Fast multiple-precision evaluation of elementary functions (1976)
| Venue: | Journal of the ACM |
| Citations: | 73 - 5 self |
BibTeX
@ARTICLE{Brent76fastmultiple-precision,
author = {Richard P. Brent},
title = {Fast multiple-precision evaluation of elementary functions},
journal = {Journal of the ACM},
year = {1976},
volume = {23},
pages = {242--251}
}
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Abstract
XI3STnXC'r. Let f(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 reqmred to multiply n-bit integers. It is shown that f(x) can be evaluated, with relative error 0(2-'), m O(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 Sehonbage-Strassen bound on M(n), it follows that an n-bit approximation to f(x) may be evaluated in O(n logS(n) log log(n)) operations. Special cases include the evaluation of constants such as f, e, and e'. The algorithms depend on the theory of elhptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations. Itsr wol~os Ar~o en~As~s: multiple-precision arithmetic, analytic complexity, arithmetic-geometric mean, computational complexity, elementary function, elliptic integral, evaluation of x, exponentml, Landen transformation, logarithm, trigonometric funetmn CR CATEGORIES: 5.12, 5.15, 5.25 1.
