Fast Multiple-Precision Evaluation of Elementary Functions

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