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Pi: A Source Book pp 424–433Cite as

Fast Multiple-Precision Evaluation of Elementary Functions

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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 → x 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 f(x) may be evaluated in O(n log2(n) log log(n)) operations. Special cases include the evaluation of constants such as π, e, and e π. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.

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Author notes

  1. Richard P. Brent

    Present address: Computer Centre, Australian National University, Box 4, 2600, Canberra, ACT, Australia

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  1. Australian National University, Canberra, Australia

    Richard P. Brent

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  1. Richard P. Brent
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© 1997 Springer Science+Business Media New York

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Brent, R.P. (1997). Fast Multiple-Precision Evaluation of Elementary Functions. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2736-4_47

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  • DOI: https://doi.org/10.1007/978-1-4757-2736-4_47

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-2738-8

  • Online ISBN: 978-1-4757-2736-4

  • eBook Packages: Springer Book Archive

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