Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.1 or any later
version published by the Free Software Foundation; with no Invariant Sections,
with the FrontCover Texts being "A GNU Manual", and with the BackCover
Texts being "You have freedom to copy and modify this GNU Manual, like GNU
software". A copy of the license is included in GNU Free Documentation License.
This library 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 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 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 MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LIB.
MPFR is a portable library written in C for arbitrary precision arithmetic
on floatingpoint numbers. It is based on the GNU MP library.
It aims to extend the class of floatingpoint numbers provided by the
GNU MP library by a precise semantics. The main differences
with the mpf
class from GNU MP are:
mpfr
code is portable, i.e. the result of any operation
does not depend (or should not) on the machine word size
mp_bits_per_limb
(32 or 64 on most machines);
mpfr
provides the four rounding modes from the IEEE 7541985
standard.
In particular, with a precision of 53 bits, mpfr
should be able
to exactly reproduce all computations with doubleprecision machine
floatingpoint
numbers (double
type in C), except the default exponent range
is much wider and subnormal numbers are not implemented.
This version of MPFR is released under the GNU Lesser General Public License. It is permitted to link MPFR to nonfree programs, as long as when distributing them the MPFR source code and a means to relink 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.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
Here are the steps needed to install the library on Unix systems
(more details are provided in the INSTALL
file):
make
program, plus some other
standard Unix utility programs.
MPFR needs some internal GMP header files that are not installed.
So, keep the GMP build directory as is, at least until you have
built MPFR.
./configure withgmpinclude=GMPBUILD withgmplib=GMPINSTALL/lib
where GMPBUILD
is the GMP build directory and GMPINSTALL
the
directory where you have installed GMP. Because of the internal header files
required by MPFR, the option withgmp=GMPINSTALL
is not sufficient
and should not be used.
If you get error messages, you might check that you use the same compiler
and compile options as for GNU MP (see the INSTALL
file).
make
This will compile MPFR, and create a library archive file libmpfr.a
in the working directory. No dynamic library is provided yet.
make check
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.)
make install
This will copy the files mpfr.h
and mpf2mpfr.h
to the directory
/usr/local/include
, the file libmpfr.a
to the directory
/usr/local/lib
, and the file mpfr.info
to the directory
/usr/local/info
(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:
mpfr.info
or info
Create an info version of the manual, in mpfr.info
.
mpfr.dvi
or dvi
Create a DVI version of the manual, in mpfr.dvi
.
mpfr.ps
Create a Postscript version of the manual, in mpfr.ps
.
clean
Delete all object files and archive files, but not the configuration files.
distclean
Delete all files not included in the distribution.
uninstall
Delete all files copied by make install
.
MPFR suffers from all bugs from the GNU MP library, plus many many more.
Please report other problems to mpfr@loria.fr
.
See Reporting Bugs.
Some bug fixes are available on the MPFR web page
<http://www.mpfr.org/
>.
If you think you have found a bug in the MPFR library, first have a look on the
MPFR web page <http://www.mpfr.org/
>: perhaps this bug is already known,
in which case you may find there a workaround for it.
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. 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 printed 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
.
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 won't 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>
A floatingpoint number or float for short, is an arbitrary
precision mantissa with a limited precision exponent. The C data type
for such objects is mpfr_t
. A floatingpoint number can have
three special values: NotaNumber (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 7541985 standard,
zero is signed, i.e. there are both +0 and 0; the behavior
is the same as in the IEEE 7541985 standard and it is generalized to
the other functions supported by MPFR.
The precision is the number of bits used to represent the mantissa
of a floatingpoint number;
the corresponding C data type is mp_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 and MPFR_PREC_MAX
is equal to ULONG_MAX
/2.
The rounding mode specifies the way to round the result of a
floatingpoint operation, in case the exact result can not be represented
exactly in the destination mantissa;
the corresponding C data type is mp_rnd_t
.
A limb means the part of a multiprecision number that fits in a single
word. (We chose this word because a limb of the human body is analogous to a
digit, only larger, and containing several digits.) Normally a limb contains
32 or 64 bits. The C data type for a limb is mp_limb_t
.
There is only one class of functions in the MPFR library:
mpfr_
. The associated type is mpfr_t
.
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 floatingpoint multiplication,
mpfr_mul
, can be used like this: mpfr_mul (x, x, x, rnd_mode)
.
This
computes the square of x with rounding mode rnd_mode
and puts the result back in x.
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 don't need to be concerned about allocating additional space for MPFR variables, since any variable has a mantissa of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floatingpoint variable will have the same allocated space during all its life.
A header file mpf2mpfr.h
is included in the distribution of MPFR for
compatibility with the GNU MP class MPF.
After 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.
All operations are then performed with the default MPFR rounding mode,
which can be reset with mpfr_set_default_rounding_mode
.
mp_rnd_t __gmpfr_default_rounding_mode  Global Variable 
The default rounding mode (to nearest initially). 
The latest version of MPFR is available from <http://www.mpfr.org/
>
or <http://www.loria.fr/projets/mpfr/
>.
The floatingpoint functions expect arguments of type mpfr_t
.
The MPFR floatingpoint functions have an interface that is similar to the
GNU MP
integer functions. The function prefix for floatingpoint operations is
mpfr_
.
There is one significant characteristic of floatingpoint numbers that has motivated a difference between this function class and other GNU MP function classes: the inherent inexactness of floatingpoint arithmetic. The user has to specify the precision for 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 from 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 floatingpoint functions are intended to be a smooth extension of the IEEE 7541985 arithmetic. The results obtained on one computer should not differ from the results obtained on a computer with a different word size.
MPFR does not keep track of the accuracy of a computation. This is left to the user or to a higher layer. 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 following four rounding modes are supported:
GMP_RNDN
: round to nearest
GMP_RNDZ
: round towards zero
GMP_RNDU
: round towards plus infinity
GMP_RNDD
: round towards minus infinity
The round to nearest
mode works as in the IEEE 7541985 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 5/2, 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
exactly 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).
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 GMP_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 (NotaNumber) always corresponds to an exact return value.
The opposite of a returned ternary value is guaranteed to be representable
in an int
.
void mpfr_set_default_rounding_mode (mp_rnd_t rnd)  Function 
Sets the default rounding mode to rnd. The default rounding mode is to nearest initially. 
int mpfr_prec_round (mpfr_t x, mp_prec_t prec, mp_rnd_t rnd)  Function 
Rounds x according to rnd with precision prec, which
must be an integer between MPFR_PREC_MIN and MPFR_PREC_MAX
(otherwise the behavior is undefined).
If prec is greater or equal to the precision of x, then new
space is allocated for the mantissa, and it is filled with zeros.
Otherwise, the mantissa is rounded to precision prec with the given
direction. In both cases, the precision of x is changed to prec.

int mpfr_round_prec (mpfr_t x, mp_rnd_t rnd, mp_prec_t prec)  Function 
[This function is obsolete. Please use mpfr_prec_round instead.]

const char * mpfr_print_rnd_mode (mp_rnd_t rnd)  Function 
Returns the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) corresponding to the rounding mode rnd or a null pointer if rnd is an invalid rounding mode. 
Note: Overflow handling is still experimental and currently implemented very partially. If an overflow occurs internally at the wrong place, anything can happen (crash, wrong results, etc).
mp_exp_t mpfr_get_emin (void)  Function 
mp_exp_t mpfr_get_emax (void)  Function 
Return the (current) smallest and largest exponents allowed for a floatingpoint variable. The smallest positive value of a floatingpoint 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. 
int mpfr_set_emin (mp_exp_t exp)  Function 
int mpfr_set_emax (mp_exp_t exp)  Function 
Set the smallest and largest exponents allowed for a floatingpoint variable.
Return a nonzero 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 floatingpoint 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.

int mpfr_check_range (mpfr_t x, int t, mp_rnd_t rnd)  Function 
This function forces x to be in the current range of acceptable values, t being the current ternary value: negative if x is smaller than the exact value, positive if x is larger than the exact value and zero if x is exact (before the call). 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 rounded result is equal to the exact one, a positive value if the rounded result is larger than the exact one, a negative value if the rounded result is smaller than the exact one. 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. 
void mpfr_clear_underflow (void)  Function 
void mpfr_clear_overflow (void)  Function 
void mpfr_clear_nanflag (void)  Function 
void mpfr_clear_inexflag (void)  Function 
Clear the underflow, overflow, invalid, and inexact flags. 
void mpfr_clear_flags (void)  Function 
Clear all global flags (underflow, overflow, inexact, invalid). 
int mpfr_underflow_p (void)  Function 
int mpfr_overflow_p (void)  Function 
int mpfr_nanflag_p (void)  Function 
int mpfr_inexflag_p (void)  Function 
Return the corresponding (underflow, overflow, invalid, inexact) flag, which is nonzero iff the flag is set. 
void mpfr_set_default_prec (mp_prec_t prec)  Function 
Set the default precision to be exactly prec bits. The
precision of a variable means the number of bits used to store its mantissa.
All
subsequent calls to mpfr_init will use this precision, but previously
initialized variables are unaffected.
This default precision is set to 53 bits initially.
The precision can be any integer between MPFR_PREC_MIN and
MPFR_PREC_MAX .

mp_prec_t mpfr_get_default_prec (void)  Function 
Returns the default MPFR precision in bits. 
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.
void mpfr_init (mpfr_t x)  Function 
Initialize x and set its value to NaN.
Normally, a variable should be initialized once only
or at least be cleared, using 
void mpfr_init2 (mpfr_t x, mp_prec_t prec)  Function 
Initialize x, set its precision to be exactly
prec bits and its value to NaN.
Normally, a variable should be initialized once only or at
least be cleared, using 
void mpfr_clear (mpfr_t x)  Function 
Free the space occupied by x. Make sure to call this function for all
mpfr_t variables when you are done with them.

Here is an example on how to initialize floatingpoint 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); }
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 NewtonRaphson, making the computation precision closely match the actual accurate part of the numbers.
void mpfr_set_prec (mpfr_t x, mp_prec_t prec)  Function 
Reset the precision of x to be exactly prec bits,
and set its value to NaN.
The previous value stored in x is lost. It is equivalent to
a call to mpfr_clear(x) followed by a call to
mpfr_init2(x, prec) , but more efficient as no allocation is done in
case the current allocated space for the mantissa of x is enough.
The precision prec can be any integer between MPFR_PREC_MIN and
MPFR_PREC_MAX .
In case you want to keep the previous value stored in x,
use 
mp_prec_t mpfr_get_prec (mpfr_t x)  Function 
Return the precision actually used for assignments of x, i.e. the number of bits used to store its mantissa. 
void mpfr_set_prec_raw (mpfr_t x, mp_prec_t prec)  Function 
Reset the precision of x to be exactly prec bits.
The only difference with mpfr_set_prec is that prec is assumed to
be small enough so that the mantissa fits into the current allocated memory
space for x. Otherwise the behavior is undefined.

These functions assign new values to already initialized floats (see Initialization Functions).
int mpfr_set (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_set_ui (mpfr_t rop, unsigned long int op, mp_rnd_t rnd)  Function 
int mpfr_set_si (mpfr_t rop, long int op, mp_rnd_t rnd)  Function 
int mpfr_set_d (mpfr_t rop, double op, mp_rnd_t rnd)  Function 
int mpfr_set_ld (mpfr_t rop, long double op, mp_rnd_t rnd)  Function 
int mpfr_set_z (mpfr_t rop, mpz_t op, mp_rnd_t rnd)  Function 
int mpfr_set_q (mpfr_t rop, mpq_t op, mp_rnd_t rnd)  Function 
int mpfr_set_f (mpfr_t rop, mpf_t op, mp_rnd_t rnd)  Function 
Set the value of rop from op, rounded
towards the given direction rnd.
Note that the input 0 is converted to +0 by mpfr_set_ui ,
mpfr_set_si , mpfr_set_z , mpfr_set_q and
mpfr_set_f , regardless of the rounding mode.

int mpfr_set_str (mpfr_t x, const char *s, int base, mp_rnd_t rnd)  Function 
Set x to the value of the whole string s in base base
(between 2 and 36), rounded in direction rnd.
See the documentation of mpfr_inp_str for a detailed description
of the valid string formats.
This function returns 0 if the entire string up to the final \0 is a
valid number in base base; otherwise it returns 1.

void mpfr_set_inf (mpfr_t x, int sign)  Function 
void mpfr_set_nan (mpfr_t x)  Function 
Set the variable x to infinity or NaN (NotaNumber) respectively.
In mpfr_set_inf , x is set to plus infinity iff sign is
nonnegative.

void mpfr_swap (mpfr_t x, mpfr_t y)  Function 
Swap the values x and y efficiently. Warning: the
precisions are exchanged too; in case the precisions are different,
mpfr_swap is thus not equivalent to three mpfr_set calls
using a third auxiliary variable.

int mpfr_init_set (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_ui (mpfr_t rop, unsigned long int op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_si (mpfr_t rop, signed long int op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_d (mpfr_t rop, double op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_ld (mpfr_t rop, long double op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_z (mpfr_t rop, mpz_t op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_q (mpfr_t rop, mpq_t op, mp_rnd_t rnd)  Macro 
int mpfr_init_set_f (mpfr_t rop, mpf_t op, mp_rnd_t rnd)  Macro 
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 .

int mpfr_init_set_str (mpfr_t x, const char *s, int base, mp_rnd_t rnd)  Function 
Initialize x and set its value from
the string s in base base,
rounded in the direction rnd.
See mpfr_set_str .

double mpfr_get_d (mpfr_t op, mp_rnd_t rnd)  Function 
long double mpfr_get_ld (mpfr_t op, mp_rnd_t rnd)  Function 
Convert op to a double (respectively long double ),
using the rounding mode rnd.

double mpfr_get_d1 (mpfr_t op)  Function 
Convert op to a double , using the default MPFR rounding mode
(see function mpfr_set_default_rounding_mode ). This function is
obsolete.

double mpfr_get_d_2exp (long *exp, mpfr_t op, mp_rnd_t rnd)  Function 
Return d and set exp such that 0.5<=abs(d)<1 and d times 2 raised to exp equals op rounded to double precision, using the given rounding mode. 
long mpfr_get_si (mpfr_t op, mp_rnd_t rnd)  Function 
unsigned long mpfr_get_ui (mpfr_t op, mp_rnd_t op)  Function 
Convert op to a long or unsigned long , after rounding
it with respect to rnd.
If op is NaN or Inf, or too big for the return type,
the result is undefined.
See also 
mp_exp_t mpfr_get_z_exp (mpz_t z, mpfr_t op)  Function 
Put the scaled mantissa of op (regarded as an integer, with the
precision of op) into z, and return the exponent exp
(which may be outside the current exponent range) such that op
exactly equals
z multiplied by two exponent exp.
If the exponent is not representable in the mp_exp_t type, the
behavior is undefined.

char * mpfr_get_str (char *str, mp_exp_t *expptr, int base, size_t n, mpfr_t op, mp_rnd_t rnd)  Function 
Convert op to a string of digits in base base, with rounding in
direction rnd. The base may vary
from 2 to 36.
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 n is zero, the number of digits of the mantissa is determined automatically from the precision of op and the value of base. Warning: this functionality may disappear or change in future versions. Otherwise generate exactly n significant digits, which must be at least 2. If str is a null pointer, space for the mantissa is allocated using
the current allocation function, and a pointer to the string is returned.
The block will be If str is not a null pointer, it should point to a block of storage large enough for the mantissa, i.e., at least n + 2. The extra two bytes are for a possible minus sign, and for the terminating null character. If n is 0, note that the space requirements for str in this case will be impossible for the user to predetermine. Therefore, one needs to pass a null pointer for the string argument whenever n is 0. If the input number is an ordinary number, the exponent is written through the pointer expptr (the current minimal exponent for 0). A pointer to the string is returned, unless there is an error, in which case a null pointer is returned. 
int mpfr_fits_ulong_p (mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_fits_slong_p (mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_fits_uint_p (mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_fits_sint_p (mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_fits_ushort_p (mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_fits_sshort_p (mpfr_t op, mp_rnd_t rnd)  Function 
Return nonzero if op would fit in the respective C data type, when rounded to an integer in the direction rnd. 
int mpfr_add (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_add_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_add_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)  Function 
int mpfr_add_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)  Function 
Set rop to op1 + op2 rounded in the direction rnd. 
int mpfr_sub (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_ui_sub (mpfr_t rop, unsigned long int op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_sub_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_sub_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)  Function 
int mpfr_sub_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)  Function 
Set rop to op1  op2 rounded in the direction rnd. 
int mpfr_mul (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_mul_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_mul_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)  Function 
int mpfr_mul_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)  Function 
Set rop to op1 times op2 rounded in the direction rnd. 
int mpfr_div (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_ui_div (mpfr_t rop, unsigned long int op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_div_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_div_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)  Function 
int mpfr_div_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)  Function 
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). 
int mpfr_sqrt (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_sqrt_ui (mpfr_t rop, unsigned long int op, mp_rnd_t rnd)  Function 
Set rop to the square root of op rounded in the direction rnd. Return 0 if rop is 0 (to be consistent with the IEEE 7541985 standard). Set rop to NaN if op is negative. 
int mpfr_cbrt (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the cubic root (defined over the real numbers) of op rounded in the direction rnd. 
int mpfr_pow (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
int mpfr_pow_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_pow_si (mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd)  Function 
int mpfr_ui_pow_ui (mpfr_t rop, unsigned long int op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_ui_pow (mpfr_t rop, unsigned long int op1, mpfr_t op2, mp_rnd_t rnd)  Function 
Set rop to op1 raised to op2,
rounded in the direction rnd.
Special values are currently handled as described in the ISO C99 standard
for the pow function (note this may change in future versions):

int mpfr_neg (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to op rounded in the direction rnd. Just changes the sign if rop and op are the same variable. 
int mpfr_abs (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the absolute value of op, rounded in the direction rnd. 
int mpfr_mul_2ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_mul_2si (mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd)  Function 
int mpfr_mul_2exp (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
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.
[mpfr_mul_2exp is kept for upward compatibility.]

int mpfr_div_2ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
int mpfr_div_2si (mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd)  Function 
int mpfr_div_2exp (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)  Function 
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.
[mpfr_div_2exp is kept for upward compatibility.]

int mpfr_cmp (mpfr_t op1, mpfr_t op2)  Function 
int mpfr_cmp_ui (mpfr_t op1, unsigned long int op2)  Function 
int mpfr_cmp_si (mpfr_t op1, signed long int op2)  Function 
int mpfr_cmp_d (mpfr_t op1, double op2)  Function 
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 (NotaNumber), the behavior is undefined. 
int mpfr_cmp_ui_2exp (mpfr_t op1, unsigned long int op2, mp_exp_t e)  Function 
int mpfr_cmp_si_2exp (mpfr_t op1, long int op2, mp_exp_t e)  Function 
Compare op1 and op2 multiplied by two to the power e. Similar as above. 
int mpfr_cmpabs (mpfr_t op1, mpfr_t op2)  Function 
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 (NotaNumber), the behavior is undefined. 
int mpfr_eq (mpfr_t op1, mpfr_t op2, unsigned long int op3)  Function 
Return nonzero if op1 and op2 are both nonzero 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 , but does not
make much sense.

int mpfr_nan_p (mpfr_t op)  Function 
int mpfr_inf_p (mpfr_t op)  Function 
int mpfr_number_p (mpfr_t op)  Function 
Return nonzero if op is respectively NotaNumber (NaN), an infinity, an ordinary number (i.e. neither NotaNumber nor an infinity). Return zero otherwise. 
void mpfr_reldiff (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
Compute the relative difference between op1 and op2 and store the result in rop. This function does not guarantee the exact rounding on the relative difference; it just computes op1op2/op1, using the rounding mode rnd for all operations and the precision of rop. 
int mpfr_sgn (mpfr_t op)  Macro 
Return a positive value if op > 0, zero if op = 0,
and a negative value if op < 0.
Its result is undefined when op is NaN (NotaNumber).
This function is actually implemented as a macro. It may evaluate its argument multiple times. 
int mpfr_greater_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 > op2, zero otherwise. 
int mpfr_greaterequal_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 >= op2, zero otherwise. 
int mpfr_less_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 < op2, zero otherwise. 
int mpfr_lessequal_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 <= op2, zero otherwise. 
int mpfr_lessgreater_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero 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 are NaN, or op1 = op2). 
int mpfr_equal_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 = op2, zero otherwise (i.e. op1 and/or op2 are NaN, or op1 <> op2). 
int mpfr_unordered_p (mpfr_t op1, mpfr_t op2)  Function 
Return nonzero if op1 or op2 is a NaN (i.e. they cannot be compared), zero otherwise. 
All those functions, except explicitly stated, return zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise.
int mpfr_log (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_log2 (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_log10 (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the natural logarithm of op, log2(op) or log10(op), respectively, rounded in the direction rnd. 
int mpfr_exp (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the exponential of op, rounded in the direction rnd. 
int mpfr_exp2 (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to 2 power of op, rounded in the direction rnd. 
int mpfr_cos (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_sin (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_tan (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the cosine of op, sine of op, tangent of op, rounded in the direction rnd. 
int mpfr_sin_cos (mpfr_t sop, mpfr_t cop, mpfr_t op, mp_rnd_t rnd)  Function 
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. Return 0 iff both results are exact. 
int mpfr_acos (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_asin (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_atan (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the arccosine, arcsine or arctangent of op, rounded in the direction rnd. 
int mpfr_cosh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_sinh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_tanh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the hyperbolic cosine, sine or tangent of op, rounded in the direction rnd. 
int mpfr_acosh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_asinh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_atanh (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the inverse hyperbolic cosine, sine or tangent of op, rounded in the direction rnd. 
int mpfr_fac_ui (mpfr_t rop, unsigned long int op, mp_rnd_t rnd)  Function 
Set rop to the factorial of the unsigned long int op,
rounded in the direction rnd.

int mpfr_log1p (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the logarithm of one plus op, rounded in the direction rnd. 
int mpfr_expm1 (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the exponential of op minus one, rounded in the direction rnd. 
int mpfr_gamma (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the value of the Gamma function on op, rounded in the direction rnd. 
int mpfr_zeta (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the value of the Riemann Zeta function on op, rounded in the direction rnd. 
int mpfr_erf (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
Set rop to the value of the error function on op, rounded in the direction rnd. 
int mpfr_fma (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mp_rnd_t rnd)  Function 
Set rop to op1 times op2 + op3, rounded in the direction rnd. 
int mpfr_agm (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
Set rop to the arithmeticgeometric mean of op1 and op2, rounded in the direction rnd. The arithmeticgeometric 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]. 
int mpfr_const_log2 (mpfr_t rop, mp_rnd_t rnd)  Function 
int mpfr_const_pi (mpfr_t rop, mp_rnd_t rnd)  Function 
int mpfr_const_euler (mpfr_t rop, mp_rnd_t rnd)  Function 
Set rop to the logarithm of 2, the value of Pi, the value of Euler's constant 0.577..., respectively, rounded in the direction rnd. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. There is currently no way to free the cache. 
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 argument 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.
size_t mpfr_out_str (FILE *stream, int base, size_t n, mpfr_t op, mp_rnd_t rnd)  Function 
Output op on stream stream, as a string of digits in
base base, rounded in direction rnd.
The base may vary from 2 to 36. Print n significant digits exactly,
or if n is 0, the maximum number of digits accurately representable
by op (this feature may disappear).
In addition to the significant digits, a decimal point at the right of the
first digit and a trailing exponent in base 10, in the form Return the number of bytes written, or if an error occurred, return 0. 
size_t mpfr_inp_str (mpfr_t rop, FILE *stream, int base, mp_rnd_t rnd)  Function 
Input a string in base base from stream stream,
rounded in direction rnd, and put the
read float in rop. The string is of the form M@N or, if the
base is 10 or less, alternatively MeN or MEN , or, if the base
is 16, alternatively MpB or MPB .
M is the mantissa in the specified base, N is the exponent
written in decimal for the specified base, and in base 16, B is the
binary exponent written in decimal (i.e. it indicates the power of 2 by
which the mantissa is to be scaled).
The argument base may be in the range 2 to 36.
Special values can be read as follows (the case does not matter):
Return the number of bytes read, or if an error occurred, return 0. 
int mpfr_rint (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
int mpfr_ceil (mpfr_t rop, mpfr_t op)  Function 
int mpfr_floor (mpfr_t rop, mpfr_t op)  Function 
int mpfr_round (mpfr_t rop, mpfr_t op)  Function 
int mpfr_trunc (mpfr_t rop, mpfr_t op)  Function 
Set rop to op rounded to an integer.
mpfr_rint rounds to the nearest representable integer in the
given rounding mode, mpfr_ceil rounds
to the next higher or equal representable integer, mpfr_floor to
the next lower or equal representable integer, mpfr_round to the
nearest representable integer, rounding halfway cases away from zero,
and mpfr_trunc to the next representable integer towards 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 
int mpfr_frac (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)  Function 
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).

int mpfr_integer_p (mpfr_t op)  Function 
Return nonzero iff op is an integer. 
void mpfr_nexttoward (mpfr_t x, mpfr_t y)  Function 
If x or y is NaN, set x to NaN. Otherwise, if x is different from y, replace x by the next floatingpoint number (with the precision of x and the current exponent range) in the direction of y, if there is one (the infinite values are seen as the smallest and largest floatingpoint numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated. 
void mpfr_nextabove (mpfr_t x)  Function 
Equivalent to mpfr_nexttoward where y is plus infinity.

void mpfr_nextbelow (mpfr_t x)  Function 
Equivalent to mpfr_nexttoward where y is minus infinity.

int mpfr_min (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
Set rop to the minimum 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. 
int mpfr_max (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)  Function 
Set rop to the 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. 
int mpfr_urandomb (mpfr_t rop, gmp_randstate_t state)  Function 
Generate a uniformly distributed random float in the interval 0 <= rop < 1. Return 0, unless the exponent is not in the current exponent range, in which case rop is set to NaN and a nonzero value is returned. 
void mpfr_random (mpfr_t rop)  Function 
Generate a uniformly distributed random float in the interval
0 <= rop < 1.
This function is deprecated; mpfr_urandomb should be used instead.

void mpfr_random2 (mpfr_t rop, mp_size_t size, mp_exp_t exp)  Function 
Generate a random float of at most size limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval exp to exp. This function is useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger cornercase bugs. Negative random numbers are generated when size is negative. Put +0 in rop when size if zero. 
The following types and
functions were mainly designed for the implementation of mpfr
,
but may be useful for users too.
However no upward compatibility is guaranteed.
You may need to include mpfrimpl.h
to use them.
The mpfr_t
type consists of four fields.
The _mpfr_prec
field is used to store the precision of
the variable (in bits); this is not less than MPFR_PREC_MIN
.
The _mpfr_size
field is used to store the number of
allocated limbs, with the high bits reserved to store
the sign (bit 31), the NaN flag (bit 30),
and the Infinity flag (bit 29);
thus bits 0 to 28 remain for the number of allocated limbs, with a maximal
value of 536870911.
A NaN is indicated by the NaN flag set, and the other fields are
undefined.
An Infinity is indicated by the NaN flag clear and the Infinity flag set;
the sign bit of an Infinity indicates the sign, the limb data
and the exponent are undefined.
The _mpfr_exp
field stores the exponent.
An exponent of 0 means a radix point just above the most significant
limb. Nonzero values n are a multiplier 2^n relative to that
point.
Finally, the _mpfr_d
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
/BITS_PER_MP_LIMB
).
Zeros are represented by the most significant limb being zero, other
limb data and the exponent are undefined (this implies that the
corresponding objects may contain invalid values, thus should not be
evaluated even if they are not taken into account).
Nonzero 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 zero.
When the precision has been lowered by mpfr_set_prec
, the space
allocated at _mpfr_d
remains as given by _mpfr_size
, but
_mpfr_prec
indicates how much of that space is actually used.
int mpfr_add_one_ulp (mpfr_t x, mp_rnd_t rnd)  Function 
Add one unit in last place (ulp) to x if x is finite
and positive, subtract one ulp if x is finite and negative;
otherwise, x is not changed.
The return value is zero unless an overflow occurs, in which case the
mpfr_add_one_ulp function behaves like a conventional addition.

int mpfr_sub_one_ulp (mpfr_t x, mp_rnd_t rnd)  Function 
Subtract one ulp to x if x is finite and positive, add one
ulp if x is finite and negative; otherwise, x is not changed.
The return value is zero unless an underflow occurs, in which case the
mpfr_sub_one_ulp function behaves like a conventional subtraction.

int mpfr_can_round (mpfr_t b, mp_exp_t err, mp_rnd_t rnd1, mp_rnd_t rnd2, mp_prec_t prec)  Function 
Assuming b is an approximation of an unknown number x in direction rnd1 with error at most two to the power E(b)err where E(b) is the exponent of b, returns a nonzero value if one is able to round exactly x to precision prec with direction rnd2, and 0 otherwise (including for NaN and Inf). This function does not modify its arguments. 
mp_exp_t mpfr_get_exp (mpfr_t x)  Function 
Get the exponent of x, assuming that x is a nonzero ordinary number. 
int mpfr_set_exp (mpfr_t x, mp_exp_t e)  Function 
Set the exponent of x if e is in the current exponent range, and return 0 (even if x is not a nonzero ordinary number); otherwise, return a nonzero value. 
void mpfr_set_str_binary (mpfr_t x, const char *s)  Function 
Set x to the value of the binary number in string s, which has to
be of the
form +/xxxx.xxxxxxEyy. The exponent is read in decimal, but is interpreted
as the power of two to be multiplied by the mantissa.
The mantissa length of s has to be less or equal to the precision of
x, otherwise an error occurs.
If s starts with N , it is interpreted as NaN (NotaNumber);
if it starts with I after the sign, it is interpreted as infinity,
with the corresponding sign.

void mpfr_print_binary (mpfr_t float)  Function 
Output float on stdout in raw binary format (the exponent is written in decimal, yet). 
The main developers consist of Guillaume Hanrot, Vincent Lefèvre, Kevin Ryde and Paul Zimmermann.
We would like to thank JeanMichel 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 their suggestions for an easy integration into GNU MP, and Nathalie Revol for her careful reading of a previous version of this documentation.
Sylvie Boldo from ENSLyon, France,
contributed the functions mpfr_agm
and mpfr_log
.
Emmanuel Jeandel, from ENSLyon too,
contributed the generic hypergeometric code in
generic.c
, as well as the 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
,
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base2 exponential, and the factorial function. Fabrice Rouillier
contributed the original version of mul_ui.c
, the gmp_op.c
file, and helped to the Windows porting.
JeanLuc Rémy contributed the mpfr_zeta
code.
Ludovic Meunier helped in the design of the mpfr_erf
code.
The development of the MPFR library would not have been possible without the continuous support of LORIA, INRIA and INRIA Lorraine. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002.
Copyright © 2000 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 021111307, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
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This License applies to any manual or other work that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you".
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mpfr.h
: MPFR Basics
mp_prec_t
: MPFR Basics
mp_rnd_t
: MPFR Basics
mpfr_abs
: Basic Arithmetic Functions
mpfr_acos
: Special Functions
mpfr_acosh
: Special Functions
mpfr_add
: Basic Arithmetic Functions
mpfr_add_one_ulp
: Internals
mpfr_add_q
: Basic Arithmetic Functions
mpfr_add_ui
: Basic Arithmetic Functions
mpfr_add_z
: Basic Arithmetic Functions
mpfr_agm
: Special Functions
mpfr_asin
: Special Functions
mpfr_asinh
: Special Functions
mpfr_atan
: Special Functions
mpfr_atanh
: Special Functions
mpfr_can_round
: Internals
mpfr_cbrt
: Basic Arithmetic Functions
mpfr_ceil
: Miscellaneous Functions
mpfr_check_range
: Exceptions
mpfr_clear
: Initialization Functions
mpfr_clear_flags
: Exceptions
mpfr_clear_inexflag
: Exceptions
mpfr_clear_nanflag
: Exceptions
mpfr_clear_overflow
: Exceptions
mpfr_clear_underflow
: Exceptions
mpfr_cmp
: Comparison Functions
mpfr_cmp_d
: Comparison Functions
mpfr_cmp_si
: Comparison Functions
mpfr_cmp_si_2exp
: Comparison Functions
mpfr_cmp_ui
: Comparison Functions
mpfr_cmp_ui_2exp
: Comparison Functions
mpfr_cmpabs
: Comparison Functions
mpfr_const_euler
: Special Functions
mpfr_const_log2
: Special Functions
mpfr_const_pi
: Special Functions
mpfr_cos
: Special Functions
mpfr_cosh
: Special Functions
mpfr_div
: Basic Arithmetic Functions
mpfr_div_2exp
: Basic Arithmetic Functions
mpfr_div_2si
: Basic Arithmetic Functions
mpfr_div_2ui
: Basic Arithmetic Functions
mpfr_div_q
: Basic Arithmetic Functions
mpfr_div_ui
: Basic Arithmetic Functions
mpfr_div_z
: Basic Arithmetic Functions
mpfr_eq
: Comparison Functions
mpfr_equal_p
: Comparison Functions
mpfr_erf
: Special Functions
mpfr_exp
: Special Functions
mpfr_exp2
: Special Functions
mpfr_expm1
: Special Functions
mpfr_fac_ui
: Special Functions
mpfr_fits_sint_p
: Conversion Functions
mpfr_fits_slong_p
: Conversion Functions
mpfr_fits_sshort_p
: Conversion Functions
mpfr_fits_uint_p
: Conversion Functions
mpfr_fits_ulong_p
: Conversion Functions
mpfr_fits_ushort_p
: Conversion Functions
mpfr_floor
: Miscellaneous Functions
mpfr_fma
: Special Functions
mpfr_frac
: Miscellaneous Functions
mpfr_gamma
: Special Functions
mpfr_get_d
: Conversion Functions
mpfr_get_d1
: Conversion Functions
mpfr_get_d_2exp
: Conversion Functions
mpfr_get_default_prec
: Initialization Functions
mpfr_get_emax
: Exceptions
mpfr_get_emin
: Exceptions
mpfr_get_exp
: Internals
mpfr_get_ld
: Conversion Functions
mpfr_get_prec
: Initialization Functions
mpfr_get_si
: Conversion Functions
mpfr_get_str
: Conversion Functions
mpfr_get_ui
: Conversion Functions
mpfr_get_z_exp
: Conversion Functions
mpfr_greater_p
: Comparison Functions
mpfr_greaterequal_p
: Comparison Functions
mpfr_inexflag_p
: Exceptions
mpfr_inf_p
: Comparison Functions
mpfr_init
: Initialization Functions
mpfr_init2
: Initialization Functions
mpfr_init_set
: Combined Initialization and Assignment Functions
mpfr_init_set_d
: Combined Initialization and Assignment Functions
mpfr_init_set_f
: Combined Initialization and Assignment Functions
mpfr_init_set_ld
: Combined Initialization and Assignment Functions
mpfr_init_set_q
: Combined Initialization and Assignment Functions
mpfr_init_set_si
: Combined Initialization and Assignment Functions
mpfr_init_set_str
: Combined Initialization and Assignment Functions
mpfr_init_set_ui
: Combined Initialization and Assignment Functions
mpfr_init_set_z
: Combined Initialization and Assignment Functions
mpfr_inp_str
: Input and Output Functions
mpfr_integer_p
: Miscellaneous Functions
mpfr_less_p
: Comparison Functions
mpfr_lessequal_p
: Comparison Functions
mpfr_lessgreater_p
: Comparison Functions
mpfr_log
: Special Functions
mpfr_log10
: Special Functions
mpfr_log1p
: Special Functions
mpfr_log2
: Special Functions
mpfr_max
: Miscellaneous Functions
mpfr_min
: Miscellaneous Functions
mpfr_mul
: Basic Arithmetic Functions
mpfr_mul_2exp
: Basic Arithmetic Functions
mpfr_mul_2si
: Basic Arithmetic Functions
mpfr_mul_2ui
: Basic Arithmetic Functions
mpfr_mul_q
: Basic Arithmetic Functions
mpfr_mul_ui
: Basic Arithmetic Functions
mpfr_mul_z
: Basic Arithmetic Functions
mpfr_nan_p
: Comparison Functions
mpfr_nanflag_p
: Exceptions
mpfr_neg
: Basic Arithmetic Functions
mpfr_nextabove
: Miscellaneous Functions
mpfr_nextbelow
: Miscellaneous Functions
mpfr_nexttoward
: Miscellaneous Functions
mpfr_number_p
: Comparison Functions
mpfr_out_str
: Input and Output Functions
mpfr_overflow_p
: Exceptions
mpfr_pow
: Basic Arithmetic Functions
mpfr_pow_si
: Basic Arithmetic Functions
mpfr_pow_ui
: Basic Arithmetic Functions
mpfr_prec_round
: Rounding Modes
mpfr_print_binary
: Internals
mpfr_print_rnd_mode
: Rounding Modes
mpfr_random
: Miscellaneous Functions
mpfr_random2
: Miscellaneous Functions
mpfr_reldiff
: Comparison Functions
mpfr_rint
: Miscellaneous Functions
mpfr_round
: Miscellaneous Functions
mpfr_round_prec
: Rounding Modes
mpfr_set
: Assignment Functions
mpfr_set_d
: Assignment Functions
mpfr_set_default_prec
: Initialization Functions
mpfr_set_default_rounding_mode
: Rounding Modes
mpfr_set_emax
: Exceptions
mpfr_set_emin
: Exceptions
mpfr_set_exp
: Internals
mpfr_set_f
: Assignment Functions
mpfr_set_inf
: Assignment Functions
mpfr_set_ld
: Assignment Functions
mpfr_set_nan
: Assignment Functions
mpfr_set_prec
: Initialization Functions
mpfr_set_prec_raw
: Initialization Functions
mpfr_set_q
: Assignment Functions
mpfr_set_si
: Assignment Functions
mpfr_set_str
: Assignment Functions
mpfr_set_str_binary
: Internals
mpfr_set_ui
: Assignment Functions
mpfr_set_z
: Assignment Functions
mpfr_sgn
: Comparison Functions
mpfr_sin
: Special Functions
mpfr_sin_cos
: Special Functions
mpfr_sinh
: Special Functions
mpfr_sqrt
: Basic Arithmetic Functions
mpfr_sqrt_ui
: Basic Arithmetic Functions
mpfr_sub
: Basic Arithmetic Functions
mpfr_sub_one_ulp
: Internals
mpfr_sub_q
: Basic Arithmetic Functions
mpfr_sub_ui
: Basic Arithmetic Functions
mpfr_sub_z
: Basic Arithmetic Functions
mpfr_swap
: Assignment Functions
mpfr_t
: MPFR Basics
mpfr_tan
: Special Functions
mpfr_tanh
: Special Functions
mpfr_trunc
: Miscellaneous Functions
mpfr_ui_div
: Basic Arithmetic Functions
mpfr_ui_pow
: Basic Arithmetic Functions
mpfr_ui_pow_ui
: Basic Arithmetic Functions
mpfr_ui_sub
: Basic Arithmetic Functions
mpfr_underflow_p
: Exceptions
mpfr_unordered_p
: Comparison Functions
mpfr_urandomb
: Miscellaneous Functions
mpfr_zeta
: Special Functions