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Pré-Publication, Document De Travail Année : 2010

Augmented precision square roots, 2-D norms, and discussion on correctly rounding $\sqrt{x^2+y^2}$

Nicolas Brisebarre
Mioara Maria Joldes
Erik Martin-Dorel
Jean-Michel Muller

Résumé

Define an “augmented precision” algorithm as an algorithm that returns, in precision-p floating-point arithmetic, its result as the unevaluated sum of two floating-point numbers, with a relative error of the order of 2^(−2p). Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2D-norm sqrt(x^2 + y^2). Then we give tight lower bounds on the minimum distance (in ulps) between sqrt(x^2 + y^2) and a midpoint when sqrt(x^2 + y^2) is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctly-rounded 2D-norms.
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Dates et versions

ensl-00545591 , version 1 (10-12-2010)
ensl-00545591 , version 2 (14-11-2011)

Identifiants

  • HAL Id : ensl-00545591 , version 1

Citer

Nicolas Brisebarre, Mioara Maria Joldes, Peter Kornerup, Erik Martin-Dorel, Jean-Michel Muller. Augmented precision square roots, 2-D norms, and discussion on correctly rounding $\sqrt{x^2+y^2}$. 2010. ⟨ensl-00545591v1⟩
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