The GNU Multiple Precision Floating-Point Reliable Library (GNU MPFR) is a

GNU GNU () is an extensive collection of free software (383 packages as of January 2022), which can be used as an operating system or can be used in parts with other operating systems. The use of the completed GNU tools led to the family of operat ...

portable C

library A library is a collection of materials, books or media that are accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vir ...

for

arbitrary-precision In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...

binary

floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...

computation with correct rounding, based on

GNU Multi-Precision Library GNU Multiple Precision Arithmetic Library (GMP) is a free library for arbitrary-precision arithmetic, operating on signed integers, rational numbers, and floating-point numbers. There are no practical limits to the precision except the ones imp ...

Library

MPFR's computation is both efficient and has a well-defined semantics: the functions are completely specified on all the possible operands and the results do not depend on the platform. This is done by copying the ideas from the ANSI/IEEE-754 standard for fixed-precision floating-point arithmetic (correct rounding and exceptions, in particular). More precisely, its main features are: * Support for special numbers:

signed zero Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...

s (+0 and −0), infinities and

not-a-number Nan or NAN may refer to: Places China * Nan County, Yiyang, Hunan, China * Nan Commandery, historical commandery in Hubei, China Thailand * Nan Province ** Nan, Thailand, the administrative capital of Nan Province * Nan River People Given name ...

(a single NaN is supported: MPFR does not differentiate between quiet NaNs and signaling NaNs). * Each number has its own

precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...

(in bits since MPFR uses

radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...

2). The floating-point results are correctly rounded to the precision of the target variable, in one of the five supported rounding modes (including the four from

IEEE 754-1985 IEEE 754-1985 was an industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the ...

). * Supported functions: MPFR implements all mathematical functions from

C99 C99 (previously known as C9X) is an informal name for ISO/IEC 9899:1999, a past version of the C programming language standard. It extends the previous version ( C90) with new features for the language and the standard library, and helps impl ...

and other usual mathematical functions: the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...

and

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...

in natural base, base 2 and base 10, the log(1+x) and exp(x)−1 functions (log1p and expm1), the six

trigonometric Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

and

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

functions and their inverses, the

gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...

zeta Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label= Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived f ...

and

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...

s, the arithmetic–geometric mean, the

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

(x^y) function. All those functions are correctly rounded over their complete range. *

Subnormal number In computer science, subnormal numbers are the subset of denormalized numbers (sometimes called denormals) that fill the underflow gap around zero in floating-point arithmetic. Any non-zero number with magnitude smaller than the smallest normal n ...

s are not supported, but can be emulated with the mpfr_subnormalize function. MPFR is not able to track the

accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...

of numbers in a whole program or expression; this is not its goal.

Interval arithmetic Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...

packages like Arb, MPFI, or

Real RAM In computing, especially computational geometry, a real RAM (random-access machine) is a mathematical model of a computer that can compute with exact real numbers instead of the binary fixed point or floating point numbers used by most actual com ...

implementations like iRRAM, which may be based on MPFR, can do that for the user. MPFR is dependent upon the

GNU Multiple Precision Arithmetic Library GNU Multiple Precision Arithmetic Library (GMP) is a free library for arbitrary-precision arithmetic, operating on signed integers, rational numbers, and floating-point numbers. There are no practical limits to the precision except the ones imp ...

(GMP). MPFR is needed to build the

GNU Compiler Collection The GNU Compiler Collection (GCC) is an optimizing compiler produced by the GNU Project supporting various programming languages, hardware architectures and operating systems. The Free Software Foundation (FSF) distributes GCC as free software ...

(GCC). Other software uses MPFR, such as

ALGLIB ALGLIB is a cross-platform open source numerical analysis and data processing library. It can be used from several programming languages (C++, C#, VB.NET, Python, Delphi). ALGLIB started in 1999 and has a long history of steady development with r ...

, CGAL,

FLINT Flint, occasionally flintstone, is a sedimentary cryptocrystalline form of the mineral quartz, categorized as the variety of chert that occurs in chalk or marly limestone. Flint was widely used historically to make stone tools and start fir ...

GNOME Calculator GNOME Calculator, formerly known as gcalctool, is the software calculator integrated with the GNOME desktop environment. It is programmed in C and Vala and part of the GNOME Core Applications. Views * ''Basic'' – interface for basic arithmet ...

, the

Julia language Julia is a high-level, dynamic programming language. Its features are well suited for numerical analysis and computational science. Distinctive aspects of Julia's design include a type system with parametric polymorphism in a dynamic programm ...

implementation, the

Magma computer algebra system Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows. Introduction Magma ...

Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...

, GNU MPC, and

GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...

References

External links

Official MPFR web site
{{DEFAULTSORT:Mpfr C (programming language) libraries Computer arithmetic Free software programmed in C GNU Project software Numerical libraries Software using the LGPL license