### Fast Multiple-Precision Evaluation of Elementary FunctionsFast Multiple-Precision Evaluation of Elementary Functions

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 Author Brent, Richard P. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1976 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract Let $ƒ(\textit{x})$ be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let $\textit{M}(\textit{n})$ be the number of single-precision operations required to multiply $\textit{n}-bit$ integers. It is shown that $ƒ(\textit{x})$ can be evaluated, with relative error $\textit{&Ogr;}(2-\textit{n}),$ in $\textit{&Ogr;}(\textit{M}(\textit{n})log$ $(\textit{n}))$ operations as $\textit{n}$ → ∞, for any floating-point number $\textit{x}$ (with an $\textit{n}-bit$ fraction) in a suitable finite interval. From the Schönhage-Strassen bound on $\textit{M}(\textit{n}),$ it follows that an $\textit{n}-bit$ approximation to $ƒ(\textit{x})$ may be evaluated in $\textit{&Ogr;}(\textit{n}$ $log2(\textit{n})$ log $log(\textit{n}))$ operations. Special cases include the evaluation of constants such as π, $\textit{e},$ and $\textit{e}π.$ The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1976-04-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 23 Issue Number 2 Page Count 10 Starting Page 242 Ending Page 251

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Source: ACM Digital Library