Efficient implementation of elementary functions in the medium-precision range

TL;DR

This paper presents a highly efficient implementation of elementary transcendental functions for medium-precision arithmetic up to 4096 bits, outperforming existing libraries like MPFR in speed while maintaining rigorous error bounds.

Contribution

The authors introduce a novel implementation combining table-based argument reduction, rectangular splitting, and GMP's mpn layer to significantly accelerate elementary function computations at medium precision.

Findings

Achieves up to 30x speedup over MPFR for atan

Provides rigorous error bounds for all functions

Efficiently handles variable precision up to 4096 bits

Abstract

We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.