Journal of Automated Reasoning

, Volume 57, Issue 3, pp 187–217 | Cite as

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

  • Érik Martin-DorelEmail author
  • Guillaume Melquiond


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.


Interval arithmetic Formal proof Decision procedure Coq proof assistant Floating-point arithmetic  Nonlinear arithmetic 



We would like to thank the people from the ANR TaMaDi project for initiating and greatly contributing to the CoqApprox project.

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Université Toulouse 3, Institut de Recherche en Informatique de Toulouse, UMR 5505 CNRSIRIT, Université Paul SabatierToulouse Cedex 9France
  2. 2.Inria Saclay–Île-de-France, LRI, UMR 8623 CNRS Inria Univ. Paris-Sud, Univ. Paris SaclayOrsay CedexFrance

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