Affiliation:
1. Computer Centre, Australian National University, Box 4, Canberra, ACT 2600, Australia
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
Ο
(2
-
n
), in
Ο
(
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
Ο
(
n
log
2
(
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.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
253 articles.
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