  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.
