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arXiv · 2112.14321

A Correctly Rounded Newton Step for the Reciprocal Square Root

Carlos F. Borges
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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.

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Carlos F. Borges. 2021-12-28. A Correctly Rounded Newton Step for the Reciprocal Square Root. https://arxiv.org/abs/2112.14321

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