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The reciprocal square root is an important computation for which manysophisticated algorithms exist (see for example \cite{Moroz,863046,863031} andthe references therein). A common theme is the use of Newton's method to refinethe estimates. In this paper we develop a correctly rounded Newton step thatcan be used to improve the accuracy of a naive calculation (using methodssimilar to those developed in \cite{borges}) . The approach relies on the useof the fused multiply-add (FMA) which is widely available in hardware on avariety of modern computer architectures. We then introduce the notion of {\emweak rounding} and prove that our proposed algorithm meets this standard. Wethen show how to leverage the exact Newton step to get a Halley's methodcompensation which requires one additional FMA and one additionalmultiplication. This method appears to give correctly rounded resultsexperimentally and we show that it can be combined with a square root freemethod for estimating the reciprocal square root to get a method that is bothvery fast (in computing environments with a slow square root) and,experimentally, highly accurate.

A Correctly Rounded Newton Step for the Reciprocal Square Root