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...

Celý popis

Uloženo v:
Podrobná bibliografie
Vydáno v:Proceedings - Symposium on Computer Arithmetic s. 18 - 25
Hlavní autoři: Borges, Carlos F., Jeannerod, Claude-Pierre, Muller, Jean-Michel
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: IEEE 01.09.2022
Témata:
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
On-line přístup:Získat plný text
Tagy: Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
Popis
Shrnutí: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