This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 2.2.1.
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 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.
This library is free; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.
Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LIB.
MPFR is a portable library written in C for arbitrary precision arithmetic
on 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:
mpfr code is portable, i.e. the result of any operation
does not depend (or should not) on the machine word size
mp_bits_per_limb (32 or 64 on most machines);
mpfr provides the four rounding modes from the IEEE 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.
Everyone should read MPFR Basics. If you need to install the library yourself, you need to read Installing MPFR, too.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
Here are the steps needed to install the library on Unix systems (more details are provided in the INSTALL file):
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).
This will compile MPFR, and create a library archive file libmpfr.a. A dynamic library may be produced too (see configure).
This will make sure MPFR was built correctly. If you get error messages, please report this to `mpfr@loria.fr'. (See Reporting Bugs, for information on what to include in useful bug reports.)
This will copy the files mpfr.h and mpf2mpfr.h to the directory /usr/local/include, the file libmpfr.a to the directory /usr/local/lib, and the file mpfr.info to the directory /usr/local/info (or if you passed the `--prefix' option to configure, using the prefix directory given as argument to `--prefix' instead of /usr/local).
There are some other useful make targets:
Create an info version of the manual, in mpfr.info.
Create a DVI version of the manual, in mpfr.dvi.
Create a Postscript version of the manual, in mpfr.ps.
Delete all object files and archive files, but not the configuration files.
Delete all files not included in the distribution.
Delete all files copied by `make install'.
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/.
The latest version of MPFR is available from http://www.mpfr.org/.
If you think you have found a bug in the MPFR library, first have a look on the MPFR web page http://www.mpfr.org/: perhaps this bug is already known, in which case you may find there a workaround for it. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find.
There are a few things you should think about when you put your bug report together.
You have to send us a test case that makes it possible for us to reproduce the bug. Include instructions on how to run the test case.
You also have to explain what is wrong; if you get a crash, or if the results printed are incorrect and in that case, in what way.
Please include compiler version information in your bug report. This can be extracted using `cc -V' on some machines, or, if you're using gcc, `gcc -v'. Also, include the output from `uname -a' and the MPFR version (the GMP version may be useful too).
If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we won't do anything about it (aside of chiding you to send better bug reports).
Send your bug report to: `mpfr@loria.fr'.
If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.
All declarations needed to use MPFR are collected in the include file mpfr.h. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library:
#include <mpfr.h>
A 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.
There is only one class of functions in the MPFR library:
mpfr_. The associated type is mpfr_t.
As a general rule, all MPFR functions expect output arguments before input arguments. This notation is based on an analogy with the assignment operator.
MPFR allows you to use the same variable for both input and output in the same
expression. For example, the main function for 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.
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
correctly rounded, i.e. MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this variable.
The input variables are regarded as exact (in particular, their precision
does not affect the result).
As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of the target in the rounding to nearest mode, and less than 1 ulp of the target in the directed rounding modes (a ulp is the weight of the least significant represented bit of the target after rounding).
Unless documented otherwise, functions returning an int return
a ternary value.
If the ternary value is zero, it means that the value stored in the
destination variable is the exact result of the corresponding mathematical
function. If the ternary value is positive (resp. negative), it means
the value stored in the destination variable is greater (resp. lower)
than the exact result. For example with the 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.
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.
The value of errno may be set to non-zero by any MPFR function or macro, whether or not there is an error.
An mpfr_t object must be initialized before storing the first value in
it. The functions mpfr_init and mpfr_init2 are used for that
purpose.
Initialize x, set its precision to be exactly prec bits and its value to NaN. (Warning: the corresponding
mpffunctions initialize to zero instead.)Normally, a variable should be initialized once only or at least be cleared, using
mpfr_clear, between initializations. To change the precision of a variable which has already been initialized, usempfr_set_prec. The precision prec must be an integer betweenMPFR_PREC_MINandMPFR_PREC_MAX(otherwise the behavior is undefined).
Free the space occupied by x. Make sure to call this function for all
mpfr_tvariables when you are done with them.
Initialize x and set its value to NaN.
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 tompfr_set_default_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_initwill use this precision, but previously initialized variables are unaffected. This default precision is set to 53 bits initially. The precision can be any integer betweenMPFR_PREC_MINandMPFR_PREC_MAX.
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.
Reset the precision of x to be exactly prec bits, and set its value to NaN. The previous value stored in x is lost. It is equivalent to a call to
mpfr_clear(x)followed by a call tompfr_init2(x, prec), but more efficient as no allocation is done in case the current allocated space for the mantissa of x is enough. The precision prec can be any integer betweenMPFR_PREC_MINandMPFR_PREC_MAX.In case you want to keep the previous value stored in x, use
mpfr_prec_roundinstead.
Return the precision actually used for assignments of x, i.e. the number of bits used to store its mantissa.
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.
Set the value of rop from op, rounded towards the given direction rnd. Note that the input 0 is converted to +0 by
mpfr_set_ui,mpfr_set_si,mpfr_set_sj,mpfr_set_uj,mpfr_set_z,mpfr_set_qandmpfr_set_f, regardless of the rounding mode. If the system doesn't support the IEEE-754 standard,mpfr_set_dandmpfr_set_ldmight not preserve the signed zeros.mpfr_set_qmight not be able to work if the numerator (or the denominator) can not be representable as ampfr_t.Note: If you want to store a floating-point constant to a
mpfr_t, you should usempfr_set_str(or one of the MPFR constant functions, such asmpfr_const_pifor Pi) instead ofmpfr_set_dormpfr_set_ld. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary number before MPFR can work with it.
Set the value of rop from op multiplied by two to the power e, rounded towards the given direction rnd. Note that the input 0 is converted to +0.
Set rop to the value of the whole string s in base base, rounded in the direction rnd. See the documentation of
mpfr_strtofrfor a detailed description of the valid string formats. This function returns 0 if the entire string up to the final null character is a valid number in base base; otherwise it returns −1, and rop may have changed.
Read a floating-point number from a string nptr in base base, rounded in the direction rnd. If successful, the result is stored in rop and
*endptr points to the character just after those parsed. If str doesn't start with a valid number then rop is set to zero and the value of nptr is stored in the location referenced by endptr.Parsing follows the standard C
strtodfunction. This means optional leading whitespace, an optional+or-, mantissa digits with an optional decimal point, and an optional exponent consisting of aneorE(if base <= 10) or@, an optional sign, and digits. The decimal point can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). A hexadecimal mantissa can be given with a leading0xor0X, in which caseporPmay introduce an optional binary exponent, indicating the power of 2 by which the mantissa is to be scaled. A binary mantissa can be given with a leading0bor0B, in which casee,E,p,Por@may introduce the binary exponent. The exponent is always written in base 10.In addition,
infinity,inf(if base <= 10) or@inf@with an optional sign, ornan,nan(n-char-sequence)(if base <= 10),@nan@or@nan@(n-char-sequence)all case insensitive (as Latin letters), can be given. An-char-sequenceis a non-empty string containing only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, b, ..., z, A, B, ..., Z, _).There must be at least one digit in the mantissa for the number to be valid. If an exponent has no digits it's ignored and parsing stops after the mantissa. If an
0x,0X,0bor0Bis not followed by hexadecimal/binary digits, parsing stops after the first0: the subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-white-space character, that is of the expected form. The subject sequence contains no characters if the input string is not of the expected form.Note that in the hex format the exponent
Prepresents a power of 2, whereas@represents a power of the base (i.e. 16).If the argument base is different from 0, it must be in the range 2 to 36. Case is ignored; uppercase and lowercase letters have the same value.
If
baseis 0, then it tries to identify the used base: if the mantissa begins with the0xprefix, it assumes that base is 16. If it begins with0b, it assumes that base is 2. Otherwise, it assumes it is 10.It returns a usual ternary value. If endptr is not a null pointer, a pointer to the character after the last character used in the conversion is stored in the location referenced by endptr.
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 nonnegative.
Swap the values x and y efficiently. Warning: the precisions are exchanged too; in case the precisions are different,
mpfr_swapis thus not equivalent to threempfr_setcalls using a third auxiliary variable.
Initialize rop and set its value from op, rounded in the direction rnd. The precision of rop will be taken from the active default precision, as set by
mpfr_set_default_prec.
Initialize x and set its value from the string s in base base, rounded in the direction rnd. See
mpfr_set_str.
Convert op to a
double(respectivelylong double), using the rounding mode rnd. If the system doesn't support IEEE 754 standard, this function might not preserve the signed zeros.
Return d and set exp such that 0.5<=abs(d)<1 and d times 2 raised to exp equals op rounded to double precision, using the given rounding mode. If op is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and exp is set to 0. If op is NaN or an infinity, then the corresponding double-precision value is returned, and exp is undefined.
Convert op to a
long, anunsigned long, anintmax_tor anuintmax_t(respectively) after rounding it with respect to rnd. If op is NaN, the result is undefined. If op is too big for the return type, it returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow. The flag erange is set too. See alsompfr_fits_slong_p,mpfr_fits_ulong_p,mpfr_fits_intmax_pandmpfr_fits_uintmax_p.
Put the scaled mantissa of op (regarded as an integer, with the precision of op) into rop, and return the exponent exp (which may be outside the current exponent range) such that op exactly equals rop multiplied by two exponent exp. If the exponent is not representable in the
mp_exp_ttype, the behavior is undefined.
Convert op to a
mpz_t, after rounding it with respect to rnd. If op is NaN or Inf, the result is undefined.
Convert op to a
mpf_t, after rounding it with respect to rnd. Return zero iff no error occurred, in particular a non-zero value is returned if op is NaN or Inf, which do not exist inmpf.
Convert op to a string of digits in base b, with rounding in the direction rnd. The base may vary from 2 to 36.
The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number −3.1416 would be returned as "−31416" in the string and 1 written at expptr. If rnd is to nearest, and op is exactly in the middle of two possible outputs, the one with an even last digit is chosen (for an odd base, this may not correspond to an even mantissa).
If n is zero, the number of digits of the mantissa is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of op. More precisely, the chosen precision is 1 + ceil(n*log(2)/log(b)). This is the minimal precision depending on n and b only that satisfies the above property.
If str is a null pointer, space for the mantissa is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use
mpfr_free_str.If str is not a null pointer, it should point to a block of storage large enough for the mantissa, i.e., at least
max(n+ 2, 7). The extra two bytes are for a possible minus sign, and for the terminating null character.If n is 0, note that the space requirements for str in this case will be impossible for the user to predetermine. Therefore, one needs to pass a null pointer for the string argument whenever n is 0.
If the input number is an ordinary number, the exponent is written through the pointer expptr (the current minimal exponent for 0).
A pointer to the string is returned, unless there is an error, in which case a null pointer is returned.
Free a string allocated by
mpfr_get_strusing the current unallocation function (preliminary interface). The block is assumed to bestrlen(str)+1bytes. For more information about how it is done: see Custom Allocation (GNU MP).
Return non-zero if op would fit in the respective C data type, when rounded to an integer in the direction rnd.
Set rop to op1 + op2 rounded in the direction rnd. For types having no signed zero, it is considered unsigned (i.e. (+0) + 0 = (+0) and (-0) + 0 = (-0)).
Set rop to op1 - op2 rounded in the direction rnd. For types having no signed zero, it is considered unsigned (i.e. (+0) − 0 = (+0), (−0) − 0 = (−0), 0 − (+0) = (−0) and 0 − (−0) = (+0)).
Set rop to op1 times op2 rounded in the direction rnd. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive).
Set rop to the square of op rounded in the direction rnd.
Set rop to op1/op2 rounded in the direction rnd. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive).
Set rop to the square root of op rounded in the direction rnd. Return −0 if op is −0 (to be consistent with the IEEE 754-1985 standard). Set rop to NaN if op is negative.
Set rop to the cubic root (resp. the kth root) of op rounded in the direction rnd. An odd (resp. even) root of a negative number (including −Inf) returns a negative number (resp. NaN). The kth root of −0 is defined to be −0, whatever the parity of k.
Set rop to op1 raised to op2, rounded in the direction rnd. Special values are currently handled as described in the ISO C99 standard for the
powfunction (note this may change in future versions):
pow(±0,y)returns plus or minus infinity for y a negative odd integer.pow(±0,y)returns plus infinity for y negative and not an odd integer.pow(±0,y)returns plus or minus zero for y a positive odd integer.pow(±0,y)returns plus zero for y positive and not an odd integer.pow(-1, ±inf)returns 1.pow(+1,y)returns 1 for any y, even a NaN.pow(x,y)returns NaN for finite negative x and finite non-integer y.pow(x, -inf)returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1.pow(x, +inf)returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1.pow(-inf,y)returns minus zero for y a negative odd integer.pow(-inf,y)returns plus zero for y negative and not an odd integer.pow(-inf,y)returns minus infinity for y a positive odd integer.pow(-inf,y)returns plus infinity for y positive and not an odd integer.pow(+inf,y)returns plus zero for y negative, and plus infinity for y positive.
Set rop to -op rounded in the direction rnd. Just changes the sign if rop and op are the same variable.
Set rop to the absolute value of op, rounded in the direction rnd. Just changes the sign if rop and op are the same variable.
Set rop to op1 times 2 raised to op2 rounded in the direction rnd. Just increases the exponent by op2 when rop and op1 are identical.
Set rop to op1 divided by 2 raised to op2 rounded in the direction rnd. Just decreases the exponent by op2 when rop and op1 are identical.
Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. Both op1 and op2 are considered to their full own precision, which may differ. If one of the operands is NaN (Not-a-Number), return zero and set the erange flag.
Note: These functions may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions (e.g.,
mpfr_equal_pfor the equality) described below; they behave like the IEEE-754 comparisons, in particular when one or both arguments are NaN. But only floating-point numbers can be compared (you may need to do a conversion first).
Compare op1 and op2 multiplied by two to the power e. Similar as above.
Compare |op1| and |op2|. Return a positive value if |op1| > |op2|, zero if |op1| = |op2|, and a negative value if |op1| < |op2|. If one of the operands is NaN (Not-a-Number), return zero and set the erange flag.
Return non-zero if op is respectively Not-a-Number (NaN), an infinity, an ordinary number (i.e. neither Not-a-Number nor an infinity) or zero. Return zero otherwise.
Return a positive value if op > 0, zero if op = 0, and a negative value if op < 0. Its result is undefined when op is NaN (Not-a-Number).
Return non-zero if op1 > op2, zero otherwise.
Return non-zero if op1 >= op2, zero otherwise.
Return non-zero if op1 <= op2, zero otherwise.
Return non-zero if op1 < op2 or op1 > op2 (i.e. neither op1, nor op2 is NaN, and op1 <> op2), zero otherwise (i.e. op1 and/or op2 are NaN, or op1 = op2).
Return non-zero if op1 = op2, zero otherwise (i.e. op1 and/or op2 are NaN, or op1 <> op2).
Return non-zero if op1 or op2 is a NaN (i.e. they cannot be compared), zero otherwise.
All those functions, except explicitly stated, return zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise.
Set rop to the natural logarithm of op, log2(op) or log10(op), respectively, rounded in the direction rnd.
Set rop to the exponential of op, to 2 power of op or to 10 power of op, respectively, rounded in the direction rnd.
Set rop to the cosine of op, sine of op, tangent of op, rounded in the direction rnd.
Set rop to the secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set simultaneously sop to the sine of op and cop to the cosine of op, rounded in the direction rnd with the corresponding precisions of sop and cop. Return 0 iff both results are exact.
Set rop to the arc-cosine, arc-sine or arc-tangent of op, rounded in the direction rnd.
Set rop to the arc-tangent2 of y and x, rounded in the direction rnd: if
x > 0,atan2(y, x) = atan (y/x); ifx < 0,atan2(y, x) = sign(y)*(PI - atan (abs(y/x))).
atan2(y, 0)does not raise any floating-point exception. Special values are currently handled as described in the ISO C99 standard for theatan2function (note this may change in future versions):
atan2(+0, -0)returns +Pi.atan2(-0, -0)returns -Pi.atan2(+0, +0)returns +0.atan2(-0, +0)returns −0.atan2(+0, x)returns +Pi for x < 0.atan2(-0, x)returns -Pi for x < 0.atan2(+0, x)returns +0 for x > 0.atan2(-0, x)returns −0 for x > 0.atan2(y, 0)returns -Pi/2 for y < 0.atan2(y, 0)returns +Pi/2 for y > 0.atan2(+INF, -INF)returns +3*Pi/4.atan2(-INF, -INF)returns -3*Pi/4.atan2(+INF, +INF)returns +Pi/4.atan2(-INF, +INF)returns -Pi/4.atan2(+INF, x)returns +Pi/2 for finite x.atan2(-INF, x)returns -Pi/2 for finite x.atan2(y, -INF)returns +Pi for finite y > 0.atan2(y, -INF)returns -Pi for finite y < 0.atan2(y, +INF)returns +0 for finite y > 0.atan2(y, +INF)returns −0 for finite y < 0.
Set rop to the hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.
Set rop to the hyperbolic secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.
Set rop to the inverse hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.
Set rop to the factorial of the
unsigned long intop, rounded in the direction rnd.
Set rop to the logarithm of one plus op, rounded in the direction rnd.
Set rop to the exponential of op minus one, rounded in the direction rnd.
Set y to the exponential integral of x, rounded in the direction rnd. For positive x, the exponential integral is the sum of Euler's constant, of the logarithm of x, and of the sum for k from 1 to infinity of x to the power k, divided by k and factorial(k). For negative x, the returned value is NaN.
Set rop to the value of the Gamma function on op, rounded in the direction rnd. When op is a negative integer, NaN is returned.
Set rop to the value of the logarithm of the Gamma function on op, rounded in the direction rnd. When −2k−1 <= x <= −2k, k being a non-negative integer, NaN is returned. Note that on these negative values, this function is different from the corresponding function defined in some languages, such as
lgammain ISO C99; a new functionmpfr_lgammawill be added in a future MPFR version to match this alternative definition.
Set rop to the value of the Riemann Zeta function on op, rounded in the direction rnd.
Set rop to the value of the error function on op, rounded in the direction rnd.
Set rop to the value of the complementary error function on op, rounded in the direction rnd.
Set rop to op1 times op2 + op3, rounded in the direction rnd.
Set rop to the arithmetic-geometric mean of op1 and op2, rounded in the direction rnd. The arithmetic-geometric mean is the common limit of the sequences u[n] and v[n], where u[0]=op1, v[0]=op2, u[n+1] is the arithmetic mean of u[n] and v[n], and v[n+1] is the geometric mean of u[n] and v[n]. If any operand is negative, the return value is NaN.
Set rop to the Euclidean norm of x and y, i.e. the square root of the sum of the squares of x and y, rounded in the direction rnd. Special values are currently handled as described in Section F.9.4.3 of the ISO C99 standard, for the
hypotfunction (note this may change in future versions): If x or y is an infinity, then plus infinity is returned in rop, even if the other number is NaN.
Set rop to the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the direction rnd. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use
mpfr_free_cache.
Free the cache used by the functions computing constants if needed (currently
mpfr_const_log2,mpfr_const_pi,mpfr_const_eulerandmpfr_const_catalan).
Set ret to the sum of all elements of tab whose size is n, rounded in the direction rnd. Warning, tab is a table of pointers to mpfr_t, not a table of mpfr_t (preliminary interface). The returned
intvalue is zero when the computed value is the exact value, and non-zero when this cannot be guaranteed, without giving the direction of the error as the other functions do.
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.
Output op on stream stream, as a string of digits in base base, rounded in the direction rnd. The base may vary from 2 to 36. Print n significant digits exactly, or if n is 0, enough digits so that op can be read back exactly (see
mpfr_get_str).In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form `eNNN', are printed. If base is greater than 10, `@' will be used instead of `e' as exponent delimiter.
Return the number of bytes written, or if an error occurred, return 0.
Input a string in base base from stream stream, rounded in the direction rnd, and put the read float in rop.
This function reads a word (defined as a sequence of characters between whitespace) and parses it using
mpfr_set_str(it may change). See the documentation ofmpfr_strtofrfor a detailed description of the valid string formats.Return the number of bytes read, or if an error occurred, return 0.
Set rop to op rounded to an integer.
mpfr_rintrounds to the nearest representable integer in the given rounding mode,mpfr_ceilrounds to the next higher or equal representable integer,mpfr_floorto the next lower or equal representable integer,mpfr_roundto the nearest representable integer, rounding halfway cases away from zero, andmpfr_truncto the next representable integer towards zero.The returned value is zero when the result is exact, positive when it is greater than the original value of op, and negative when it is smaller. More precisely, the returned value is 0 when op is an integer representable in rop, 1 or −1 when op is an integer that is not representable in rop, 2 or −2 when op is not an integer.
Note that
mpfr_roundis different frommpfr_rintcalled with the rounding to the nearest mode (where halfway cases are rounded to an even integer or mantissa). Note also that no double rounding is performed; for instance, 4.5 (100.1 in binary) is rounded bympfr_roundto 4 (100 in binary) in 2-bit precision, thoughround(4.5)is equal to 5 and 5 (101 in binary) is rounded to 6 (110 in binary) in 2-bit precision.
Set rop to op rounded to an integer.
mpfr_rint_ceilrounds to the next higher or equal integer,mpfr_rint_floorto the next lower or equal integer,mpfr_rint_roundto the nearest integer, rounding halfway cases away from zero, andmpfr_rint_truncto the next integer towards zero. If the result is not representable, it is rounded in the direction rnd. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function).
Set rop to the fractional part of op, having the same sign as op, rounded in the direction rnd (unlike in
mpfr_rint, rnd affects only how the exact fractional part is rounded, not how the fractional part is generated).
If x or y is NaN, set x to NaN. Otherwise, if x is different from y, replace x by the next floating-point number (with the precision of x and the current exponent range) in the direction of y, if there is one (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated.
Set rop to the minimum of op1 and op2. If op1 and op2 are both NaN, then rop is set to NaN. If op1 or op2 is NaN, then rop is set to the numeric value. If op1 and op2 are zeros of different signs, then rop is set to −0.
Set rop to the maximum of op1 and op2. If op1 and op2 are both NaN, then rop is set to NaN. If op1 or op2 is NaN, then rop is set to the numeric value. If op1 and op2 are zeros of different signs, then rop is set to +0.
Generate a uniformly distributed random float in the interval 0 <= rop < 1. Return 0, unless the exponent is not in the current exponent range, in which case rop is set to NaN and a non-zero value is returned. The second argument is a
gmp_randstate_tstructure which should be created using the GMPgmp_randinitfunction, see the GMP manual.
Generate a uniformly distributed random float in the interval 0 <= rop < 1. This function is deprecated;
mpfr_urandombshould be used instead.
Generate a random float of at most size limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval −exp to exp. This function is useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when size is negative. Put +0 in rop when size if zero.
Get the exponent of x, assuming that x is a non-zero ordinary number. The behavior for NaN, Infinity or Zero is undefined.
Set the exponent of x if e is in the current exponent range, and return 0 (even if x is not a non-zero ordinary number); otherwise, return a non-zero value.
Return the MPFR version, as a null-terminated string.
MPFR_VERSIONis the version of MPFR as a preprocessing constant.MPFR_VERSION_MAJOR,MPFR_VERSION_MINORandMPFR_VERSION_PATCHLEVELare respectively the major, minor and patch level of MPFR version, as preprocessing constants.MPFR_VERSION_STRINGis the version as a string constant, which can be compared to the result ofmpfr_get_versionto check at run time the header file and library used match:if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Error, header and library files do not match\n");
Create an integer in the same format as used by
MPFR_VERSIONfrom the given major, minor and patchlevel. Here is an example of how to check the MPFR version at compile time:#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(2,1,0))) # error "Wrong MPFR version." #endif
Set the default rounding mode to rnd. The default rounding mode is to nearest initially.
Round x according to rnd with precision prec, which must be an integer between
MPFR_PREC_MINandMPFR_PREC_MAX(otherwise the behavior is undefined). If prec is greater or equal to the precision of x, then new space is allocated for the mantissa, and it is filled with zeros. Otherwise, the mantissa is rounded to precision prec with the given direction. In both cases, the precision of x is changed to prec.
[This function is obsolete. Please use
mpfr_prec_roundinstead.]
Return 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.
Return the (current) smallest and largest exponents allowed for a floating-point variable. The smallest positive value of a floating-point variable is one half times 2 raised to the smallest exponent and the largest value has the form (1 - epsilon) times 2 raised to the largest exponent.
Set the smallest and largest exponents allowed for a floating-point variable. Return a non-zero value when exp is not in the range accepted by the implementation (in that case the smallest or largest exponent is not changed), and zero otherwise. If the user changes the exponent range, it is her/his responsibility to check that all current floating-point variables are in the new allowed range (for example using
mpfr_check_range), otherwise the subsequent behavior will be undefined, in the sense of the ISO C standard.
Return the minimum and maximum of the smallest and largest exponents allowed for
mpfr_set_eminandmpfr_set_emax. These values are implementation dependent; it is possible to create a non portable program by writingmpfr_set_emax(mpfr_get_emax_max())andmpfr_set_emin(mpfr_get_emin_min())since the values of the smallest and largest exponents become implementation dependent.
This function forces x to be in the current range of acceptable values, t being the current ternary value: negative if x is smaller than the exact value, positive if x is larger than the exact value and zero if x is exact (before the call). It generates an underflow or an overflow if the exponent of x is outside the current allowed range; the value of t may be used to avoid a double rounding. This function returns zero if the rounded result is equal to the exact one, a positive value if the rounded result is larger than the exact one, a negative value if the rounded result is smaller than the exact one. Note that unlike most functions, the result is compared to the exact one, not the input value x, i.e. the ternary value is propagated.
This function rounds x emulating subnormal number arithmetic: if x is outside the subnormal exponent range, it just propagates the ternary value t; otherwise, it rounds x to precision
EXP(x)-emin+1according to rounding mode rnd and previous ternary value t, avoiding double rounding problems.PREC(x)is not modified by this function. rnd and t must be the used rounding mode for computing x and the returned ternary value when computing x. The subnormal exponent range is fromemintoemin+PREC(x)-1. This functions assumes thatemax-emin >= PREC(x). Note that unlike most functions, the result is compared to the exact one, not the input value x, i.e. the ternary value is propagated. This is a preliminary interface.
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 (1024);
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 correct rounding in the subnormal range, which may not be the case for your hardware.
Clear the underflow, overflow, invalid, inexact and erange flags.
Set the underflow, overflow, invalid, inexact and erange flags.
Clear all global flags (underflow, overflow, inexact, invalid, erange).
Return the corresponding (underflow, overflow, invalid, inexact, erange) flag, which is non-zero iff the flag is set.
All the given interfaces are preliminary. They might change incompatibly in future revisions.
This macro declares name as an automatic variable of type
mpfr_t, initializes it and sets its precision to be exactly prec bits and its value to NaN. name must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than usingmpfr_init2but has some drawbacks:
- You must not call
mpfr_clearwith variables created with this macro (The storage is allocated at the point of declaration and deallocated when the brace-level is exited.).- You can not change their precision.
- You should not create variables with huge precision with this macro.
- Your compiler must support `Non-Constant Initializers' (standard in C++ and ISO C99) and `Token Pasting' (standard in ISO C89). If prec is not a compiler constant, your compiler must support `Variable-length automatic arrays' (standard in ISO C99). `GCC 2.95.3' supports all these features.
Initialize all the
mpfr_tvariables of the givenva_list, set their precision to be the default precision and their value to NaN. Seempfr_initfor more details. Theva_listis assumed to be composed only of typempfr_t. It begins from x. It ends when it encounters a null pointer.
Initialize all the
mpfr_tvariables of the givenva_list, set their precision to be exactly prec bits and their value to NaN. Seempfr_init2for more details. Theva_listis assumed to be composed only of typempfr_t. It begins from x. It ends when it encounters a null pointer.
Free the space occupied by all the
mpfr_tvariables of the givenva_list. Seempfr_clearfor more details. Theva_listis assumed to be composed only of typempfr_t. It begins from x. It ends when it encounters a null pointer.
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);
}
A header file mpf2mpfr.h is included in the distribution of MPFR for
compatibility with the GNU MP class MPF.
After inserting the following two lines after the #include <gmp.h>
line,
#include <mpfr.h> #include <mpf2mpfr.h>any program written for MPF can be compiled directly with MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with
mpfr_set_default_rounding_mode.
Warning: the mpf_init and mpf_init2 functions initialize
to zero, whereas the corresponding mpfr functions initialize to NaN:
this is useful to detect uninitialized values, but is slightly incompatible
with mpf.
Reset the precision of x to be exactly prec bits. The only difference with
mpfr_set_precis that prec is assumed to be small enough so that the mantissa fits into the current allocated memory space for x. Otherwise the behavior is undefined.
Return non-zero if op1 and op2 are both non-zero ordinary numbers with the same exponent and the same first op3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with
mpf, but does not make much sense.
Compute the relative difference between op1 and op2 and store the result in rop. This function does not guarantee the correct rounding on the relative difference; it just computes |op1-op2|/op1, using the rounding mode rnd for all operations and the precision of rop.
See
mpfr_mul_2uiandmpfr_div_2ui. These functions are only kept for compatibility with MPF.
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:
mpfr_t
on the stack.
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.
Return the needed size in bytes to store the mantissa of a FP number of precision prec.
Initialize a mantissa of precision prec. mantissa must be an area of
mpfr_custom_get_size (prec)bytes at least and be suitably aligned for an array ofmp_limb_t.
Perform a dummy initialization of a
mpfr_tand set it to:In all cases, it uses mantissa directly for further computing involving x. It will not allocate anything. A FP number initialized with this function cannot be resized using
- if
ABS(kind) == MPFR_NAN_LIND, x is set to NaN;- if
ABS(kind) == MPFR_INF_KIND, x is set to the infinity of signsign(kind);- if
ABS(kind) == MPFR_ZERO_KIND, x is set to the zero of signsign(kind);- if
ABS(kind) == MPFR_REGULAR_KIND, x is set to a regular number:x = sign(kind)*mantissa*2^expmpfr_set_prec, or cleared usingmpfr_clear! mantissa must have been initialized withmpfr_custom_initusing the same precision prec.
Return the current kind of a
mpfr_tas used bympfr_custom_init_set. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
Return a pointer to the mantissa used by a
mpfr_tinitialized withmpfr_custom_init_set. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
Return the exponent of x, assuming that x is a non-zero ordinary number. The return value for NaN, Infinity or Zero is unspecified but doesn't produced any trap. The behavior of this function for any
mpfr_tnot initialized withmpfr_custom_init_setis undefined.
Inform MPFR that the mantissa has moved due to a garbage collect and update its new position to
new_position. However the application has to move the mantissa and thempfr_titself. The behavior of this function for anympfr_tnot initialized withmpfr_custom_init_setis undefined.
See the test suite for examples.
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.
_mpfr_prec field is used to store the precision of
the variable (in bits); this is not less than MPFR_PREC_MIN.
_mpfr_sign field is used to store the sign of the variable.
_mpfr_exp field stores the exponent.
An exponent of 0 means a radix point just above the most significant
limb. Non-zero values n are a multiplier 2^n relative to that
point.
A NaN, an Infinity and a Zero are indicated by a special value of the exponent.
_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.
Assuming b is an approximation of an unknown number x in the direction rnd1 with error at most two to the power E(b)-err where E(b) is the exponent of b, return a non-zero value if one is able to round correctly x to precision prec with the direction rnd2, and 0 otherwise (including for NaN and Inf). This function does not modify its arguments.
Convert op to a
double, using the default MPFR rounding mode (see functionmpfr_set_default_rounding_mode). This function is obsolete.
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.
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*mpfr_custom_get_mantissa: Custom Interfacemp_prec_t: MPFR Basicsmp_rnd_t: MPFR Basicsmpfr_abs: Basic Arithmetic Functionsmpfr_acos: Special Functionsmpfr_acosh: Special Functionsmpfr_add: Basic Arithmetic Functionsmpfr_add_q: Basic Arithmetic Functionsmpfr_add_si: Basic Arithmetic Functionsmpfr_add_ui: Basic Arithmetic Functionsmpfr_add_z: Basic Arithmetic Functionsmpfr_agm: Special Functionsmpfr_asin: Special Functionsmpfr_asinh: Special Functionsmpfr_atan: Special Functionsmpfr_atan2: Special Functionsmpfr_atanh: Special Functionsmpfr_can_round: Internalsmpfr_cbrt: Basic Arithmetic Functionsmpfr_ceil: Integer Related Functionsmpfr_check_range: Exceptionsmpfr_clear: Initialization Functionsmpfr_clear_erangeflag: Exceptionsmpfr_clear_flags: Exceptionsmpfr_clear_inexflag: Exceptionsmpfr_clear_nanflag: Exceptionsmpfr_clear_overflow: Exceptionsmpfr_clear_underflow: Exceptionsmpfr_clears: Advanced Functionsmpfr_cmp: Comparison Functionsmpfr_cmp_d: Comparison Functionsmpfr_cmp_f: Comparison Functionsmpfr_cmp_ld: Comparison Functionsmpfr_cmp_q: Comparison Functionsmpfr_cmp_si: Comparison Functionsmpfr_cmp_si_2exp: Comparison Functionsmpfr_cmp_ui: Comparison Functionsmpfr_cmp_ui_2exp: Comparison Functionsmpfr_cmp_z: Comparison Functionsmpfr_cmpabs: Comparison Functionsmpfr_const_catalan: Special Functionsmpfr_const_euler: Special Functionsmpfr_const_log2: Special Functionsmpfr_const_pi: Special Functionsmpfr_cos: Special Functionsmpfr_cosh: Special Functionsmpfr_cot: Special Functionsmpfr_coth: Special Functionsmpfr_csc: Special Functionsmpfr_csch: Special Functionsmpfr_custom_get_exp: Custom Interfacempfr_custom_get_kind: Custom Interfacempfr_custom_get_size: Custom Interfacempfr_custom_init: Custom Interfacempfr_custom_init_set: Custom Interfacempfr_custom_move: Custom InterfaceMPFR_DECL_INIT: Advanced Functionsmpfr_div: Basic Arithmetic Functionsmpfr_div_2exp: Compatibility with MPFmpfr_div_2si: Basic Arithmetic Functionsmpfr_div_2ui: Basic Arithmetic Functionsmpfr_div_q: Basic Arithmetic Functionsmpfr_div_si: Basic Arithmetic Functionsmpfr_div_ui: Basic Arithmetic Functionsmpfr_div_z: Basic Arithmetic Functionsmpfr_eint: Special Functionsmpfr_eq: Compatibility with MPFmpfr_equal_p: Comparison Functionsmpfr_erangeflag_p: Exceptionsmpfr_erf: Special Functionsmpfr_erfc: Special Functionsmpfr_exp: Special Functionsmpfr_exp10: Special Functionsmpfr_exp2: Special Functionsmpfr_expm1: Special Functionsmpfr_fac_ui: Special Functionsmpfr_fits_intmax_p: Conversion Functionsmpfr_fits_sint_p: Conversion Functionsmpfr_fits_slong_p: Conversion Functionsmpfr_fits_sshort_p: Conversion Functionsmpfr_fits_uint_p: Conversion Functionsmpfr_fits_uintmax_p: Conversion Functionsmpfr_fits_ulong_p: Conversion Functionsmpfr_fits_ushort_p: Conversion Functionsmpfr_floor: Integer Related Functionsmpfr_fma: Special Functionsmpfr_frac: Integer Related Functionsmpfr_free_cache: Special Functionsmpfr_free_str: Conversion Functionsmpfr_gamma: Special Functionsmpfr_get_d: Conversion Functionsmpfr_get_d1: Internalsmpfr_get_d_2exp: Conversion Functionsmpfr_get_default_prec: Initialization Functionsmpfr_get_default_rounding_mode: Rounding Modesmpfr_get_emax: Exceptionsmpfr_get_emax_max: Exceptionsmpfr_get_emax_min: Exceptionsmpfr_get_emin: Exceptionsmpfr_get_emin_max: Exceptionsmpfr_get_emin_min: Exceptionsmpfr_get_exp: Miscellaneous Functionsmpfr_get_f: Conversion Functionsmpfr_get_ld: Conversion Functionsmpfr_get_prec: Initialization Functionsmpfr_get_si: Conversion Functionsmpfr_get_sj: Conversion Functionsmpfr_get_str: Conversion Functionsmpfr_get_ui: Conversion Functionsmpfr_get_uj: Conversion Functionsmpfr_get_version: Miscellaneous Functionsmpfr_get_z: Conversion Functionsmpfr_get_z_exp: Conversion Functionsmpfr_greater_p: Comparison Functionsmpfr_greaterequal_p: Comparison Functionsmpfr_hypot: Special Functionsmpfr_inexflag_p: Exceptionsmpfr_inf_p: Comparison Functionsmpfr_init: Initialization Functionsmpfr_init2: Initialization Functionsmpfr_init_set: Combined Initialization and Assignment Functionsmpfr_init_set_d: Combined Initialization and Assignment Functionsmpfr_init_set_f: Combined Initialization and Assignment Functionsmpfr_init_set_ld: Combined Initialization and Assignment Functionsmpfr_init_set_q: Combined Initialization and Assignment Functionsmpfr_init_set_si: Combined Initialization and Assignment Functionsmpfr_init_set_str: Combined Initialization and Assignment Functionsmpfr_init_set_ui: Combined Initialization and Assignment Functionsmpfr_init_set_z: Combined Initialization and Assignment Functionsmpfr_inits: Advanced Functionsmpfr_inits2: Advanced Functionsmpfr_inp_str: Input and Output Functionsmpfr_integer_p: Integer Related Functionsmpfr_less_p: Comparison Functionsmpfr_lessequal_p: Comparison Functionsmpfr_lessgreater_p: Comparison Functionsmpfr_lngamma: Special Functionsmpfr_log: Special Functionsmpfr_log10: Special Functionsmpfr_log1p: Special Functionsmpfr_log2: Special Functionsmpfr_max: Miscellaneous Functionsmpfr_min: Miscellaneous Functionsmpfr_mul: Basic Arithmetic Functionsmpfr_mul_2exp: Compatibility with MPFmpfr_mul_2si: Basic Arithmetic Functionsmpfr_mul_2ui: Basic Arithmetic Functionsmpfr_mul_q: Basic Arithmetic Functionsmpfr_mul_si: Basic Arithmetic Functionsmpfr_mul_ui: Basic Arithmetic Functionsmpfr_mul_z: Basic Arithmetic Functionsmpfr_nan_p: Comparison Functionsmpfr_nanflag_p: Exceptionsmpfr_neg: Basic Arithmetic Functionsmpfr_nextabove: Miscellaneous Functionsmpfr_nextbelow: Miscellaneous Functionsmpfr_nexttoward: Miscellaneous Functionsmpfr_number_p: Comparison Functionsmpfr_out_str: Input and Output Functionsmpfr_overflow_p: Exceptionsmpfr_pow: Basic Arithmetic Functionsmpfr_pow_si: Basic Arithmetic Functionsmpfr_pow_ui: Basic Arithmetic Functionsmpfr_pow_z: Basic Arithmetic Functionsmpfr_prec_round: Rounding Modesmpfr_print_rnd_mode: Rounding Modesmpfr_random: Miscellaneous Functionsmpfr_random2: Miscellaneous Functionsmpfr_reldiff: Compatibility with MPFmpfr_rint: Integer Related Functionsmpfr_rint_ceil: Integer Related Functionsmpfr_rint_floor: Integer Related Functionsmpfr_rint_round: Integer Related Functionsmpfr_rint_trunc: Integer Related Functionsmpfr_root: Basic Arithmetic Functionsmpfr_round: Integer Related Functionsmpfr_round_prec: Rounding Modesmpfr_sec: Special Functionsmpfr_sech: Special Functionsmpfr_set: Assignment Functionsmpfr_set_d: Assignment Functionsmpfr_set_default_prec: Initialization Functionsmpfr_set_default_rounding_mode: Rounding Modesmpfr_set_emax: Exceptionsmpfr_set_emin: Exceptionsmpfr_set_erangeflag: Exceptionsmpfr_set_exp: Miscellaneous Functionsmpfr_set_f: Assignment Functionsmpfr_set_inexflag: Exceptionsmpfr_set_inf: Assignment Functionsmpfr_set_ld: Assignment Functionsmpfr_set_nan: Assignment Functionsmpfr_set_nanflag: Exceptionsmpfr_set_overflow: Exceptionsmpfr_set_prec: Initialization Functionsmpfr_set_prec_raw: Compatibility with MPFmpfr_set_q: Assignment Functionsmpfr_set_si: Assignment Functionsmpfr_set_si_2exp: Assignment Functionsmpfr_set_sj: Assignment Functionsmpfr_set_sj_2exp: Assignment Functionsmpfr_set_str: Assignment Functionsmpfr_set_ui: Assignment Functionsmpfr_set_ui_2exp: Assignment Functionsmpfr_set_uj: Assignment Functionsmpfr_set_uj_2exp: Assignment Functionsmpfr_set_underflow: Exceptionsmpfr_set_z: Assignment Functionsmpfr_sgn: Comparison Functionsmpfr_si_div: Basic Arithmetic Functionsmpfr_si_sub: Basic Arithmetic Functionsmpfr_sin: Special Functionsmpfr_sin_cos: Special Functionsmpfr_sinh: Special Functionsmpfr_sqr: Basic Arithmetic Functionsmpfr_sqrt: Basic Arithmetic Functionsmpfr_sqrt_ui: Basic Arithmetic Functionsmpfr_strtofr: Assignment Functionsmpfr_sub: Basic Arithmetic Functionsmpfr_sub_q: Basic Arithmetic Functionsmpfr_sub_si: Basic Arithmetic Functionsmpfr_sub_ui: Basic Arithmetic Functionsmpfr_sub_z: Basic Arithmetic Functionsmpfr_subnormalize: Exceptionsmpfr_sum: Special Functionsmpfr_swap: Assignment Functionsmpfr_t: MPFR Basicsmpfr_tan: Special Functionsmpfr_tanh: Special Functionsmpfr_trunc: Integer Related Functionsmpfr_ui_div: Basic Arithmetic Functionsmpfr_ui_pow: Basic Arithmetic Functionsmpfr_ui_pow_ui: Basic Arithmetic Functionsmpfr_ui_sub: Basic Arithmetic Functionsmpfr_underflow_p: Exceptionsmpfr_unordered_p: Comparison Functionsmpfr_urandomb: Miscellaneous FunctionsMPFR_VERSION: Miscellaneous FunctionsMPFR_VERSION_MAJOR: Miscellaneous FunctionsMPFR_VERSION_MINOR: Miscellaneous FunctionsMPFR_VERSION_NUM: Miscellaneous FunctionsMPFR_VERSION_PATCHLEVEL: Miscellaneous FunctionsMPFR_VERSION_STRING: Miscellaneous Functionsmpfr_zero_p: Comparison Functionsmpfr_zeta: Special Functions