Certified, Efficient and Sharp Univariate Taylor Models in COQ

Martin-Dorel, Érik; Rideau, Laurence; Théry, Laurent; Mayero, Micaela; Pasca, Ioana

doi:10.1109/synasc.2013.33

Cited by 10 publications

(19 citation statements)

References 10 publications

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“…Our next priority will be to combine the CoqHensel machinery presented here with that of the CoqApprox library for certified polynomial approximation [7,26]. We will then get a validation for the whole SLZ algorithm for solving the Table Maker's Dilemma.…”

Section: Discussionmentioning

confidence: 99%

“…This different wording is due to some Coq technicalities, on which we will elaborate in Section 5. Note also that unlike Problem 2 where the required property (20) was an equivalence, here we only require an implication (26) saying that S is a superset of the ISValP solutions. Indeed we are especially interested in formally proving that no ISValP solution has been forgotten.…”

Section: The Problem At Stakementioning

confidence: 99%

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Formally Verified Certificate Checkers for Hardest-to-Round Computation

et al. 2014

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In order to derive efficient and robust floating-point implementations of a given function f , it is crucial to compute its hardest-to-round points, i.e. the floating-point numbers x such that f (x) is closest to the midpoint of two consecutive floating-point numbers. Depending on the floating-point format one is aiming at, this can be highly computationally intensive. In this paper, we show how certificates based on Hensel's lemma can be added to an algorithm using lattice basis reduction so that the result of a computation can be formally checked in the Coq proof assistant.

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Section: Discussionmentioning

confidence: 99%

Section: The Problem At Stakementioning

confidence: 99%

Formally Verified Certificate Checkers for Hardest-to-Round Computation

et al. 2014

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“…Independently developed, the Coq-Approx library was built on top of CoqInterval's formalization of interval arithmetic. It was designed to compute formally-verified polynomial approximations of univariate functions which can then be fed to programs solving the Table-Maker's Dilemma [4,17].…”

Section: Background and Scopementioning

confidence: 99%

“…One of the approaches is based on automatic differentiation "Auto-diff" (see Section 4.2). Another approach is based on rigorous polynomial approximations using "Taylor models" (see Section 4.3); they are built using the CoqApprox library [4,17], now part of CoqInterval.…”

Section: Architecture Of Coqintervalmentioning

confidence: 99%

“…The CoqApprox library gathers certified algorithms for univariate Taylor models, which can be executed inside the logic of Coq [4,17]. The main data type involved in the formalization is a record that combines a polynomial and an interval; it is parameterized by a type for representing polynomials and another type for intervals.…”

Section: Main Features Of the Coqapprox Formalizationmentioning

confidence: 99%

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Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Martin-Dorel

Melquiond

2015

J Autom Reasoning

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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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Taylor Series Revisited

Thirioux

Maffart

2019

Theoretical Aspects of Computing – ICTAC 2019

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We propose a renovated approach around the use of Taylor expansions to provide polynomial approximations. We introduce a coinductive type scheme and finely-tuned operations that altogether constitute an algebra, where our multivariate Taylor expansions are firstclass objects. As for applications, beyond providing classical expansions of integro-differential and algebraic expressions mixed with elementary functions, we demonstrate that solving ODE and PDE in a direct way, without external solvers, is also possible. We also discuss the possibility of computing certified errors within our scheme.

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Certified, Efficient and Sharp Univariate Taylor Models in COQ

Cited by 10 publications

References 10 publications

Formally Verified Certificate Checkers for Hardest-to-Round Computation

Formally Verified Certificate Checkers for Hardest-to-Round Computation

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Taylor Series Revisited

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