Fast Multiple-Precision Evaluation of Elementary Functions

Abstract

Let ƒ(x) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M(n) be the number of single-precision operations required to multiply n-bit integers. It is shown that ƒ(x) can be evaluated, with relative error Ο(2-n), in Ο(M(n)log (n)) operations as n → ∞, for any floating-point number x (with an n-bit fraction) in a suitable finite interval. From the Schönhage-Strassen bound on M(n), it follows that an n-bit approximation to ƒ(x) may be evaluated in Ο(n log2(n) log log(n)) operations. Special cases include the evaluation of constants such as π, e, and eπ. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations. © 1976, ACM. All rights reserved.