# Fast Algorithms for High-Precision Computation of Elementary Functions

@inproceedings{Brent2006FastAF, title={Fast Algorithms for High-Precision Computation of Elementary Functions}, author={Richard P. Brent}, year={2006} }

Open problem: Is γ or exp(γ) rational ? Since the regular continued fraction gives best rational approximations, continued fraction computations can give theorems of the form: If x is rational, say x = p/q, then |q| > B for some (very large) bound B. To obtain a result like this with given bound B, we need to compute x with absolute error O(1/B2). Using this method we know that, if γ or exp(γ) is rational, say p/q, then |q| must be very large. Thus, in some sense these numbers are “unlikely” to…

## Topics from this paper

## 9 Citations

Modern Computer Arithmetic

- Computer Science
- 2010

Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics…

Modern Computer Arithmetic (version 0.5.1)

- Computer Science, MathematicsArXiv
- 2010

This book focuses on algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions.

Matters Computational: Ideas, Algorithms, Source Code

- Computer Science
- 2010

This book provides algorithms and ideas for computationalists on low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields.

Sampling from Arbitrary Centered Discrete Gaussians for Lattice-Based Cryptography

- Computer ScienceACNS
- 2017

This work study how to reduce the number of centers for which the authors have to precompute tables and propose a non-centered CDT algorithm with practicable size of precomputed tables as fast as its centered variant.

Efficient evaluation methods of elementary functions suitable for SIMD computation

- Computer ScienceComputer Science - Research and Development
- 2010

Efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches.

978-0-521-19469-3-Modern Computer Arithmetic

- Computer Science
- 2008

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, for teaching and research institutions in France or abroad.

Time-Bounded Reachability for Initialized Hybrid Automata with Linear Differential Inclusions and Rectangular Constraints

- Mathematics, Computer ScienceFORMATS
- 2014

The main result is that while the control state reachability problem for such automata is undecidable, the time-bounded reachable problem is decidable.

An ECMA-55 Minimal BASIC Compiler for x86-64 Linux

- Computer ScienceComput.
- 2014

This compiler is designed to be small, simple, and easy to understand for people who want to study a compiler that actually implements full error checking on floating point on x86-64 CPUs even if those people have little programming experience.

Contents / Preface

- Journal of Inherited Metabolic Disease
- 2004

grouping The abstracts have been grouped according to their subject matter. This has not always been easy, since there is often a degree of overlap. If authors feel they have been misallocated,…

## References

SHOWING 1-10 OF 19 REFERENCES

REMOVING REDUNDANCY IN HIGH-PRECISION NEWTON ITERATION

- 2004

This paper presents new algorithms for several high-precision operations in the power series ring C[[x]]. Compared to computing n coefficients of a product in C[[x]], computing n coefficients of a…

Fast Algorithms for Manipulating Formal Power Series

- Mathematics, Computer ScienceJACM
- 1978

This paper shows that the composition and reversion problems are equivalent (up to constant factors), and gives algorithms which require only order (n log n) ~/2 operations in many cases of practical importance.

On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials

- Mathematics, Computer ScienceSIAM J. Comput.
- 1973

Algorithms which use only $O(\sqrt n )$ nonscalar multiplications to evaluate polynomials of degree n, and proofs that at least $\sqrt ...

Fast Multiple-Precision Evaluation of Elementary Functions

- Mathematics, Computer ScienceJACM
- 1976

It is shown that ƒ(x) can be evaluated, with relative error, in the Schönhage-Strassen bound on M(n), the number of single-precision operations required to multiply n-bit integers.

Efficient multiple-precision evaluation of elementary functions

- Mathematics
- 1989

Let M(t) denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in O(tI/3M(t)) time.

High-precision floating-point arithmetic in scientific computation

- Computer ScienceComputing in Science & Engineering
- 2005

Software packages have yielded interesting scientific results that suggest numeric precision in scientific computations could be as important to program design as algorithms and data structures.

Modern Computer Arithmetic, book in preparation

- Modern Computer Arithmetic, book in preparation
- 2006

English translation in Problems of Information Transmission

- English translation in Problems of Information Transmission
- 1991

Fast evaluation of transcendental functions (in Russian)

- Probl. Peredachi Inf. 27, 4 (1991), 87–110. English translation in Problems of Information Transmission 27
- 1991

Monographies etet´etÉtudes de la Société Mathématique du Canada

- Monographies etet´etÉtudes de la Société Mathématique du Canada
- 1987