Formal Verification of a Floating-Point Expansion Renormalization Algorithm

Sylvie Boldo 1, 2 Mioara Joldes 3 Jean-Michel Muller 3, 4, 5 Valentina Popescu 4
2 TOCCATA - Certified Programs, Certified Tools, Certified Floating-Point Computations
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
5 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Many numerical problems require a higher computing precision than the one offered by standard floating-point formats. A common way of extending the precision is to use floating-point expansions. As the problems may be critical and as the algorithms used have very complex proofs (many sub-cases), a formal guarantee of correctness is a wish that can now be fulfilled, using interactive theorem proving. In this article we give a formal proof in Coq for one of the algorithms used as a basic brick when computing with floating-point expansions, the renormaliza-tion, which is usually applied after each operation. It is a critical step needed to ensure that the resulted expansion has the same property as the input one, and is more " compressed ". The formal proof uncovered several gaps in the pen-and-paper proof and gives the algorithm a very high level of guarantee.
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Sylvie Boldo, Mioara Joldes, Jean-Michel Muller, Valentina Popescu. Formal Verification of a Floating-Point Expansion Renormalization Algorithm. 8th International Conference on Interactive Theorem Proving (ITP'2017), Sep 2017, Brasilia, Brazil. ⟨hal-01512417⟩

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