Error Bounds on Complex Floating-Point Multiplication (original) (raw)

Rapport (Rapport De Recherche) Année : 2006

Résumé

Given floating-point arithmetic with ttt-digit base-$\beta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values z0z_0z0 and z1z_1z1 can be computed with maximum absolute error absz0absz1frac12beta1−tsqrt5\abs{z_0} \abs{z_1} \frac{1}{2} \beta^{1 - t} \sqrt{5}absz_0absz_1frac12beta1−tsqrt5. In particular, this provides relative error bounds of 2−24sqrt52^{-24} \sqrt{5}2−24sqrt5 and 2−53sqrt52^{-53} \sqrt{5}2−53sqrt5 for {IEEE 754} single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for {IEEE 754} single and double precision arithmetic.

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https://inria.hal.science/inria-00120352

Soumis le : lundi 18 décembre 2006-17:48:45

Dernière modification le : mercredi 18 mars 2026-11:52:02

Archivage à long terme le : jeudi 20 septembre 2012-16:00:49

Dates et versions

inria-00120352 , version 1 (18-12-2006)

inria-00120352 , version 2 (19-12-2006)

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Richard P. Brent, Colin Percival, Paul Zimmermann. Error Bounds on Complex Floating-Point Multiplication. [Research Report] RR-6068, 2006. ⟨inria-00120352v1⟩

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