Computing elementary functions using multi-prime argument reduction

TL;DR

This paper introduces a fast, high-precision algorithm for elementary functions using multi-prime argument reduction, leveraging Diophantine combinations and optimized formulas for improved computational efficiency.

Contribution

It presents a novel multi-prime argument reduction method with a fast integer relation algorithm, achieving roughly twice the speed of previous methods at very high precision.

Findings

Achieves approximately twofold speedup over previous algorithms

Supports computations from thousands to millions of bits of precision

Provides optimized Machin-like formulas for key precomputations

Abstract

We describe an algorithm for arbitrary-precision computation of the elementary functions (exp, log, sin, atan, etc.) which, after a cheap precomputation, gives roughly a factor-two speedup over previous state-of-the-art algorithms at precision from a few thousand bits up to millions of bits. Following an idea of Sch{\"o}nhage, we perform argument reduction using Diophantine combinations of logarithms of primes; our contribution is to use a large set of primes instead of a single pair, aided by a fast algorithm to solve the associated integer relation problem. We also list new, optimized Machin-like formulas for the necessary logarithm and arctangent precomputations.