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

GNU GNU ( ) is an extensive collection of free software (394 packages ), 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 operating systems popu ...

portable C

library A library is a collection of Book, books, and possibly other Document, materials and Media (communication), media, that is accessible for use by its members and members of allied institutions. Libraries provide physical (hard copies) or electron ...

for arbitrary-precision binary

floating-point In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...

computation with correct rounding, based on GNU Multi-Precision Library.

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 equivalent. 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 n ...

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

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

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 is a historic 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, ...

). * Supported functions: MPFR implements all mathematical functions from C99 and other usual mathematical functions: the

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

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 * Ex ...

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 concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field ...

and

hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

functions and their inverses, the

gamma Gamma (; uppercase , lowercase ; ) 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 normally repr ...

zeta Zeta (, ; uppercase Ζ, lowercase ζ; , , 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 from the Phoenician alphabet, Phoenician letter zay ...

and

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

s, the

arithmetic–geometric mean In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms f ...

, the power (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 arithmetic underflow, underflow gap around zero in floating-point arithmetic. Any non-zero number with magnitude smaller than ...

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''. ''Precision'' is how close the measurements are to each other. The ...

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 mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numeri ...

packages like Arb, MPFI, or Real RAM 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 software, free library for arbitrary-precision arithmetic, operating on Sign (mathematics), signed integers, Rational data type, rational numbers, and Floating-point arithmetic, floating-p ...

(GMP). MPFR is needed to build the

GNU Compiler Collection The GNU Compiler Collection (GCC) is a collection of compilers from the GNU Project that support various programming languages, Computer architecture, hardware architectures, and operating systems. The Free Software Foundation (FSF) distributes ...

(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, Java). ALGLIB started in 1999 and has a long history of steady developm ...

, 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. Historically, flint was widely used to make stone tools and start ...

, GNOME Calculator, the Julia language 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 soapberry family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated si ...

, GNU MPC, and

GNU Octave GNU Octave is a scientific programming language for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly ...

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 GNU Lesser General Public License