Emulation of a FMA and correctly-rounded sums: proved algorithms using rounding to odd

Sylvie Boldo 1 Guillaume Melquiond 2
1 PROVAL - Proof of Programs
UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR
2 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Rounding to odd is a non-standard rounding on floating-point numbers. By using it for some intermediate values instead of rounding to nearest, correctly rounded results can be obtained at the end of computations. We present an algorithm to emulate the fused multiply-and-add operator. We also present an iterative algorithm for computing the correctly rounded sum of a set floating-point numbers under mild assumptions. A variation on both previous algorithms is the correctly rounded sum of any three floating-point numbers. This leads to efficient implementations, even when this rounding is not available. In order to guarantee the correctness of these properties and algorithms, we formally proved them using the Coq proof checker.
Keywords : Coq proof assistant floating-point arithmetic Floating-point Rounding to odd Accurate summation FMA Formal proof Coq
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Sylvie Boldo, Guillaume Melquiond. Emulation of a FMA and correctly-rounded sums: proved algorithms using rounding to odd. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2008, 57 (4), pp.462-471. ⟨10.1109/TC.2007.70819⟩. ⟨inria-00080427v2⟩

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