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

1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : 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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Preprints, Working Papers, ...
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00545591
Contributor : Jean-Michel Muller Connect in order to contact the contributor
Submitted on : Friday, December 10, 2010 - 4:50:56 PM
Last modification on : Thursday, September 29, 2022 - 2:58:07 PM
Long-term archiving on: : Monday, November 5, 2012 - 1:15:15 PM

### File

CompensatedNorms.pdf
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### Identifiers

• HAL Id : ensl-00545591, version 1

### Citation

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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