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)

1 Inria Saclay - Ile de France
2 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
3 CNRS - Centre National de la Recherche Scientifique
4 LIP - Laboratoire de l'Informatique du Parallélisme
5 ARIC - Arithmetic and Computing

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.

Mots clés: Coq; formal proof; floating-point expansions; floating-point arithmetic; multiple-precision arithmetic

Domaines: Arithmétique des ordinateurs; Logique en informatique [cs.LO]

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⟩