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Conference Papers Year : 2022

High-level algorithms for correctly-rounded reciprocal square roots

Abstract

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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Dates and versions

hal-03728088 , version 1 (20-07-2022)

Identifiers

  • HAL Id : hal-03728088 , version 1

Cite

Carlos F. Borges, Claude-Pierre Jeannerod, Jean-Michel Muller. High-level algorithms for correctly-rounded reciprocal square roots. 29th IEEE Symposium on Computer Arithmetic (ARITH 2022), Sep 2022, Lyon (virtual meeting due to the COVID pandemic), France. ⟨hal-03728088⟩
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