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...

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Bibliographic Details
Published in:Proceedings - Symposium on Computer Arithmetic pp. 18 - 25
Main Authors: Borges, Carlos F., Jeannerod, Claude-Pierre, Muller, Jean-Michel
Format: Conference Proceeding
Language:English
Published: IEEE 01.09.2022
Subjects:
accuracy analysis
correct rounding
Digital arithmetic
error-free transform
Floating-point arithmetic
fused multiply-add
Newton method
reciprocal square root
Robustness
Testing
Transforms
ISSN:2576-2265
Online Access:Get full text
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Summary: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.
ISSN:2576-2265
DOI:10.1109/ARITH54963.2022.00013