     START-INFO-DIR-ENTRY
     * mpfr: (mpfr.info).               Multiple Precision Floating-Point Reliable Library.
     END-INFO-DIR-ENTRY

This file documents MPFR, a library for reliable multiple precision floating-point arithmetic

Copyright (C) 1999-2001, Free Software Foundation.

Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the Foundation.

MPFR
****

This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 2002

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 Library General Public License that accompany the source code. See the file COPYING.

Introduction to MPFR
********************

MPFR is a portable library written in C for arbitrary precision arithmetic on reliable 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 reliable floating-point numbers. It may replace the GNU MP floating-point numbers in a future release. The main differences with the mpf class 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 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 range of values is much wider (the exponent has at least 32 bits instead of 11).

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 is provided.

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.

Installing MPFR
***************

To build MPFR, you first have to install GNU MP (version 3.0 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.

Here are the steps needed to install the library on Unix systems:

 1. In most cases, ./configure --with-gmp=/usr/local/gmp should work, where the directory /usr/local/gmp is where you have installed GNU MP. When you install GNU MP, you have to copy the files config.h, gmp-impl.h, gmp-mparam.h and longlong.h from the GNU MP source directory to /usr/local/gmp/include; these additional files are needed by MPFR. An alternative way is ./configure --with-gmp-include=dir1 --with-gmp-lib=dir2, where dir1 contains the GNU MP header files, and dir2 the GNU MP library files. If you get error messages, your machine might not be supported.

     If you want to compile in a separate object directory, cd to that directory, and prefix the configure command with the path to the MPFR source directory. Not all make programs have the necessary features to support this.  In particular, SunOS and Solaris make have bugs that make them unable to build from a separate object directory.  Use GNU make instead. Also, the generated makefiles use some pattern matching rules that are not supported by all make programs. Again, use GNU make instead.

 2. make

     This will compile MPFR, and create a library archive file libmpfr.a in the working directory.

 3. 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.)

 4. make install

     This will copy the files mpfr.h and mpf2mpfr.h, and libmpfr.a, to the directories /usr/local/include and /usr/local/lib respectively (or if you passed the --prefix option to configure, to the directory given as argument to --prefix).

There are some other useful make targets:

  * 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.

Known Build Problems
====================

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.

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 will find 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 what `which cc`, 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.

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"

Nomenclature and Types
======================

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 a value which cannot be represented in the floating-point format, like 0 divided by 0, or Infinity minus Infinity.

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 and MPFR_PREC_MAX is equal to ULONG_MAX.

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.

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.  There are about 50 functions is this class.

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.

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"

any program written for MPF can be linked 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.

 -- Global Variable: mp_rnd_t __gmp_default_rounding_mode
     The default rounding mode (to nearest initially).

Getting the Latest Version of MPFR
==================================

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

Floating-point Functions
************************

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 MPFR function classes: the inherent inexactness of floating-point arithmetic.  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 from the precision of variables used as input on average.

The precision of a calculation is defined as follows: Compute the requested operation exactly (with ``infinite precision''), and round the result to the destination variable precision with the given rounding mode. Even if the user has asked for a very high precision, MP will not calculate with superfluous digits.  For example, if two low-precision numbers of nearly equal magnitude are added, the precision of the result will be limited to what is required to represent the result accurately.

The MPFR floating-point functions are intended to be a smooth extension of the IEEE P754 arithmetic. The results obtained on one computer should not differ from the results obtained on a computer with a different word size.

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 P754 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, which is represented by (101) in binary, is rounded to (100)=4 with a precision of two bits, and not to (110)=6. This rule avoids the drift phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (section 4.2.2, pages 221-222).

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. If this value is zero, it means that the value stored in the destination variable is the exact result of the corresponding mathematical function. If the returned 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 returned 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 inexact return value.

 -- Function: void mpfr_set_default_rounding_mode (mp_rnd_t RND)
     Sets the default rounding mode to RND. The default rounding mode is to nearest initially.

 -- Function: int mpfr_round_prec (mpfr_t X, mp_rnd_t RND, mp_prec_t PREC)
     Rounds X according to RND with precision PREC, which may be different from that of X. If PREC is greater or equal to the precision of X, then new space is allocated for the mantissa, and it is filled with zeroes. Otherwise, the mantissa is rounded to precision PREC with the given direction. In both cases, the precision of X is changed to PREC. The returned value is zero when the result is exact, positive when it is greater than the original value of X, and negative when it is smaller. The precision PREC can be any integer between MPFR_PREC_MIN and MPFR_PREC_MAX.

 -- Function: void mpfr_set_machine_rnd_mode (mp_rnd_t RND)
     Set the machine rounding mode to RND. This function is provided only when the operating system supports the ISOC9X standard interface for setting rounding modes (i.e. through the header file <fenv.h>).

 -- Function: char * mpfr_print_rnd_mode (mp_rnd_t RND)
     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.

Exceptions
==========

 -- Function: mp_exp_t mpfr_get_emin (void)
 -- Function: mp_exp_t mpfr_get_emax (void)
     Return the (current) smallest and largest exponents allowed for a floating-point variable.

 -- Function: int mpfr_set_emin (mp_exp_t EXP)
 -- Function: int mpfr_set_emax (mp_exp_t EXP)
     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 behaviour will be undefined, in the sense of the ISO C standard.

 -- Function: int mpfr_check_range (mpfr_t X, mp_rnd_t RND)
     Return zero if the exponent of X is in the current allowed range (see mpfr_get_emin and mpfr_get_emax), otherwise reset X according to the current floating-point system and the rounding mode RND, and return a positive value if the rounded result is larger than the original one, a negative value otherwise (the result cannot be exact in that case).

 -- Function: void mpfr_clear_underflow (void)
 -- Function: void mpfr_clear_overflow (void)
 -- Function: void mpfr_clear_nanflag (void)
 -- Function: void mpfr_clear_inexflag (void)
     Clear the underflow, overflow, invalid, and inexact flags.

 -- Function: void mpfr_clear_flags (void)
     Clear all global flags (underflow, overflow, inexact, invalid).

 -- Function: int mpfr_underflow_p (void)
 -- Function: int mpfr_overflow_p (void)
 -- Function: int mpfr_nanflag_p (void)
 -- Function: int mpfr_inexflag_p (void)
     Return the corresponding (underflow, overflow, invalid, inexact) flag, which is non-zero iff the flag is set.

Initialization and Assignment Functions
=======================================

 -- Function: void mpfr_set_default_prec (mp_prec_t PREC)
     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.

 -- Function: mp_prec_t mpfr_get_default_prec ()
     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.

 -- Function: void mpfr_init (mpfr_t X)
     Initialize X, and set its value to zero. Normally, a variable should be initialized once only or at least be cleared, using mpfr_clear, between initializations.  The precision of X is the default precision, which can be changed by a call to mpfr_set_default_prec.

 -- Function: void mpfr_init2 (mpfr_t X, mp_prec_t PREC)
     Initialize X, set its precision to be exactly PREC bits, and set its value to zero. Normally, a variable should be initialized once only or at least be cleared, using mpfr_clear, between initializations. To change the precision of a variable which has already been initialized, use mpfr_set_prec instead. The precision PREC can be any integer between MPFR_PREC_MIN and MPFR_PREC_MAX.

 -- Function: void mpfr_clear (mpfr_t X)
     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 floating-point variables:

     {
       mpfr_t x, y;
       mpfr_init (x);			/* use default precision */
       mpfr_init2 (y, 256);		/* precision exactly 256 bits */
       ...
       /* Unless the program is about to exit, do ... */
       mpfr_clear (x);
       mpfr_clear (y);
     }

The following two 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.

 -- Function: int mpfr_set_prec (mpfr_t X, mp_prec_t PREC)
     Reset the precision of X to be exactly PREC bits. 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 mpfr_round_prec instead.

 -- Function: mp_prec_t mpfr_get_prec (mpfr_t X)
     Return the precision actually used for assignments of X, i.e. the number of bits used to store its mantissa.

 -- Function: void mpfr_set_prec_raw (mpfr_t X, unsigned long int P)
     Reset the precision of X to be exactly PREC bits. The only difference with mpfr_set_prec is that P is assumed to be small enough so that the mantissa fits into the current allocated memory space for X. Otherwise an error will occur.

Assignment Functions
====================

These functions assign new values to already initialized floats (see Initializing Floats).

 -- Function: 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_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)
     Set the value of ROP from OP, rounded to the precision of ROP towards the given direction RND. Please note that even a long int may have to be rounded, if the destination precision is less than the machine word width. The return value is zero when ROP=OP, positive when ROP>OP, and negative when ROP<OP. For mpfr_set_d, be careful that the input number OP may not be exactly representable as a double-precision number (this happens for 0.1 for instance), in which case it is first rounded by the C compiler to a double-precision number, and then only to a mpfr floating-point number.

 -- Function: int mpfr_set_str (mpfr_t X, char *S, int BASE, mp_rnd_t RND)
     Set X to the value of the string S in base BASE (between 2 and 36), rounded in direction RND to the precision of X. The exponent is read in decimal. This function returns -1 if an internal overflow occurred (for instance, because the exponent is too large). Otherwise it returns 0 if the base is valid and if the entire string up to the final '\0' is a valid number in base BASE, and 1 if the input is incorrect.

 -- Function: void mpfr_set_str_raw (mpfr_t X, char *S)
     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 (Not-a-Number); if it starts with I after the sign, it is interpreted as infinity, with the corresponding sign.

 -- Function: int mpfr_set_f (mpfr_t X, mpf_t Y, mp_rnd_t RND)
     Set X to the GNU MP floating-point number Y, rounded with the RND mode and the precision of X. The returned value is zero when X=Y, positive when X>Y, and negative when X<Y.

 -- Function: void mpfr_set_inf (mpfr_t X, int SIGN)
 -- Function: void mpfr_set_nan (mpfr_t X)
     Set the variable X to infinity or NaN (Not-a-Number) respectively. In mpfr_set_inf, X is set to plus infinity iff SIGN is positive.

 -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y)
     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.

Combined Initialization and Assignment Functions
================================================

 -- Macro: 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_f (mpfr_t ROP, 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)
     Initialize ROP and set its value from OP, rounded to direction RND. The precision of ROP will be taken from the active default precision, as set by mpfr_set_default_prec. The return value if zero if ROP=OP, positive if ROP>OP, and negative when ROP<OP.

 -- Function: int mpfr_init_set_str (mpfr_t X, char *S, int BASE, mp_rnd_t RND)
     Initialize X and set its value from the string S in base BASE, rounded to direction RND. See mpfr_set_str.

Conversion Functions
====================

 -- Function: double mpfr_get_d2 (mpfr_t OP, mp_rnd_t RND)
     Convert OP to a double, using the rounding mode RND.

 -- Function: double mpfr_get_d (mpfr_t OP)
     Convert OP to a double, using the default MPFR rounding mode (see function mpfr_set_default_rounding_mode).

 -- Function: mp_exp_t mpfr_get_z_exp (mpz_t Z, mpfr_t OP)
     Puts the mantissa of OP into Z, and returns the exponent EXP such that OP equals Z multiplied by two exponent EXP.

 -- Function: char * mpfr_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpfr_t OP, mp_rnd_t RND)
     Convert OP to a string of digits in base BASE, with rounding in direction RND. The base may vary from 2 to 36.  Generate exactly N_DIGITS significant digits.

     If N_DIGITS is 0, it writes the maximum possible number of digits giving an exact rounding in the given base BASE with the direction RND. In other words, if OP was the exact rounding of a real number in direction RND, then the written value is also an exact rounding in base BASE of that real number with the same precision. An error occurs when one is unable to determine the leading digit, which can happen especially if the precision of OP is small.

     If STR is a null pointer, space for the mantissa is allocated using the default allocation function, and a pointer to the string is returned. In that case, the user should her/himself free the corresponding memory with (*_mp_free_func)(s, strlen(s) + 1).

     If STR is not a null pointer, it should point to a block of storage large enough for the mantissa, i.e., N_DIGITS + 2 or more. The extra two bytes are for a possible minus sign, and for the terminating null character.

     If the input number is a real number, the exponent is written through the pointer EXPPTR (the current minimal exponent for 0).

     If N_DIGITS 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_DIGITS is 0.

     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.

     A pointer to the string is returned, unless there is an error, in which case a null pointer is returned.

Basic Arithmetic Functions
==========================

 -- Function: 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)
     Set ROP to OP1 + OP2 rounded in the direction RND. The return value is zero if ROP is exactly OP1 + OP2, positive if ROP is larger than OP1 + OP2, and negative if ROP is smaller than OP1 + OP2.

 -- Function: 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)
     Set ROP to OP1 - OP2 rounded in the direction RND. The return value is zero if ROP is exactly OP1 - OP2, positive if ROP is larger than OP1 - OP2, and negative if ROP is smaller than OP1 - OP2.

 -- Function: 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)
     Set ROP to OP1 times OP2 rounded in the direction RND. Return 0 if the result is exact, a positive value if ROP>OP1*OP2, a negative value otherwise.

Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This gives the user the possibility to handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions.

 -- Function: 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)
     Set ROP to OP1/OP2 rounded in the direction RND. These functions return 0 if the division is exact, a positive value when ROP is larger than OP1 divided by OP2, and a negative value otherwise.

 -- Function: 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)
     Set ROP to the square root of OP rounded in the direction RND. Set ROP to NaN if OP is negative. Return 0 if the operation is exact, a non-zero value otherwise.

 -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned 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)
     Set ROP to OP1 raised to OP2. The computation is done by binary exponentiation. Return 0 if the result is exact, a non-zero value otherwise (but the sign of the return value has no meaning).

 -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mp_rnd_t RND)
     Set ROP to OP1 raised to OP2, rounded to the direction RND with the precision of ROP. Return zero iff the result is exact, a positive value when the result is greater than OP1 to the power OP2, and a negative value when it is smaller.

 -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mp_rnd_t RND)
     Set ROP to OP1 raised to the power OP2, rounded to the direction RND with the precision of ROP. Return zero iff the result is exact.

 -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mp_rnd_t RND)
     Set ROP to OP1 raised to the power OP2, rounded to the direction RND with the precision of ROP. If OP1 is negative then ROP is set to NaN, even if OP2 is an integer. Return zero iff the result is exact.

 -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to -OP rounded in the direction RND. Just changes the sign if ROP and OP are the same variable.

 -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to the absolute value of OP, rounded in the direction RND. Return 0 if the result is exact, a positive value if ROP is larger than the absolute value of OP, and a negative value otherwise.

 -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mp_rnd_t RND)
     Set ROP to OP1 times 2 raised to OP2 rounded to the direction RND. Just increases the exponent by OP2 when ROP and OP1 are identical. Return zero when ROP=OP1, a positive value when ROP>OP1, and a negative value when ROP<OP1.

 -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mp_rnd_t RND)
     Set ROP to OP1 divided by 2 raised to OP2 rounded to the direction RND. Just decreases the exponent by OP2 when ROP and OP1 are identical. Return zero when ROP=OP1, a positive value when ROP>OP1, and a negative value when ROP<OP1.

Comparison Functions
====================

 -- Function: 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)
     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. In case OP1 and OP2 are of same sign but different, the absolute value returned is one plus the absolute difference of their exponents. It is not allowed that one of the operands is NaN (Not-a-Number).

 -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, int E)
 -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, int E)
     Compare OP1 and OP2 multiplied by two to the power E.

 -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int op3)
     Return non-zero if the first OP3 bits of OP1 and OP2 are equal, zero otherwise.  I.e., tests if OP1 and OP2 are approximately equal.

 -- Function: int mpfr_nan_p (mpfr_t OP)
     Return non-zero if OP is Not-a-Number (NaN), zero otherwise.

 -- Function: int mpfr_inf_p (mpfr_t OP)
     Return non-zero if OP is plus or minus infinity, zero otherwise.

 -- Function: int mpfr_number_p (mpfr_t OP)
     Return non-zero if OP is an ordinary number, i.e. neither Not-a-Number nor plus or minus infinity.

 -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mp_rnd_t RND)
     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 abs(OP1-OP2)/OP1, using the rounding mode RND for all operations.

 -- Function: int mpfr_sgn (mpfr_t OP)
     Return a positive value if OP > 0, zero if OP = 0, and a negative value if OP < 0. Its result is not specified when OP is NaN (Not-a-Number).

Special Functions
=================

 -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to the natural logarithm of OP, rounded to the direction RND with the precision of ROP. Return zero when the result is exact (this occurs in fact only when OP is 0, 1, or +infinity) and a non-zero value otherwise (except for rounding to nearest, the sign of the return value is that of ROP-log(OP).

 -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to the exponential of OP, rounded to the direction RND with the precision of ROP. Return zero when the result is exact (this occurs in fact only when OP is -infinity, 0, or +infinity), a positive value when the result is greater than the exponential of OP, and a negative value when it is smaller.

 -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to 2 power of OP, rounded to the direction RND with the precision of ROP. Return zero iff the result is exact (this occurs in fact only when OP is -infinity, 0, or +infinity), a positive value when the result is greater than the exponential of OP, and a negative value when it is smaller.

 -- Function: int mpfr_cos (mpfr_t COP, mpfr_t OP, mp_rnd_t RND)
 -- Function: int mpfr_sin (mpfr_t SOP, mpfr_t OP, mp_rnd_t RND)
 -- Function: int mpfr_tan (mpfr_t TOP, mpfr_t OP, mp_rnd_t RND)
     Set COP to the cosine of OP, SOP to the sine of OP, TOP to the tangent of OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact (this occurs in fact only when OP is 0 i.e. the sine is 0, the cosine is 1, and the tangent is 0).

 -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mp_rnd_t RND)
     Set simultaneously SOP to the sine of OP and COP to the cosine of OP, rounded to the direction RND with their corresponding precisions. Return 0 iff both results are exact.

 -- Function: 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)
     Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact.

 -- Function: int mpfr_cosh (mpfr_t COP, mpfr_t OP, mp_rnd_t RND)
 -- Function: int mpfr_sinh (mpfr_t SOP, mpfr_t OP, mp_rnd_t RND)
 -- Function: int mpfr_tanh (mpfr_t TOP, mpfr_t OP, mp_rnd_t RND)
     Set COP to the hyperbolic cosine of OP, SOP to the hyperbolic sine of OP, TOP to the hyperbolic tangent of OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact (this occurs in fact only when OP is 0 i.e. the result is 1).

 -- Function: 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)
     Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact.

 -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, mp_rnd_t RND)
     Set ROP to the factorial of the unsigned long int OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact.

 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to the logarithm of one plus OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact (this occurs in fact only when OP is 0 i.e. the result is 0).

 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND)
     Set ROP to the exponential of OP minus one, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact (this occurs in fact only when OP is 0 i.e. the result is 0).

 -- 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)
     Set ROP to the log[t] (t=2 or 10)(log x / log t) of OP, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact (this occurs in fact only when OP is 1 i.e. the result is 0).

 -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OPX,mpfr_t OPY,mpfr_t OPZ, mp_rnd_t RND)
     Set ROP to OPX * OPY + OPZ, rounded to the direction RND with the precision of ROP. Return 0 iff the result is exact, a positive value if ROP is larger than OPX * OPY + OPZ, and a negative value otherwise.

 -- Function: void mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mp_rnd_t RND)
     Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded to the direction RND with the precision of ROP.

 -- Function: void mpfr_const_log2 (mpfr_t ROP, mp_rnd_t RND)
     Set ROP to the logarithm of 2 rounded to the direction RND with the precision of ROP. This function stores the computed value to avoid another calculation if a lower or equal precision is requested.

 -- Function: void mpfr_const_pi (mpfr_t ROP, mp_rnd_t RND)
     Set ROP to the value of Pi rounded to the direction RND with the precision of ROP. This function uses the Borwein, Borwein, Plouffe formula which directly gives the expansion of Pi in base 16.

 -- Function: void mpfr_const_euler (mpfr_t ROP, mp_rnd_t RND)
     Set ROP to the value of Euler's constant 0.577... rounded to the direction RND with the precision of ROP.

Input and Output Functions
==========================

Functions that perform input from a standard input/output stream, and functions that output to a standard 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, it is a good idea to include stdio.h before mpfr.h, since that will allow mpfr.h to define prototypes for these functions.

 -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N_DIGITS, mpfr_t OP, mp_rnd_t RND)
     Output OP on stdio stream STREAM, as a string of digits in base BASE, rounded to direction RND. The base may vary from 2 to 36.  Print at most N_DIGITS significant digits, or if N_DIGITS is 0, the maximum number of digits accurately representable by OP.

     In addition to the significant digits, a decimal point at the right of the first digit and a trailing exponent, 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.

 -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, mp_rnd_t RND)
     Input a string in base BASE from stdio 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. M is the mantissa and N is the exponent.  The mantissa is always in the specified base.  The exponent is in decimal.

     The argument BASE may be in the range 2 to 36.

     Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if BASE is 0.  This is so that numbers like 0.23 are not interpreted as octal.

     Return the number of bytes read, or if an error occurred, return 0.

 -- Function: void mpfr_print_binary (mpfr_t FLOAT)
     Output FLOAT on stdout in raw binary format (the exponent is in decimal, yet). The last bits from the least significant limb which do not belong to the mantissa are printed between square brackets; they should always be zero.

Miscellaneous Functions
=======================

 -- Function: 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)
     Set ROP to OP rounded to an integer. mpfr_ceil rounds to the next higher representable integer, mpfr_floor to the next lower, mpfr_round to the nearest representable integer, rounding halfway cases away from zero, and mpfr_trunc to the representable integer towards zero. mpfr_rint behaves like one of these four functions, depending on the rounding mode. 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.

 -- Function: void mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE)
     Generate a uniformly distributed random float in the interval 0 <= X < 1.

 -- Function: void mpfr_random (mpfr_t ROP)
     Generate a uniformly distributed random float in the interval 0 <= X < 1.

 -- Function: void mpfr_random2 (mpfr_t ROP, mp_size_t MAX_SIZE, mp_exp_t MAX_EXP)
     Generate a random float of at most MAX_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 corner-case bugs. Negative random numbers are generated when MAX_SIZE is negative.

Internals
=========

These 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 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. 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. The _mpfr_exp field stores the exponent, and the _mpfr_d is a pointer to the limbs, least significant limbs stored first. The number zero is represented with its most significant limb set to zero, i.e. _mpfr_d[n-1] where n=ceil(_mpfr_prec/BITS_PER_MP_LIMB).

 -- Function: int mpfr_add_one_ulp (mpfr_t X, mp_rnd_t RND)
     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.

 -- Function: int mpfr_sub_one_ulp (mpfr_t X, mp_rnd_t RND)
     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.

 -- Function: int mpfr_can_round (mpfr_t B, mp_exp_t ERR, mp_rnd_t RND1, mp_rnd_t RND2, mp_prec_t PREC)
     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 1 if one is able to round exactly X to precision PREC with direction RND2, and 0 otherwise. This function does not modify its arguments.

Contributors
************

We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjorn 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 this documentation.

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, David Daney contributed the hyperbolic and inverse hyperbolic functions, the base-2 exponential, and the factorial function.

References
**********

  * Torbjorn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 3.1, 2000.

  * 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.