MPFR

This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 2.2.0.

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005 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 Front-Cover Texts being “A GNU Manual”, and with the Back-Cover 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.

• Copying: MPFR Copying Conditions (LGPL).
• Introduction to MPFR: Brief introduction to MPFR.
• Installing MPFR: How to configure and compile the MPFR library.
• Reporting Bugs: How to usefully report bugs.
• MPFR Basics: What every MPFR user should now.
• MPFR Interface: MPFR functions and macros.
• Contributors
• References
• GNU Free Documentation License
• Concept Index
• Function Index

MPFR Copying Conditions

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.

1 Introduction to MPFR

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 extend the class of floating-point numbers provided by the GNU MP library by a precise semantics. The main differences with the mpf class from GNU MP are:

• the 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);
• the precision in bits can be set exactly to any valid value for each variable (including very small precision);
• mpfr provides the four rounding modes from the IEEE 754-1985 standard.

In particular, with a precision of 53 bits, mpfr should be able to exactly reproduce all computations with double-precision machine floating-point numbers (double type in C), 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. It is permitted to link MPFR to 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.

1.1 How to use this Manual

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.

2 Installing MPFR

2.1 How to install

Here are the steps needed to install the library on Unix systems (more details are provided in the INSTALL file):

1. To build MPFR, you first have to install GNU MP (version 4.1 or higher) on your computer. You need a C compiler, preferably GCC, but any reasonable compiler should work. And you need a standard Unix make program, plus some other standard Unix utility programs.
2. In the MPFR build directory, type ./configure

   This will prepare the build and setup the options according to your system. 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).

3. make

   This will compile MPFR, and create a library archive file libmpfr.a. A dynamic library may be produced too (see configure).

4. 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.)

5. 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).

2.2 Other make targets

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.

2.3 Known Build Problems

MPFR suffers from all bugs from the GNU MP library, plus 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/.

2.4 Getting the Latest Version of MPFR

The latest version of MPFR is available from http://www.mpfr.org/.

3 Reporting Bugs

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.

4 MPFR Basics

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>

4.1 Nomenclature and Types

A floating-point 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 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 754-1985 standard, zero is signed, i.e. there are both +0 and −0; the behavior is the same as in the IEEE 754-1985 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 floating-point 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.

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 mantissa; the corresponding C data type is mp_rnd_t.

A limb means the part of a multi-precision 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.

4.2 Function Classes

There is only one class of functions in the MPFR library:

1. Functions for floating-point arithmetic, with names beginning with mpfr_. The associated type is mpfr_t.

4.3 MPFR Variable Conventions

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_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 floating-point variable will have the same allocated space during all its life.

4.4 Rounding modes.

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 754-1985 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 (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 a 1 (or any other value specified in this manual) for special cases (like acos(0)) should return an overflow or an underflow if 1 is not representable in the current exponent range.

5 MPFR Interface

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 integer functions. The function prefix for floating-point operations is mpfr_.

There is one significant characteristic of floating-point numbers that has motivated a difference between this function class and other GNU MP function classes: the inherent inexactness of floating-point 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 floating-point functions are intended to be a smooth extension of the IEEE 754-1985 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.

• Initialization Functions
• Assignment Functions
• Combined Initialization and Assignment Functions
• Conversion Functions
• Basic Arithmetic Functions
• Comparison Functions
• Special Functions
• Input and Output Functions
• Integer Related Functions
• Miscellaneous Functions
• Rounding Modes
• Exceptions
• Advanced Functions
• Compatibility with MPF
• Custom interface
• Internals

5.1 Initialization Functions

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.

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);
     }

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.

5.2 Assignment Functions

These functions assign new values to already initialized floats (see Initialization Functions). When using 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.

5.3 Combined Initialization and Assignment Functions

5.4 Conversion Functions

5.5 Basic Arithmetic Functions

5.6 Comparison Functions

5.7 Special Functions

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.

5.8 Input and Output Functions

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.

5.9 Integer Related Functions

5.10 Miscellaneous Functions

5.11 Rounding Modes

5.12 Exceptions

This is an example of how to emulate double IEEE-754 arithmetic using MPFR:

     {
       mpfr_t xa, xb;
       int i;
       volatile double a, b;
     
       mpfr_set_default_prec (53);
       mpfr_set_emin (-1073);
       mpfr_set_emax (1021);
     
       mpfr_init (xa); mpfr_init (xb);
     
       b = 34.3; mpfr_set_d (xb, b, GMP_RNDN);
       a = 0x1.1235P-1021; mpfr_set_d (xa, a, GMP_RNDN);
     
       a /= b;
       i = mpfr_div (xa, xa, xb, GMP_RNDN);
       i = mpfr_subnormalize (xa, i, GMP_RNDN);
     
       mpfr_clear (xa); mpfr_clear (xb);
     }

Warning: this emulates a double IEEE-754 arithmetic with exact rounding in the subnormal range, which may not be the case for your hardware.

5.13 Advanced Functions

All the given interfaces are preliminary. They might change incompatibly in future revisions.

Here is an example of how to use multiple initialization functions:

     {
       mpfr_t x, y, z, t;
       mpfr_inits2 (256, x, y, z, t, (void *) 0);
       ...
       mpfr_clears (x, y, z, t, (void *) 0);
     }

5.14 Compatibility with MPF

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>

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.

5.15 Custom interface

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 them to use MPFR in two ways:

• Either they directly store the MPFR FP number as a mpfr_t on the stack.
• Either they store their own representation of a FP number on the stack and construct a new temporary mpfr_t each time it is needed.

Each function is this interface is also implemented as a macro for efficiency reasons.

Note 1: MPFR functions may still initialize temporary FP numbers using standard mpfr_init. 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.

Note 3: This interface is preliminary.

See the test suite for examples.

5.16 Internals

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 mpfr-impl.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_sign field is used to store the sign of the variable.
• The _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 a special value of the exponent.
• 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/mp_bits_per_limb). Non-singular 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.

Contributors

The main developers consist of Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann.

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. Kevin Ryde did a tremendous job for the portability of MPFR, and integrating it into GMP 4.x; alas the GMP developers decided in January 2004 not to include MPFR any more.

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 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, 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 original version of mul_ui.c, the gmp_op.c file, and helped to the Windows porting. Jean-Luc 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.

References

• Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 4.1.2, 2002.
• IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages.
• Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.
• Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation", Birkhauser, Boston, 1997.

Appendix A GNU Free Documentation License

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Concept Index

• Accuracy: MPFR Interface
• Advanced Functions: Advanced Functions
• Arithmetic functions: Basic Arithmetic Functions
• Assignment functions: Assignment Functions
• Basic arithmetic functions: Basic Arithmetic Functions
• Combined initialization and assignment functions: Combined Initialization and Assignment Functions
• Comparison functions: Comparison Functions
• Compatibility with MPF: Compatibility with MPF
• Conditions for copying MPFR: Copying
• Conversion functions: Conversion Functions
• Copying conditions: Copying
• Custom interface: Custom interface
• Exceptions: Exceptions
• FDL, GNU Free Documentation License: GNU Free Documentation License
• Float arithmetic functions: Basic Arithmetic Functions
• Float comparisons functions: Comparison Functions
• Float functions: MPFR Interface
• Float input and output functions: Input and Output Functions
• Floating-point functions: MPFR Interface
• Floating-point number: MPFR Basics
• GNU Free Documentation License: GNU Free Documentation License
• I/O functions: Input and Output Functions
• Initialization functions: Initialization Functions
• Input functions: Input and Output Functions
• Installation: Installing MPFR
• Integer Related Functions: Integer Related Functions
• Internals: Internals
• Limb: MPFR Basics
• Miscellaneous float functions: Miscellaneous Functions
• mpfr.h: MPFR Basics
• Output functions: Input and Output Functions
• Precision: MPFR Interface
• Precision: MPFR Basics
• Reporting bugs: Reporting Bugs
• Rounding Modes: Rounding Modes
• Rounding Modes: MPFR Basics
• Special functions: Special Functions

Function and Type Index

• *mpfr_custom_get_mantissa: Custom interface
• 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_q: Basic Arithmetic Functions
• mpfr_add_si: 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_atan2: Special Functions
• mpfr_atanh: Special Functions
• mpfr_can_round: Internals
• mpfr_cbrt: Basic Arithmetic Functions
• mpfr_ceil: Integer Related Functions
• mpfr_check_range: Exceptions
• mpfr_clear: Initialization Functions
• mpfr_clear_erangeflag: Exceptions
• mpfr_clear_flags: Exceptions
• mpfr_clear_inexflag: Exceptions
• mpfr_clear_nanflag: Exceptions
• mpfr_clear_overflow: Exceptions
• mpfr_clear_underflow: Exceptions
• mpfr_clears: Advanced Functions
• mpfr_cmp: Comparison Functions
• mpfr_cmp_d: Comparison Functions
• mpfr_cmp_f: Comparison Functions
• mpfr_cmp_ld: Comparison Functions
• mpfr_cmp_q: 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_cmp_z: Comparison Functions
• mpfr_cmpabs: Comparison Functions
• mpfr_const_catalan: Special 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_cot: Special Functions
• mpfr_coth: Special Functions
• mpfr_csc: Special Functions
• mpfr_csch: Special Functions
• mpfr_custom_get_exp: Custom interface
• mpfr_custom_get_kind: Custom interface
• mpfr_custom_get_size: Custom interface
• mpfr_custom_init: Custom interface
• mpfr_custom_init_set: Custom interface
• mpfr_custom_move: Custom interface
• MPFR_DECL_INIT: Advanced Functions
• mpfr_div: Basic Arithmetic Functions
• mpfr_div_2exp: Compatibility with MPF
• mpfr_div_2si: Basic Arithmetic Functions
• mpfr_div_2ui: Basic Arithmetic Functions
• mpfr_div_q: Basic Arithmetic Functions
• mpfr_div_si: Basic Arithmetic Functions
• mpfr_div_ui: Basic Arithmetic Functions
• mpfr_div_z: Basic Arithmetic Functions
• mpfr_eint: Special Functions
• mpfr_eq: Compatibility with MPF
• mpfr_equal_p: Comparison Functions
• mpfr_erangeflag_p: Exceptions
• mpfr_erf: Special Functions
• mpfr_erfc: Special Functions
• mpfr_exp: Special Functions
• mpfr_exp10: Special Functions
• mpfr_exp2: Special Functions
• mpfr_expm1: Special Functions
• mpfr_fac_ui: Special Functions
• mpfr_fits_intmax_p: Conversion 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_uintmax_p: Conversion Functions
• mpfr_fits_ulong_p: Conversion Functions
• mpfr_fits_ushort_p: Conversion Functions
• mpfr_floor: Integer Related Functions
• mpfr_fma: Special Functions
• mpfr_frac: Integer Related Functions
• mpfr_free_cache: Special Functions
• mpfr_free_str: Conversion Functions
• mpfr_gamma: Special Functions
• mpfr_get_d: Conversion Functions
• mpfr_get_d1: Internals
• mpfr_get_d_2exp: Conversion Functions
• mpfr_get_default_prec: Initialization Functions
• mpfr_get_default_rounding_mode: Rounding Modes
• mpfr_get_emax: Exceptions
• mpfr_get_emax_max: Exceptions
• mpfr_get_emax_min: Exceptions
• mpfr_get_emin: Exceptions
• mpfr_get_emin_max: Exceptions
• mpfr_get_emin_min: Exceptions
• mpfr_get_exp: Miscellaneous Functions
• mpfr_get_f: Conversion Functions
• mpfr_get_ld: Conversion Functions
• mpfr_get_prec: Initialization Functions
• mpfr_get_si: Conversion Functions
• mpfr_get_sj: Conversion Functions
• mpfr_get_str: Conversion Functions
• mpfr_get_ui: Conversion Functions
• mpfr_get_uj: Conversion Functions
• mpfr_get_version: Miscellaneous Functions
• mpfr_get_z: Conversion Functions
• mpfr_get_z_exp: Conversion Functions
• mpfr_greater_p: Comparison Functions
• mpfr_greaterequal_p: Comparison Functions
• mpfr_hypot: Special 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_inits: Advanced Functions
• mpfr_inits2: Advanced Functions
• mpfr_inp_str: Input and Output Functions
• mpfr_integer_p: Integer Related Functions
• mpfr_less_p: Comparison Functions
• mpfr_lessequal_p: Comparison Functions
• mpfr_lessgreater_p: Comparison Functions
• mpfr_lngamma: Special 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: Compatibility with MPF
• mpfr_mul_2si: Basic Arithmetic Functions
• mpfr_mul_2ui: Basic Arithmetic Functions
• mpfr_mul_q: Basic Arithmetic Functions
• mpfr_mul_si: 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_pow_z: Basic Arithmetic Functions
• mpfr_prec_round: Rounding Modes
• mpfr_print_rnd_mode: Rounding Modes
• mpfr_random: Miscellaneous Functions
• mpfr_random2: Miscellaneous Functions
• mpfr_reldiff: Compatibility with MPF
• mpfr_rint: Integer Related Functions
• mpfr_rint_ceil: Integer Related Functions
• mpfr_rint_floor: Integer Related Functions
• mpfr_rint_round: Integer Related Functions
• mpfr_rint_trunc: Integer Related Functions
• mpfr_root: Basic Arithmetic Functions
• mpfr_round: Integer Related Functions
• mpfr_round_prec: Rounding Modes
• mpfr_sec: Special Functions
• mpfr_sech: Special Functions
• 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_erangeflag: Exceptions
• mpfr_set_exp: Miscellaneous Functions
• mpfr_set_f: Assignment Functions
• mpfr_set_inexflag: Exceptions
• mpfr_set_inf: Assignment Functions
• mpfr_set_ld: Assignment Functions
• mpfr_set_nan: Assignment Functions
• mpfr_set_nanflag: Exceptions
• mpfr_set_overflow: Exceptions
• mpfr_set_prec: Initialization Functions
• mpfr_set_prec_raw: Compatibility with MPF
• mpfr_set_q: Assignment Functions
• mpfr_set_si: Assignment Functions
• mpfr_set_si_2exp: Assignment Functions
• mpfr_set_sj: Assignment Functions
• mpfr_set_sj_2exp: Assignment Functions
• mpfr_set_str: Assignment Functions
• mpfr_set_ui: Assignment Functions
• mpfr_set_ui_2exp: Assignment Functions
• mpfr_set_uj: Assignment Functions
• mpfr_set_uj_2exp: Assignment Functions
• mpfr_set_underflow: Exceptions
• mpfr_set_z: Assignment Functions
• mpfr_sgn: Comparison Functions
• mpfr_si_div: Basic Arithmetic Functions
• mpfr_si_sub: Basic Arithmetic Functions
• mpfr_sin: Special Functions
• mpfr_sin_cos: Special Functions
• mpfr_sinh: Special Functions
• mpfr_sqr: Basic Arithmetic Functions
• mpfr_sqrt: Basic Arithmetic Functions
• mpfr_sqrt_ui: Basic Arithmetic Functions
• mpfr_strtofr: Assignment Functions
• mpfr_sub: Basic Arithmetic Functions
• mpfr_sub_q: Basic Arithmetic Functions
• mpfr_sub_si: Basic Arithmetic Functions
• mpfr_sub_ui: Basic Arithmetic Functions
• mpfr_sub_z: Basic Arithmetic Functions
• mpfr_subnormalize: Exceptions
• mpfr_sum: Special Functions
• mpfr_swap: Assignment Functions
• mpfr_t: MPFR Basics
• mpfr_tan: Special Functions
• mpfr_tanh: Special Functions
• mpfr_trunc: Integer Related 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_VERSION: Miscellaneous Functions
• MPFR_VERSION_MAJOR: Miscellaneous Functions
• MPFR_VERSION_MINOR: Miscellaneous Functions
• MPFR_VERSION_NUM: Miscellaneous Functions
• MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions
• MPFR_VERSION_STRING: Miscellaneous Functions
• mpfr_zero_p: Comparison Functions
• mpfr_zeta: Special Functions