# Contributions to the Formal Verification of Arithmetic Algorithms

2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : The Floating-Point (FP) implementation of a real-valued function is performed with correct rounding if the output is always equal to the rounding of the exact value, which has many advantages. But for implementing a function with correct rounding in a reliable and efficient manner, one has to solve the Table Maker's Dilemma'' (TMD). Two sophisticated algorithms (L and SLZ) have been designed to solve this problem, relying on some long and complex calculations that are performed by some heavily-optimized implementations. Hence the motivation to provide strong guarantees on these costly pre-computations. To this end, we use the Coq proof assistant. First, we develop a library of Rigorous Polynomial Approximation'', allowing one to compute an approximation polynomial and an interval that bounds the approximation error in Coq. This formalization is a key building block for verifying the first step of SLZ, as well as the implementation of a mathematical function in general (with or without correct rounding). Then we have implemented, formally verified and made effective 3 interrelated certificates checkers in Coq, whose correctness proof derives from Hensel's lemma that we have formalized for both univariate and bivariate cases. In particular, our ISValP verifier'' is a key component for formally verifying the results generated by SLZ. Then, we have focused on the mathematical proof of augmented-precision'' FP algorithms for the square root and the Euclidean 2D norm. We give some tight lower bounds on the minimum non-zero distance between sqrt(x²+y²) and a midpoint, allowing one to solve the TMD for this bivariate function. Finally, the double-rounding'' phenomenon can typically occur when several FP precision are available, and may change the behavior of some usual small FP algorithms. We have formally verified in Coq a set of results describing the behavior of the Fast2Sum algorithm with double-roundings.
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Cited literature [168 references]

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Submitted on : Thursday, October 25, 2012 - 5:07:11 PM
Last modification on : Friday, April 20, 2018 - 3:44:26 PM
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• HAL Id : tel-00745553, version 1

### Citation

Erik Martin-Dorel. Contributions to the Formal Verification of Arithmetic Algorithms. Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2012. English. ⟨NNT : 2012ENSL0742⟩. ⟨tel-00745553⟩

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