     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, PolKA team, INRIA Lorraine and LORIA, France

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 1.0

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.LIB.

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 positive value for each variable (including one bit of 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 Library General Public License. It is permitted to link MPFR to non-free programs, as long as MPFR source code is provided when distributing the non-free program.

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 2.0.2 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 should work. 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 Slowaris 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 libmpfr.a, as well as the info files, to /usr/local (or if you passed the --prefix option to configure, to the directory given as argument to --prefix).

There are some other useful make targets:

  * doc

     Create a DVI version of the manual, in mpfr.dvi and a set of info files, in mpfr.info, mpfr.info-1, mpfr.info-2, etc.

  * ps

     Create a Postscript version of the manual, in mpfr.ps.

  * html

     Create a HTML version of the manual, in mpfr.html.

  * 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.

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.

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 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.

Getting the Latest Version of MP
================================

The latest version of the MPFR library is available from the web page of the PolKA team at http://www.loria.fr/equipes/polka.

Reporting Bugs
**************

If you think you have found a bug in the MPFR library, please investigate it 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.

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.

Global Variables and Rounding Modes
===================================

 -- Global Variable: mp_size_t __gmp_default_fp_bit_precision
     The default precision in bits (53 initially).

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

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 reprensentable 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 phenomena mentioned by Knuth in volume 2 of The Art of Computer Programming (section 4.2.2, pages 221-222).

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

 -- Function: void mpfr_round (mpfr_t X, char RND, unsigned long PREC)
     Round 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.

 -- Function: void mpfr_set_machine_rnd_mode (char RND)
     Set the machine rounding mode to RND. This function is useful for debugging purpose, but also as a common interface to all different ways of setting the rounding mode, which unfortunately differ from one operating system to another one.

 -- Function: char * mpfr_print_rnd_mode (unsigned char RND)
     Returns the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) corresponding to the rounding mode RND.

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

 -- Function: void mpfr_set_default_prec (unsigned long int 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 mpf_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 positive integer, even a precision of 1 is possible.

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. No special value is set. 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, unsigned long int PREC)
     Initialize X and set its precision to be exactly PREC bits.  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.

 -- 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: void mpfr_set_prec (mpfr_t X, unsigned long int 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.

     In case you want to keep the previous value stored in X, use mpfr_round instead.

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

Assignment Functions
--------------------

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

 -- Function: void mpfr_set (mpfr_t ROP, mpfr_t OP, char RND)
 -- Function: void mpfr_set_ui (mpfr_t ROP, unsigned long int OP, char RND)
 -- Function: void mpfr_set_si (mpfr_t ROP, long int OP, char RND
 -- Function: void mpfr_set_d (mpfr_t ROP, double OP, char RND)
 -- Function: void mpfr_set_z (mpfr_t ROP, mpz_t OP, char 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.

 -- 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.

 -- Function: void mpfr_set_f (mpfr_t X, mpf_t Y, char RND)
     Set X to the GNU MP floating-point number Y, rounded with the RND mode and the precision of X.

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

 -- Function: double mpfr_get_d (mpfr_t OP)
     Convert OP to a double, using the current machine rounding mode.

 -- Function: char * mpfr_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpfr_t OP, char 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 prints 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 printed 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 NULL, space for the mantissa is allocated using the default allocation function, and a pointer to the string is returned.

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

     The exponent is written through the pointer EXPPTR.

     If N_DIGITS is 0, note that the space requirements for STR in this case will be impossible for the user to predetermine.  Therefore, you need to pass NULL 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.

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

 -- Function: void mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, char RND)
     Set ROP to OP1 + OP2 rounded in the direction RND.

 -- Function: void mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, char RND)
     Set ROP to OP1 - OP2 rounded in the direction RND.

 -- Function: void mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, char RND)
 -- Function: void mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, char RND)
     Set ROP to OP1 times OP2 rounded in the direction RND.

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: void mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, char RND)
 -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, char RND)
     Set ROP to OP1/OP2 rounded in the direction RND.

 -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, char 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: void mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, char RND)
 -- Function: void mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, unsigned long int OP2, char RND)
     Set ROP to OP1 raised to OP2. The computation is done by binary exponentiation, each multiplication being rounded to direction RND. Hence the relative error with respect to the exact result may be larger than one unit in last place (ulp). In other words, this function does not provide an exact rounding.

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

 -- Function: void mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, char 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.

 -- Function: void mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2)
     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.

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.

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

 -- Macro: int SIGN (mpfr_t OP)
     Return +1 if OP > 0, and -1 if OP < 0. This function is actually implemented as a macro.  It evaluates its arguments multiple times. As in the IEEE 754 standard, there is a positive zero and a negative one, therefore the SIGN macro does not distinguish zero from non-zero numbers.

 -- Macro: int NOTZERO (mpfr_t OP)
     Returns zero when its argument is zero, and a non-zero value otherwise.

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

 -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, char RND)
     Set ROP to the natural logarithm 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_exp (mpfr_t ROP, mpfr_t OP, char RND)
     Set ROP to the exponential 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: void mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, char 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_log2 (mpfr_t ROP, unsigned char 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_pi (mpfr_t ROP, unsigned char 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.

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

Functions that perform input from a stdio stream, and functions that output to a stdio 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, char 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: void mpfr_print_raw (mpfr_t FLOAT)
     Output FLOAT on stdout (should be changed to allow any io stream), 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: void mpfr_random (mpfr_t ROP)
     Put in ROP a random float in the interval [0,1[. Random limbs are generated using the random system function. 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.

 -- Function: void mpfr_srandom (unsigned long SEED)
     Set the random seed used by mpfr_random to SEED.

Internal Functions
==================

These functions were mainly designed for the implementation of mpfr, but may be useful for users too.

 -- Function: Macro int FLAG_NAN (mpfr_t x)
     Returns a non-zero value iff its argument is Not a Number.

 -- Function: Macro int SET_NAN (mpfr_t x)
     Sets its argument to Not a Number.

 -- Function: Macro int ABSSIZE (mpfr_t x)
     Returns the number of limb the mantissa of X points to.

 -- Function: Macro int EXP (mpfr_t x)
     Returns the exponent of X.

 -- Function: Macro mp_limb_t* MANT (mpfr_t x)
     Returns a pointer to the mantissa of X.

 -- Function: Macro unsigned int PREC (mpfr_t x)
     Returns the precision of X.

 -- Function: Macro int CHANGE_SIGN (mpfr_t x)
     Changes the sign of X.

 -- Function: int mpfr_add_one_ulp (mpfr_t X)
     Add one unit in last place (ulp) to the mantissa of X if it is positive or zero, and subtracts one ulp otherwise. Always return 0 (result is exact).

 -- Function: int mpfr_sub_one_ulp (mpfr_t X)
     Subtract one ulp to X if it is positive or zero, and adds one ulp otherwise. Always return 0 (result is exact).

 -- Function: int mpfr_cmp2 (mpfr_t A, mpfr_t B)
     Assuming A is greater or equal to B, returns the number of cancelled bits when one subtracts B from A. Returns the precision of A when both operands are equal. Mainly used in mpfr_sub.

 -- Function: int mpfr_round_raw (mp_limb_t* Y, mp_limb_t* X, long XPREC, char NEG, long YPREC, char RND)
     Puts in mantissa beginning at Y the value of that beginning at X (with precision XPREC bits and negative iff NEG is not zero) rounded with mode RND to precision YPREC.

 -- Function: int mpfr_round_raw2 (mp_limb_t* X, long XN, char NEG, char RND, unsigned long PREC)
     Returns 0 if the multiple precision number stored in the XN limbs starting at XP (least significant limbs first) with sign NEG (positive iff NEG equals zero) is rounded similarly with direction RND and precision PREC than towards zero, and 1 otherwise. In other words, the bit returned is what has to be added to bit PREC to round to mode RND. This function does not modify the mantissa stored in XP.

 -- Function: int mpfr_can_round (mpfr_t B, unsigned long ERR, char RND1, char RND2, unsigned long 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 for his help about design issues and his suggestions for a easy integration into GNU MP.

Sylvie Boldo from ENS-Lyon, France, contributed the function mpfr_agm during her stage in the PolKA team in June and July 1999.

References
**********

  * Torbjorn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 2.0.2, 1996.

  * 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.