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Journal of Automated Reasoning

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01-10-2016

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Authors: Érik Martin-Dorel, Guillaume Melquiond

Published in: Journal of Automated Reasoning | Issue 3/2016

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Abstract

The verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq’s logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples.

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Available only for authorised users

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An interval function \(\mathbf {f}\) is isotone if, for any pair of intervals \((\mathbf {x},\mathbf {x'})\), we have \(\mathbf {x}\subseteq \mathbf {x'}\implies \mathbf {f}(\mathbf {x})\subseteq \mathbf {f}(\mathbf {x'})\) (see also [11, Definition 4.8.10]).

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Title: Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq
Authors: Érik Martin-Dorel
Guillaume Melquiond
Publication date: 01-10-2016
Publisher: Springer Netherlands
Published in: Journal of Automated Reasoning / Issue 3/2016
Print ISSN: 0168-7433
Electronic ISSN: 1573-0670
DOI: https://doi.org/10.1007/s10817-015-9350-4

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