High-level algorithms for correctly-rounded reciprocal square roots
Résumé
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.
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Cite 10.1109/ARITH54963.2022.00013 Autre Borges, C. F., Jeannerod, C.-P., & Muller, J.-M. (2022). High-level algorithms for correctly-rounded reciprocal square roots. In 2022 IEEE 29th Symposium on Computer Arithmetic (ARITH) (pp. 18–25). 2022 IEEE 29th Symposium on Computer Arithmetic (ARITH). IEEE. https://doi.org/10.1109/arith54963.2022.00013