to be exactly prec bits, where prec can be any integer between
MPFR_PREC_MINandMPFR_PREC_MAX. The precision of a variable means the number of bits used to store its significand. All subsequent calls tompfr_initormpfr_initswill use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially.
Return the current default MPFR precision in bits.
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 tompfr_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 betweenMPFR_PREC_MINandMPFR_PREC_MAX. In case you want to keep the previous value stored in x, usempfr_prec_roundinstead.
Return the precision of x, i.e., the number of bits used to store its significand.
These functions assign new values to already initialized floats (see Initialization Functions).
Set the value of rop from op, rounded toward the given direction rnd. Note that the input 0 is converted to +0 by
mpfr_set_ui,mpfr_set_si,mpfr_set_uj,mpfr_set_sj,mpfr_set_z,mpfr_set_qandmpfr_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_ldandmpfr_set_decimal64might not preserve the signed zeros. Thempfr_set_decimal64function is built only with the configure option ‘--enable-decimal-float’, which also requires ‘--with-gmp-build’, and when the compiler or system provides the ‘_Decimal64’ data type (recent versions of GCC support this data type).mpfr_set_qmight fail if the numerator (or the denominator) can not be represented as ampfr_t.Note: If you want to store a floating-point constant to a
mpfr_t, you should usempfr_set_str(or one of the MPFR constant functions, such asmpfr_const_pifor Pi) instead ofmpfr_set_flt,mpfr_set_d,mpfr_set_ldormpfr_set_decimal64. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary 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_strtofrfor a detailed description of the valid string formats. Contrary tompfr_strtofr,mpfr_set_strrequires the whole string to represent a valid floating-point number. This function returns 0 if the entire string up to the final null character is a valid number in base base; otherwise it returns −1, and rop may have changed. Note: it is preferable to usempfr_set_strif 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 withstrtod) 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
strtodfunction 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 beeorEfor 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 beporP, 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 example1p2represents 4 whereas1@2represents 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
0bor0B, base 2 is assumed. If the significand starts with0xor0X, 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,0xor0Xis not followed by a binary/hexadecimal digit, then the subject sequence stops at the character0, 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 beinfinity,inf,nanornan(n-char-sequence-opt), all case insensitive. An-char-sequence-optis 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_informpfr_set_zero, x is set to plus infinity or plus zero iff sign is nonnegative; inmpfr_set_nan, the sign bit of the result is unspecified.
Swap the values x and y efficiently. Warning: the precisions are exchanged too; in case the precisions are different,
mpfr_swapis thus not equivalent to threempfr_setcalls using a third auxiliary variable.
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.
Convert op to a
float(respectivelydouble,long doubleor_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. Thempfr_get_decimal64function is built only under some conditions: see the documentation ofmpfr_set_decimal64.
Convert op to a
long, anunsigned long, anintmax_tor anuintmax_t(respectively) after rounding it with respect to rnd. If op is NaN, 0 is returned and the erange flag is set. If op is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the erange flag is set too. See alsompfr_fits_slong_p,mpfr_fits_ulong_p,mpfr_fits_intmax_pandmpfr_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.
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
eminis returned. If op is NaN or an infinity, the erange flag is set, rop is set to 0, and the the minimal exponenteminis returned. The returned exponent may be less than the minimal exponenteminof MPFR numbers in the current exponent range; in case the exponent is not representable in thempfr_exp_ttype, the erange flag is set and the minimal value of thempfr_exp_ttype is returned.
Convert op to a
mpz_t, after rounding it with respect to rnd. If op is NaN or an infinity, the erange flag is set, rop is set to 0, and 0 is returned.
Convert op to a
mpf_t, after rounding it with respect to rnd. The erange flag is set if op is NaN or Inf, which do not exist in MPF.
Convert op to a string of digits in base b, with rounding in the direction rnd, where n is either zero (see below) or the number of significant digits output in the string; in the latter case, n must be greater or equal to 2. The base may vary from 2 to 62. If the input number is an ordinary number, the exponent is written through the pointer expptr (for input 0, the current minimal exponent is written).
The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number −3.1416 would be returned as "−31416" in the string and 1 written at expptr. If rnd is to nearest, and op is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of op. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal.
If n is zero, the number of digits of the significand is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of op. More precisely, in most cases, the chosen precision of str is the minimal precision m depending only on p = PREC(op) and b that satisfies the above property, i.e., m = 1 + ceil(p*log(2)/log(b)), with p replaced by p−1 if b is a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is when p equals 186564318007 for bases 7 and 49).
If str is a null pointer, space for the significand is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use
mpfr_free_str.If str is not a null pointer, it should point to a block of storage large enough for the significand, i.e., at least
max(n+ 2, 7). The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for-@Inf@plus the terminating null character.A pointer to the string is returned, unless there is an error, in which case a null pointer is returned.
Free a string allocated by
mpfr_get_strusing the current unallocation function. The block is assumed to bestrlen(str)+1bytes. For more information about how it is done: see Section “Custom Allocation” in GNU MP.
Return non-zero if op would fit in the respective C data type, respectively
unsigned long,long,unsigned int,int,unsigned short,short,uintmax_t,intmax_t, when rounded to an integer in the direction rnd.
Set rop to op1 + op2 rounded in the direction rnd. For types having no signed zero, it is considered unsigned (i.e., (+0) + 0 = (+0) and (−0) + 0 = (−0)). The
mpfr_add_dfunction assumes that the radix of thedoubletype is a power of 2, with a precision at most that declared by the C implementation (macroIEEE_DBL_MANT_DIG, and if not defined 53 bits).
Set rop to op1 - op2 rounded in the direction rnd. For types having no signed zero, it 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_dapply tompfr_d_subandmpfr_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 zero, it is considered positive). The same restrictions than for
mpfr_add_dapply tompfr_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 zero, it is considered positive). The same restrictions than for
mpfr_add_dapply tompfr_d_divandmpfr_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.
Set rop to the cubic root (resp. the kth root) of op rounded in the direction rnd. For k odd (resp. even) and op negative (including −Inf), set rop to a negative number (resp. NaN). The kth root of −0 is defined to be −0, whatever the parity of k.
Set rop to op1 raised to op2, rounded in the direction rnd. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the
powfunction:
pow(±0,y)returns plus or minus infinity for y a negative odd integer.pow(±0,y)returns plus infinity for y negative and not an odd integer.pow(±0,y)returns plus or minus zero for y a positive odd integer.pow(±0,y)returns plus zero for y positive and not an odd integer.pow(-1, ±Inf)returns 1.pow(+1,y)returns 1 for any y, even a NaN.pow(x, ±0)returns 1 for any x, even a NaN.pow(x,y)returns NaN for finite negative x and finite non-integer y.pow(x, -Inf)returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1.pow(x, +Inf)returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1.pow(-Inf,y)returns minus zero for y a negative odd integer.pow(-Inf,y)returns plus zero for y negative and not an odd integer.pow(-Inf,y)returns minus infinity for y a positive odd integer.pow(-Inf,y)returns plus infinity for y positive and not an odd integer.pow(+Inf,y)returns plus zero for y negative, and plus infinity for y positive.
Set rop to -op and the absolute value of op respectively, rounded in the direction rnd. Just changes or adjusts the sign if rop and op are the same variable, otherwise a rounding might occur if the precision of rop is less than that of op.
Set rop to the positive difference of op1 and op2, i.e., op1 - op2 rounded in the direction rnd if op1 > op2, +0 if op1 <= op2, and NaN if op1 or op2 is NaN.
Set rop to op1 times 2 raised to op2 rounded in the direction rnd. Just increases the exponent by op2 when rop and op1 are identical.
Set rop to op1 divided by 2 raised to op2 rounded in the direction rnd. Just decreases the exponent by op2 when rop and op1 are identical.
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_pfor 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.
All those functions, except explicitly stated (for example
mpfr_sin_cos), return a ternary value as defined in Section
“Rounding Modes”, i.e., zero for an
exact return value, a positive value for a return value larger than the
exact result, and a negative value otherwise.
Important note: in some domains, computing special functions (either with correct or incorrect rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument.
Set rop to the natural logarithm of op, log2(op) or log10(op), respectively, rounded in the direction rnd. Set rop to −Inf if op is −0 (i.e., the sign of the zero has no influence on the result).
Set rop to the exponential of op, to 2 power of op or to 10 power of op, respectively, rounded in the direction rnd.
Set rop to the cosine of op, sine of op, tangent of op, rounded in the direction rnd.
Set simultaneously sop to the sine of op and cop to the cosine of op, rounded in the direction rnd with the corresponding precisions of sop and cop, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if sop is exact, s=1 if sop is larger than the sine of op, s=2 if sop is smaller than the sine of op, and similarly for c and the cosine of op.
Set rop to the secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set rop to the arc-cosine, arc-sine or arc-tangent of op, rounded in the direction rnd. Note that since
acos(-1)returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= rop < \pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds forasin(-1),asin(1),atan(-Inf),atan(+Inf)or foratan(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); ifx < 0,atan2(y, x) = sign(y)*(Pi - atan (abs(y/x))), thus a number from -Pi to Pi. As foratan, 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 theatan2function:
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_cosfor a more detailed description of the return value).
Set rop to the hyperbolic secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set rop to the inverse hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.
Set rop to the factorial of op, rounded in the direction rnd.
Set rop to the logarithm of one plus op, rounded in the direction rnd.
Set rop to the exponential of op followed by a subtraction by one, rounded in the direction rnd.
Set rop to the exponential integral of op, rounded in the direction rnd. For positive op, the exponential integral is the sum of Euler's constant, of the logarithm of op, and of the sum for k from 1 to infinity of op to the power k, divided by k and factorial(k). For negative op, rop is set to NaN.
Set rop to real part of the dilogarithm of op, rounded in the direction rnd. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to op.
Set rop to the value of the Gamma function on op, rounded in the direction rnd. When op is a negative integer, rop is set to NaN.
Set rop to the value of the logarithm of the Gamma function on op, rounded in the direction rnd. When −2k−1 <= op <= −2k, k being a non-negative integer, rop is set 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 an infinity or a non-positive integer, set rop to +Inf. When op is NaN, −Inf or a negative integer, *signp is undefined, and when op is ±0, *signp is the sign of the zero.
Set rop to the value of the Digamma (sometimes also called Psi) function on op, rounded in the direction rnd. When op is a negative integer, set rop to NaN.
Set rop to the value of the Riemann Zeta function on op, rounded in the direction rnd.
Set rop to the value of the error function on op (resp. the complementary error function on op) rounded in the direction rnd.
Set rop to the value of the first kind Bessel function of order 0, (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN, rop is always set to NaN. When op is plus or minus Infinity, rop is set to +0. When op is zero, and n is not zero, rop is set to +0 or −0 depending on the parity and sign of n, and the sign of op.
Set rop to the value of the second kind Bessel function of order 0 (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN or negative, rop is always set to NaN. When op is +Inf, rop is set to +0. When op is zero, rop is set to +Inf or −Inf depending on the parity and sign of n.
Set rop to (op1 times op2) + op3 (resp. (op1 times op2) - op3) rounded in the direction rnd.
Set rop to the arithmetic-geometric mean of op1 and op2, rounded in the direction rnd. The arithmetic-geometric mean is the common limit of the sequences u_n and v_n, where u_0=op1, v_0=op2, u_(n+1) is the arithmetic mean of u_n and v_n, and v_(n+1) is the geometric mean of u_n and v_n. If any operand is negative, set rop to NaN.
Set rop to the Euclidean norm of x and y, i.e., the square root of the sum of the squares of x and y, rounded in the direction rnd. Special values are handled as described in Section F.9.4.3 of the ISO C99 and IEEE 754-2008 standards: If x or y is an infinity, then +Inf is returned in rop, even if the other number is NaN.
Set rop to the value of the Airy function Ai on x, rounded in the direction rnd. When x is NaN, rop is always set to NaN. When x is +Inf or −Inf, rop is +0. The current implementation is not intended to be used with large arguments. It works with abs(x) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version.
Set rop to the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the direction rnd. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use
mpfr_free_cache.
Free various caches used by MPFR internally, in particular the caches used by the functions computing constants (
mpfr_const_log2,mpfr_const_pi,mpfr_const_eulerandmpfr_const_catalan). You should call this function before terminating a thread, even if you did not call these functions directly (they could have been called internally).
Set rop to the sum of all elements of tab, whose size is n, rounded in the direction rnd. Warning: for efficiency reasons, tab is an array of pointers to
mpfr_t, not an array ofmpfr_t. If the returnedintvalue is zero, rop is guaranteed to be the exact sum; otherwise rop might be smaller than, equal to, or larger than the exact sum (in accordance to the rounding mode). However,mpfr_sumdoes guarantee the result is correctly rounded.
This section describes functions that perform input from an input/output
stream, and functions that output to an input/output stream.
Passing a null pointer for a stream to any of these functions will make
them read from stdin and write to stdout, respectively.
When using any of these functions, you must include the <stdio.h>
standard header before mpfr.h, to allow mpfr.h to define
prototypes for these functions.
Output op on stream stream, as a string of digits in base base, rounded in the direction rnd. The base may vary from 2 to 62. Print n significant digits exactly, or if n is 0, enough digits so that op can be read back exactly (see
mpfr_get_str).In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form ‘eNNN’, are printed. If base is greater than 10, ‘@’ will be used instead of ‘e’ as exponent delimiter.
Return the number of bytes 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 ofmpfr_strtofrfor a detailed description of the valid string formats.Return the number of bytes read, or if an error occurred, return 0.
The class of mpfr_printf functions provides formatted output in a
similar manner as the standard C printf. These functions are defined
only if your system supports ISO C variadic functions and the corresponding
argument access macros.
When using any of these functions, you must include the <stdio.h>
standard header before mpfr.h, to allow mpfr.h to define
prototypes for these functions.
The format specification accepted by mpfr_printf is an extension of the
printf one. The conversion specification is of the form:
% [flags] [width] [.[precision]] [type] [rounding] conv
‘flags’, ‘width’, and ‘precision’ have the same meaning as for
the standard printf (in particular, notice that the ‘precision’ is
related to the number of digits displayed in the base chosen by ‘conv’
and not related to the internal precision of the mpfr_t variable).
mpfr_printf accepts the same ‘type’ specifiers as GMP (except the
non-standard and deprecated ‘q’, use ‘ll’ instead), namely the
length modifiers defined in the C standard:
‘h’ short‘hh’ char‘j’ intmax_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 ‘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 unsigned 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 ‘*’ 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 or scientific float
The conversion specifier ‘b’ which displays the argument in binary is
specific to mpfr_t arguments and should not be used with other types.
Other conversion specifiers have the same meaning as for a double
argument.
In case of non-decimal output, only the significand is written in the
specified base, the exponent is always displayed in decimal.
Special values are always displayed as nan, -inf, and inf
for ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers and
NAN, -INF, and INF for ‘A’, ‘E’, ‘F’, and
‘G’ specifiers.
If the ‘precision’ field is not empty, the mpfr_t number is
rounded to the given precision in the direction specified by the rounding
mode.
If the precision is zero with rounding to nearest mode and one of the
following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’,
‘E’, tie case is rounded to even when it lies between two consecutive
values at the
wanted precision which have the same exponent, otherwise, it is rounded away
from zero.
For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the
format specification "%.0RNe".
This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses
the ‘e’ (resp. ‘E’) style.
If the precision is set to a value greater than the maximum value for an
int, it will be silently reduced down to INT_MAX.
If the ‘precision’ field is empty (as in %Re or %.RE) with
‘conv’ specifier ‘e’ and ‘E’, the number is displayed with
enough digits so that it can be read back exactly, assuming that the input and
output variables have the same precision and that the input and output
rounding modes are both rounding to nearest (as for mpfr_get_str).
The default precision for an empty ‘precision’ field with ‘conv’
specifiers ‘f’, ‘F’, ‘g’, and ‘G’ is 6.
For all the following functions, if the number of characters which ought to be
written appears to exceed the maximum limit for an int, nothing is
written in the stream (resp. to stdout, to buf, to str),
the function returns −1, sets the erange flag, and (in
POSIX system only) errno is set to EOVERFLOW.
Print to the stream stream the optional arguments under the control of the template string template. Return the number of characters written or a negative value if an error occurred.
Print to
stdoutthe optional arguments under the control of the template string template. Return the number of characters written or a negative value if an error occurred.
Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. No overlap is permitted between buf and the other arguments. Return the number of characters written in the array buf not counting the terminating null character or a negative value if an error occurred.
Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. If n is zero, nothing is written and buf may be a null pointer, otherwise, the n−1 first characters are written in buf and the n-th is a null character. Return the number of characters that would have been written had n be sufficiently large, not counting the terminating null character, or a negative value if an error occurred.
Write their output as a null terminated string in a block of memory allocated using the current allocation function. A pointer to the block is stored in str. The block of memory must be freed using
mpfr_free_str. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred.
Set rop to op rounded to an integer.
mpfr_rintrounds to the nearest representable integer in the given direction rnd,mpfr_ceilrounds to the next higher or equal representable integer,mpfr_floorto the next lower or equal representable integer,mpfr_roundto the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008), andmpfr_truncto the next representable integer toward zero.The returned value is zero when the result is exact, positive when it is greater than the original value of op, and negative when it is smaller. More precisely, the returned value is 0 when op is an integer representable in rop, 1 or −1 when op is an integer that is not representable in rop, 2 or −2 when op is not an integer.
Note that
mpfr_roundis different frommpfr_rintcalled with the rounding to nearest mode (where halfway cases are rounded to an even integer or significand). Note also that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded bympfr_rintwith 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_ceilrounds to the next higher or equal integer,mpfr_rint_floorto the next lower or equal integer,mpfr_rint_roundto the nearest integer, rounding halfway cases away from zero, andmpfr_rint_truncto the next integer toward zero. If the result is not representable, it is rounded in the direction rnd. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). Contrary tompfr_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. For example,mpfr_rint_roundwith rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8.
Set rop to the fractional part of op, having the same sign as op, rounded in the direction rnd (unlike in
mpfr_rint, rnd affects only how the exact fractional part is rounded, not how the fractional part is generated).
Set simultaneously iop to the integral part of op and fop to the fractional part of op, rounded in the direction rnd with the corresponding precision of iop and fop (equivalent to
mpfr_trunc(iop,op,rnd)andmpfr_frac(fop,op,rnd)). The variables iop and fop must be different. Return 0 iff both results are exact (seempfr_sin_cosfor a more detailed description of the return value).
Set r to the value of x - ny, rounded according to the direction rnd, where n is the integer quotient of x divided by y, defined as follows: n is rounded toward zero for
mpfr_fmod, and to the nearest integer (ties rounded to even) formpfr_remainderandmpfr_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_remquostores the low significant bits from the quotient n in *q (more precisely the number of bits in alongminus 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. Thempfr_remainderandmpfr_remquofunctions are useful for additive argument reduction.
Set the default rounding mode to rnd. The default rounding mode is to nearest initially.
Round x according to rnd with precision prec, which must be an integer between
MPFR_PREC_MINandMPFR_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_roundto 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 */
Assuming b is an approximation of an unknown number x in the direction rnd1 with error at most two to the power E(b)-err where E(b) is the exponent of b, return a non-zero value if one is able to round correctly x to precision prec with the direction rnd2, and 0 otherwise (including for NaN and Inf). This function does not modify its arguments.
If rnd1 is
MPFR_RNDN, then the sign of the error is unknown, but its absolute value is the same, so that the possible range is twice as large as with a directed rounding for rnd1.Note: if one wants to also determine the correct ternary value when rounding b to precision prec with rounding mode rnd, a useful trick is the following:
if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ...Indeed, if rnd isMPFR_RNDN, this will check if one can round to prec+1 bits with a directed rounding: if so, one can surely round to nearest to prec bits, and in addition one can determine the correct ternary value, which would not be the case when b is near from a value exactly representable on prec bits.
Return the minimal number of bits required to store the significand of x, and 0 for special values, including 0. (Warning: the returned value can be less than
MPFR_PREC_MIN.)The function name is subject to change.
Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode rnd, or a null pointer if rnd is an invalid rounding mode.
If x or y is NaN, set x to NaN. If x and y are equal, x is unchanged. Otherwise, if x is different from y, replace x by the next floating-point number (with the precision of x and the current exponent range) in the direction of y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated.
Equivalent to
mpfr_nexttowardwhere 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_tstructure which should be created using the GMPgmp_randinitfunction (see the GMP manual).
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_tstructure which should be created using the GMPgmp_randinitfunction (see the GMP manual).
Return the exponent of x, assuming that x is a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined.
Set the exponent of x if e is in the current exponent range, and return 0 (even if x is not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1).
Return a non-zero value iff op has its sign bit set (i.e., if it is negative, −0, or a NaN whose representation has its sign bit set).
Set the value of rop from op, rounded toward the given direction rnd, then set (resp. clear) its sign bit if s is non-zero (resp. zero), even when op is a NaN.
Set the value of rop from op1, rounded toward the given direction rnd, then set its sign bit to that of op2 (even when op1 or op2 is a NaN). This function is equivalent to
mpfr_setsign (rop,op1, mpfr_signbit (op2),rnd).
Return the MPFR version, as a null-terminated string.
MPFR_VERSIONis the version of MPFR as a preprocessing constant.MPFR_VERSION_MAJOR,MPFR_VERSION_MINORandMPFR_VERSION_PATCHLEVELare respectively the major, minor and patch level of MPFR version, as preprocessing constants.MPFR_VERSION_STRINGis the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result ofmpfr_get_versionto 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_VERSIONfrom 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-safeconfigure option, seeINSTALLfile), 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-floatconfigure option), return zero otherwise.
Return the (current) smallest and largest exponents allowed for a floating-point variable. The smallest positive value of a floating-point variable is one half times 2 raised to the smallest exponent and the largest value has the form (1 - epsilon) times 2 raised to the largest exponent, where epsilon depends on the precision of the considered variable.
Set the smallest and largest exponents allowed for a floating-point variable. Return a non-zero value when exp is not in the range accepted by the implementation (in that case the smallest or largest exponent is not changed), and zero otherwise. If the user changes the exponent range, it is her/his responsibility to check that all current floating-point variables are in the new allowed range (for example using
mpfr_check_range), otherwise the subsequent behavior will be undefined, in the sense of the ISO C standard.
Return the minimum and maximum of the exponents allowed for
mpfr_set_eminandmpfr_set_emaxrespectively. These values are implementation dependent, thus a program usingmpfr_set_emax(mpfr_get_emax_max())ormpfr_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_rangeis typically called (at least in MPFR functions) after restoring the flags that could have been set due to internal computations.
This function rounds x emulating subnormal number arithmetic: if x is outside the subnormal exponent range, it just propagates the ternary value t; otherwise, it rounds x to precision
EXP(x)-emin+1according to rounding mode rnd and previous ternary value t, avoiding double rounding problems. More precisely in the subnormal domain, denoting by e the value ofemin, 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 inmpfr_check_range). The subnormal exponent range is fromemintoemin+PREC(x)-1. If the result cannot be represented in the current exponent range (due to a too smallemax), 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.
This is an example of how to emulate binary double IEEE 754 arithmetic (binary64 in IEEE 754-2008) using MPFR:
{
mpfr_t xa, xb; int i; volatile double a, b;
mpfr_set_default_prec (53);
mpfr_set_emin (-1073); mpfr_set_emax (1024);
mpfr_init (xa); mpfr_init (xb);
b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN);
a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN);
a /= b;
i = mpfr_div (xa, xa, xb, MPFR_RNDN);
i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */
mpfr_clear (xa); mpfr_clear (xb);
}
Warning: this emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware.
Clear the underflow, overflow, invalid, inexact and erange flags.
Set the underflow, overflow, invalid, inexact and erange flags.
Clear all global flags (underflow, overflow, invalid, inexact, erange).
Return the corresponding (underflow, overflow, invalid, inexact, erange) flag, which is non-zero iff the flag is set.
A header file mpf2mpfr.h is included in the distribution of MPFR for
compatibility with the GNU MP class MPF.
By inserting the following two lines after the #include <gmp.h> line,
#include <mpfr.h> #include <mpf2mpfr.h>any program written for MPF can be compiled directly with MPFR without any changes (except the
gmp_printf functions will not work for arguments of type
mpfr_t).
All operations are then performed with the default MPFR rounding mode,
which can be reset with mpfr_set_default_rounding_mode.
Warning: the mpf_init and mpf_init2 functions initialize
to zero, whereas the corresponding MPFR functions initialize to NaN:
this is useful to detect uninitialized values, but is slightly incompatible
with MPF.
Reset the precision of x to be exactly prec bits. The only difference with
mpfr_set_precis 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_2uiandmpfr_div_2uirespectively. These functions are only kept for compatibility with MPF, one should prefermpfr_mul_2uiandmpfr_div_2uiotherwise.
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.
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 ofmp_limb_t(GMP type, see Internals).
Perform a dummy initialization of a
mpfr_tand set it to:In all cases, it uses significand directly for further computing involving x. It will not allocate anything. A floating-point number initialized with this function cannot be resized using
- if
ABS(kind) == MPFR_NAN_KIND, x is set to NaN;- if
ABS(kind) == MPFR_INF_KIND, x is set to the infinity of signsign(kind);- if
ABS(kind) == MPFR_ZERO_KIND, x is set to the zero of signsign(kind);- if
ABS(kind) == MPFR_REGULAR_KIND, x is set to a regular number:x = sign(kind)*significand*2^exp.mpfr_set_precormpfr_prec_round, or cleared usingmpfr_clear! The significand must have been initialized withmpfr_custom_initusing the same precision prec.
Return the current kind of a
mpfr_tas created bympfr_custom_init_set. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
Return a pointer to the significand used by a
mpfr_tinitialized withmpfr_custom_init_set. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
Return the exponent of x, assuming that x is a non-zero ordinary number. The return value for NaN, Infinity or zero is unspecified but does not produce any trap. The behavior of this function for any
mpfr_tnot initialized withmpfr_custom_init_setis 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 thempfr_titself. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
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.
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.
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).
We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version.
mpfr_add_d in MPFR 2.4.
mpfr_ai in MPFR 3.0 (incomplete, experimental).
mpfr_asprintf in MPFR 2.4.
mpfr_buildopt_decimal_p and mpfr_buildopt_tls_p in MPFR 3.0.
mpfr_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_div_d in MPFR 2.4.
mpfr_fmod in MPFR 2.4.
mpfr_fms in MPFR 2.3.
mpfr_fprintf in MPFR 2.4.
mpfr_get_flt in MPFR 3.0.
mpfr_get_patches in MPFR 2.3.
mpfr_get_z_2exp in MPFR 3.0.
This function was named mpfr_get_z_exp in previous versions;
mpfr_get_z_exp is still available via a macro in mpfr.h:
#define mpfr_get_z_exp mpfr_get_z_2exp
Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
use mpfr_get_z_exp.
mpfr_j0, mpfr_j1 and mpfr_jn in MPFR 2.3.
mpfr_lgamma in MPFR 2.3.
mpfr_li2 in MPFR 2.4.
mpfr_modf in MPFR 2.4.
mpfr_mul_d in MPFR 2.4.
mpfr_printf in MPFR 2.4.
mpfr_rec_sqrt in MPFR 2.4.
mpfr_regular_p in MPFR 3.0.
mpfr_remainder and mpfr_remquo in MPFR 2.3.
mpfr_set_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.
The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below.
mpfr_check_range changed in MPFR 2.3.2 and MPFR 2.4.
If the value is an inexact infinity, the overflow flag is now set
(in case it was lost), while it was previously left unchanged.
This is really what is expected in practice (and what the MPFR code
was expecting), so that the previous behavior was regarded as a bug.
Hence the change in MPFR 2.3.2.
mpfr_get_f changed in MPFR 3.0.
This function was returning zero, except for NaN and Inf, which do not
exist in MPF. The erange flag is now set in these cases,
and mpfr_get_f now returns the usual ternary value.
mpfr_get_si, mpfr_get_sj, mpfr_get_ui
and mpfr_get_uj changed in MPFR 3.0.
In previous MPFR versions, the cases where the erange flag
is set were unspecified.
mpfr_get_z changed in MPFR 3.0.
The return type was void; it is now int, and the usual
ternary value is returned. Thus programs that need to work with both
MPFR 2.x and 3.x must not use the return value. Even in this case,
C code using mpfr_get_z as the second or third term of
a conditional operator may also be affected. For instance, the
following is correct with MPFR 3.0, but not with MPFR 2.x:
bool ? mpfr_get_z(...) : mpfr_add(...);
On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0:
bool ? mpfr_get_z(...) : (void) mpfr_add(...);
Portable code should cast mpfr_get_z(...) to void to
use the type void for both terms of the conditional operator,
as in:
bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);
Alternatively, if ... else can be used instead of the
conditional operator.
Moreover the cases where the erange flag is set were unspecified in MPFR 2.x.
mpfr_get_z_exp changed in MPFR 3.0.
In previous MPFR versions, the cases where the erange flag
is set were unspecified.
Note: this function has been renamed to mpfr_get_z_2exp
in MPFR 3.0, but mpfr_get_z_exp is still available for
compatibility reasons.
mpfr_strtofr changed in MPFR 2.3.1 and MPFR 2.4.
This was actually a bug fix since the code and the documentation did
not match. But both were changed in order to have a more consistent
and useful behavior. The main changes in the code are as follows.
The binary exponent is now accepted even without the 0b or
0x prefix. Data corresponding to NaN can now have an optional
sign (such data were previously invalid).
mpfr_strtofr changed in MPFR 3.0.
This function now accepts bases from 37 to 62 (no changes for the other
bases). Note: if an unsupported base is provided to this function,
the behavior is undefined; more precisely, in MPFR 2.3.1 and later,
providing an unsupported base yields an assertion failure (this
behavior may change in the future).
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.)
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 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.
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.
Mathieu Dutour contributed the functions mpfr_atan and mpfr_asin,
and a previous version of mpfr_gamma;
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base-2 exponential, and the factorial function. Fabrice Rouillier
contributed the mpfr_xxx_z and mpfr_xxx_q functions,
and helped to the Microsoft Windows porting.
Jean-Luc Rémy contributed the mpfr_zeta code.
Ludovic Meunier helped in the design of the mpfr_erf code.
Damien Stehlé contributed the mpfr_get_ld_2exp function.
Sylvain Chevillard contributed the mpfr_ai function.
We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004.
The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao and Caramel project-teams at LORIA and of the Arénaire project-team 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.
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intmax_t: MPFR Basicsinttypes.h: MPFR Basicslibmpfr: MPFR Basicsstdarg.h: MPFR Basicsstdint.h: MPFR Basicsstdio.h: MPFR Basicsuintmax_t: MPFR Basicsmpfr_abs: Basic Arithmetic Functionsmpfr_acos: Special Functionsmpfr_acosh: Special Functionsmpfr_add: Basic Arithmetic Functionsmpfr_add_d: Basic Arithmetic Functionsmpfr_add_q: Basic Arithmetic Functionsmpfr_add_si: Basic Arithmetic Functionsmpfr_add_ui: Basic Arithmetic Functionsmpfr_add_z: Basic Arithmetic Functionsmpfr_agm: Special Functionsmpfr_ai: Special Functionsmpfr_asin: Special Functionsmpfr_asinh: Special Functionsmpfr_asprintf: Formatted Output Functionsmpfr_atan: Special Functionsmpfr_atan2: Special Functionsmpfr_atanh: Special Functionsmpfr_buildopt_decimal_p: Miscellaneous Functionsmpfr_buildopt_tls_p: Miscellaneous Functionsmpfr_can_round: Rounding Related Functionsmpfr_cbrt: Basic Arithmetic Functionsmpfr_ceil: Integer Related Functionsmpfr_check_range: Exception Related Functionsmpfr_clear: Initialization Functionsmpfr_clear_erangeflag: Exception Related Functionsmpfr_clear_flags: Exception Related Functionsmpfr_clear_inexflag: Exception Related Functionsmpfr_clear_nanflag: Exception Related Functionsmpfr_clear_overflow: Exception Related Functionsmpfr_clear_underflow: Exception Related Functionsmpfr_clears: Initialization Functionsmpfr_cmp: Comparison Functionsmpfr_cmp_d: Comparison Functionsmpfr_cmp_f: Comparison Functionsmpfr_cmp_ld: Comparison Functionsmpfr_cmp_q: Comparison Functionsmpfr_cmp_si: Comparison Functionsmpfr_cmp_si_2exp: Comparison Functionsmpfr_cmp_ui: Comparison Functionsmpfr_cmp_ui_2exp: Comparison Functionsmpfr_cmp_z: Comparison Functionsmpfr_cmpabs: Comparison Functionsmpfr_const_catalan: Special Functionsmpfr_const_euler: Special Functionsmpfr_const_log2: Special Functionsmpfr_const_pi: Special Functionsmpfr_copysign: Miscellaneous Functionsmpfr_cos: Special Functionsmpfr_cosh: Special Functionsmpfr_cot: Special Functionsmpfr_coth: Special Functionsmpfr_csc: Special Functionsmpfr_csch: Special Functionsmpfr_custom_get_exp: Custom Interfacempfr_custom_get_kind: Custom Interfacempfr_custom_get_significand: Custom Interfacempfr_custom_get_size: Custom Interfacempfr_custom_init: Custom Interfacempfr_custom_init_set: Custom Interfacempfr_custom_move: Custom Interfacempfr_d_div: Basic Arithmetic Functionsmpfr_d_sub: Basic Arithmetic FunctionsMPFR_DECL_INIT: Initialization Functionsmpfr_digamma: Special Functionsmpfr_dim: Basic Arithmetic Functionsmpfr_div: Basic Arithmetic Functionsmpfr_div_2exp: Compatibility with MPFmpfr_div_2si: Basic Arithmetic Functionsmpfr_div_2ui: Basic Arithmetic Functionsmpfr_div_d: Basic Arithmetic Functionsmpfr_div_q: Basic Arithmetic Functionsmpfr_div_si: Basic Arithmetic Functionsmpfr_div_ui: Basic Arithmetic Functionsmpfr_div_z: Basic Arithmetic Functionsmpfr_eint: Special Functionsmpfr_eq: Compatibility with MPFmpfr_equal_p: Comparison Functionsmpfr_erangeflag_p: Exception Related Functionsmpfr_erf: Special Functionsmpfr_erfc: Special Functionsmpfr_exp: Special Functionsmpfr_exp10: Special Functionsmpfr_exp2: Special Functionsmpfr_expm1: Special Functionsmpfr_fac_ui: Special Functionsmpfr_fits_intmax_p: Conversion Functionsmpfr_fits_sint_p: Conversion Functionsmpfr_fits_slong_p: Conversion Functionsmpfr_fits_sshort_p: Conversion Functionsmpfr_fits_uint_p: Conversion Functionsmpfr_fits_uintmax_p: Conversion Functionsmpfr_fits_ulong_p: Conversion Functionsmpfr_fits_ushort_p: Conversion Functionsmpfr_floor: Integer Related Functionsmpfr_fma: Special Functionsmpfr_fmod: Integer Related Functionsmpfr_fms: Special Functionsmpfr_fprintf: Formatted Output Functionsmpfr_frac: Integer Related Functionsmpfr_free_cache: Special Functionsmpfr_free_str: Conversion Functionsmpfr_gamma: Special Functionsmpfr_get_d: Conversion Functionsmpfr_get_d_2exp: Conversion Functionsmpfr_get_decimal64: Conversion Functionsmpfr_get_default_prec: Initialization Functionsmpfr_get_default_rounding_mode: Rounding Related Functionsmpfr_get_emax: Exception Related Functionsmpfr_get_emax_max: Exception Related Functionsmpfr_get_emax_min: Exception Related Functionsmpfr_get_emin: Exception Related Functionsmpfr_get_emin_max: Exception Related Functionsmpfr_get_emin_min: Exception Related Functionsmpfr_get_exp: Miscellaneous Functionsmpfr_get_f: Conversion Functionsmpfr_get_flt: Conversion Functionsmpfr_get_ld: Conversion Functionsmpfr_get_ld_2exp: Conversion Functionsmpfr_get_patches: Miscellaneous Functionsmpfr_get_prec: Initialization Functionsmpfr_get_si: Conversion Functionsmpfr_get_sj: Conversion Functionsmpfr_get_str: Conversion Functionsmpfr_get_ui: Conversion Functionsmpfr_get_uj: Conversion Functionsmpfr_get_version: Miscellaneous Functionsmpfr_get_z: Conversion Functionsmpfr_get_z_2exp: Conversion Functionsmpfr_greater_p: Comparison Functionsmpfr_greaterequal_p: Comparison Functionsmpfr_hypot: Special Functionsmpfr_inexflag_p: Exception Related Functionsmpfr_inf_p: Comparison Functionsmpfr_init: Initialization Functionsmpfr_init2: Initialization Functionsmpfr_init_set: Combined Initialization and Assignment Functionsmpfr_init_set_d: Combined Initialization and Assignment Functionsmpfr_init_set_f: Combined Initialization and Assignment Functionsmpfr_init_set_ld: Combined Initialization and Assignment Functionsmpfr_init_set_q: Combined Initialization and Assignment Functionsmpfr_init_set_si: Combined Initialization and Assignment Functionsmpfr_init_set_str: Combined Initialization and Assignment Functionsmpfr_init_set_ui: Combined Initialization and Assignment Functionsmpfr_init_set_z: Combined Initialization and Assignment Functionsmpfr_inits: Initialization Functionsmpfr_inits2: Initialization Functionsmpfr_inp_str: Input and Output Functionsmpfr_integer_p: Integer Related Functionsmpfr_j0: Special Functionsmpfr_j1: Special Functionsmpfr_jn: Special Functionsmpfr_less_p: Comparison Functionsmpfr_lessequal_p: Comparison Functionsmpfr_lessgreater_p: Comparison Functionsmpfr_lgamma: Special Functionsmpfr_li2: Special Functionsmpfr_lngamma: Special Functionsmpfr_log: Special Functionsmpfr_log10: Special Functionsmpfr_log1p: Special Functionsmpfr_log2: Special Functionsmpfr_max: Miscellaneous Functionsmpfr_min: Miscellaneous Functionsmpfr_min_prec: Rounding Related Functionsmpfr_modf: Integer Related Functionsmpfr_mul: Basic Arithmetic Functionsmpfr_mul_2exp: Compatibility with MPFmpfr_mul_2si: Basic Arithmetic Functionsmpfr_mul_2ui: Basic Arithmetic Functionsmpfr_mul_d: Basic Arithmetic Functionsmpfr_mul_q: Basic Arithmetic Functionsmpfr_mul_si: Basic Arithmetic Functionsmpfr_mul_ui: Basic Arithmetic Functionsmpfr_mul_z: Basic Arithmetic Functionsmpfr_nan_p: Comparison Functionsmpfr_nanflag_p: Exception Related Functionsmpfr_neg: Basic Arithmetic Functionsmpfr_nextabove: Miscellaneous Functionsmpfr_nextbelow: Miscellaneous Functionsmpfr_nexttoward: Miscellaneous Functionsmpfr_number_p: Comparison Functionsmpfr_out_str: Input and Output Functionsmpfr_overflow_p: Exception Related Functionsmpfr_pow: Basic Arithmetic Functionsmpfr_pow_si: Basic Arithmetic Functionsmpfr_pow_ui: Basic Arithmetic Functionsmpfr_pow_z: Basic Arithmetic Functionsmpfr_prec_round: Rounding Related Functionsmpfr_prec_t: MPFR Basicsmpfr_print_rnd_mode: Rounding Related Functionsmpfr_printf: Formatted Output Functionsmpfr_rec_sqrt: Basic Arithmetic Functionsmpfr_regular_p: Comparison Functionsmpfr_reldiff: Compatibility with MPFmpfr_remainder: Integer Related Functionsmpfr_remquo: Integer Related Functionsmpfr_rint: Integer Related Functionsmpfr_rint_ceil: Integer Related Functionsmpfr_rint_floor: Integer Related Functionsmpfr_rint_round: Integer Related Functionsmpfr_rint_trunc: Integer Related Functionsmpfr_rnd_t: MPFR Basicsmpfr_root: Basic Arithmetic Functionsmpfr_round: Integer Related Functionsmpfr_sec: Special Functionsmpfr_sech: Special Functionsmpfr_set: Assignment Functionsmpfr_set_d: Assignment Functionsmpfr_set_decimal64: Assignment Functionsmpfr_set_default_prec: Initialization Functionsmpfr_set_default_rounding_mode: Rounding Related Functionsmpfr_set_emax: Exception Related Functionsmpfr_set_emin: Exception Related Functionsmpfr_set_erangeflag: Exception Related Functionsmpfr_set_exp: Miscellaneous Functionsmpfr_set_f: Assignment Functionsmpfr_set_flt: Assignment Functionsmpfr_set_inexflag: Exception Related Functionsmpfr_set_inf: Assignment Functionsmpfr_set_ld: Assignment Functionsmpfr_set_nan: Assignment Functionsmpfr_set_nanflag: Exception Related Functionsmpfr_set_overflow: Exception Related Functionsmpfr_set_prec: Initialization Functionsmpfr_set_prec_raw: Compatibility with MPFmpfr_set_q: Assignment Functionsmpfr_set_si: Assignment Functionsmpfr_set_si_2exp: Assignment Functionsmpfr_set_sj: Assignment Functionsmpfr_set_sj_2exp: Assignment Functionsmpfr_set_str: Assignment Functionsmpfr_set_ui: Assignment Functionsmpfr_set_ui_2exp: Assignment Functionsmpfr_set_uj: Assignment Functionsmpfr_set_uj_2exp: Assignment Functionsmpfr_set_underflow: Exception Related Functionsmpfr_set_z: Assignment Functionsmpfr_set_z_2exp: Assignment Functionsmpfr_set_zero: Assignment Functionsmpfr_setsign: Miscellaneous Functionsmpfr_sgn: Comparison Functionsmpfr_si_div: Basic Arithmetic Functionsmpfr_si_sub: Basic Arithmetic Functionsmpfr_signbit: Miscellaneous Functionsmpfr_sin: Special Functionsmpfr_sin_cos: Special Functionsmpfr_sinh: Special Functionsmpfr_sinh_cosh: Special Functionsmpfr_snprintf: Formatted Output Functionsmpfr_sprintf: Formatted Output Functionsmpfr_sqr: Basic Arithmetic Functionsmpfr_sqrt: Basic Arithmetic Functionsmpfr_sqrt_ui: Basic Arithmetic Functionsmpfr_strtofr: Assignment Functionsmpfr_sub: Basic Arithmetic Functionsmpfr_sub_d: Basic Arithmetic Functionsmpfr_sub_q: Basic Arithmetic Functionsmpfr_sub_si: Basic Arithmetic Functionsmpfr_sub_ui: Basic Arithmetic Functionsmpfr_sub_z: Basic Arithmetic Functionsmpfr_subnormalize: Exception Related Functionsmpfr_sum: Special Functionsmpfr_swap: Assignment Functionsmpfr_t: MPFR Basicsmpfr_tan: Special Functionsmpfr_tanh: Special Functionsmpfr_trunc: Integer Related Functionsmpfr_ui_div: Basic Arithmetic Functionsmpfr_ui_pow: Basic Arithmetic Functionsmpfr_ui_pow_ui: Basic Arithmetic Functionsmpfr_ui_sub: Basic Arithmetic Functionsmpfr_underflow_p: Exception Related Functionsmpfr_unordered_p: Comparison Functionsmpfr_urandom: Miscellaneous Functionsmpfr_urandomb: Miscellaneous Functionsmpfr_vasprintf: Formatted Output FunctionsMPFR_VERSION: Miscellaneous FunctionsMPFR_VERSION_MAJOR: Miscellaneous FunctionsMPFR_VERSION_MINOR: Miscellaneous FunctionsMPFR_VERSION_NUM: Miscellaneous FunctionsMPFR_VERSION_PATCHLEVEL: Miscellaneous FunctionsMPFR_VERSION_STRING: Miscellaneous Functionsmpfr_vfprintf: Formatted Output Functionsmpfr_vprintf: Formatted Output Functionsmpfr_vsnprintf: Formatted Output Functionsmpfr_vsprintf: Formatted Output Functionsmpfr_y0: Special Functionsmpfr_y1: Special Functionsmpfr_yn: Special Functionsmpfr_zero_p: Comparison Functionsmpfr_zeta: Special Functionsmpfr_zeta_ui: Special Functions