High-level algorithms for correctly-rounded reciprocal square roots

We analyze two fast and accurate algorithms recently presented by Borges for computing
$x^{-1/2}$
in binary floating-point arithmetic (assuming that efficient and correctly-rounded FMA and square root are available). The first algorithm is based on the Newton-Raphson iteration, and the second one uses an order-3 iteration. We give attainable relative-error bounds for these two algorithms, build counterexamples showing that in very rare cases they do not provide a correctly-rounded result, and characterize precisely when such failures happen in IEEE 754 binary32 and binary64 arithmetics. We then give a generic (i.e., precision-independent) algorithm that always returns a correctly-rounded result, and show how it can be simplified and made more efficient in the important cases of binary32 and binary64.