A Correctly Rounded Newton Step for the Reciprocal Square Root
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.
Cite
@article{arxiv.2112.14321,
title = {A Correctly Rounded Newton Step for the Reciprocal Square Root},
author = {Carlos F. Borges},
journal= {arXiv preprint arXiv:2112.14321},
year = {2021}
}
Comments
arXiv admin note: text overlap with arXiv:2103.08694