# RISC Forum

When |
Jun 20, 2011
from 01:30 PM to 02:30 PM |
---|---|

Where | Seminar room |

Attendees |
all@risc |

Add event to calendar |
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Low precision methods for the elementary functions ln, exp, cos, sin,

etc. are well established in modern computer hardware. Beyond that,

ultrahigh precision methods for ln(x) etc. are nowadays based on the

AGM-iteration, see e.g. the comprehensive monograph "Pi and the AGM"

by Borwein & Borwein.

This talk presents a new idea for medium precision evaluation of

those functions, within the range of 60-3000 bits, say. Standard

domain reductions like x'= x - n.ln 2 for exp, x'= x.2^n in [1,2)

for ln, and x'= x - n.\pi/2 for cos + i.sin plus Taylor series

approximations are combined with further reductions of similar type

by diophantine combinations of incommensurable logarithms, like

z = x' -(k.ln 3 - m.ln 2) for exp, later on multiplying with 3^k.

Analogous methods apply in the case of the trigonometric functions,

where the role of such extra base 3 is then taken by Gaussian primes

2+i, 2-i, --- plus 3+2i, 3-2i for an improved design of that kind.