Fast Multiple-Precision Evaluation of Elementary Functions

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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 &Ogr;(2-n), in &Ogr;(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 &Ogr;(n log2(n) log log(n)) operations. Special cases include the evaluation of constants such as &pgr;, e, and e&pgr;. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.

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