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Abstract "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 required to multiply n-bit integers. It is shown that f(x) can be evaluated, with relative error O(2^(-n)) in O(M(n)log(n)) operations as n tends to infinity, for any floating-point number x (with an n-bit fraction) in a suitable finite interval. From the Schonhage-Strassen bound on M(n), it follows that an n-bit approximation to f(x) may be evaluated in O(n log^2(n) log log (n)) operations. Special cases include the evaluation of constants such as pi, e and e^pi. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations."

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