Fast Compensated Algorithms for the Reciprocal Square Root, the Reciprocal Hypotenuse, and Givens Rotations

TL;DR

This paper introduces simple differential compensation algorithms utilizing FMA to enhance the accuracy of reciprocal square root, reciprocal hypotenuse, and Givens rotations, combining speed and precision in modern computing environments.

Contribution

The paper presents a novel, simple compensation method using FMA to improve accuracy of reciprocal square root and related computations, extending to Givens rotations.

Findings

Enhanced accuracy with simple compensation techniques

Effective combination of fast approximate methods with FMA-based correction

Applicable to Givens rotations in numerical algorithms

Abstract

The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example \cite{863046,863031} and the references therein). In this paper we develop a simple differential compensation (much like those developed in \cite{borges}) that can be used to improve the accuracy of a naive calculation. The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then demonstrate how to combine this approach with a somewhat inaccurate but fast square root free method for estimating the reciprocal square root to get a method that is both fast (in computing environments with a slow square root) and, experimentally, highly accurate. Finally, we show how this same approach can be extended to the reciprocal hypotenuse calculation and, most importantly, to the construction of Givens rotations.