  The reciprocal square root is an important computation for which many very
sophisticated algorithms exist (see for example \cite{863046,863031} and the
references therein). In this paper we develop a simple differential
compensation (much like those developed in \cite{borges}) that can be used to
improve the accuracy of a naive calculation. 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 demonstrate how to combine this
approach with a somewhat inaccurate but fast square root free method for
estimating the reciprocal square root to get a method that is both fast (in
computing environments with a slow square root) and, experimentally, highly
accurate. Finally, we show how this same approach can be extended to the
reciprocal hypotenuse calculation and, most importantly, to the construction of
Givens rotations.
