A Correctly Rounded Newton Step for the Reciprocal Square Root

TL;DR

This paper introduces a correctly rounded Newton step for reciprocal square root calculations using fused multiply-add operations, ensuring high accuracy and efficiency, especially on hardware with slow square root computations.

Contribution

It develops a correctly rounded Newton step leveraging FMA and introduces weak rounding, improving reciprocal square root accuracy and efficiency over naive methods.

Findings

The proposed method achieves experimentally correct rounding.

It can be combined with Halley's method for enhanced accuracy.

The approach is fast on architectures with slow square root operations.

Abstract

The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example \cite{Moroz,863046,863031} and the references therein). A common theme is the use of Newton's method to refine the estimates. In this paper we develop a correctly rounded Newton step that can be used to improve the accuracy of a naive calculation (using methods similar to those developed in \cite{borges}) . 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 introduce the notion of {\em weak rounding} and prove that our proposed algorithm meets this standard. We then show how to leverage the exact Newton step to get a Halley's method compensation which requires one additional FMA and one additional multiplication. This method appears to give correctly rounded results experimentally and we show that it can be combined with a square root free method for estimating the reciprocal square root to get a method that is both very fast (in computing environments with a slow square root) and, experimentally, highly accurate.