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Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Érik Martin-Dorel, Guillaume Melquiond
2015 Journal of automated reasoning  
In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions.  ...  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.  ...  In fact, the correctness of a modern library might depend on the proof of hundreds of tight bounds on univariate functions, for table-based implementations [22] .  ... 
doi:10.1007/s10817-015-9350-4 fatcat:4ges4bruovhqbphmqstyfhqkz4

Rigorous Polynomial Approximation Using Taylor Models in Coq [chapter]

Nicolas Brisebarre, Mioara Joldeş, Érik Martin-Dorel, Micaela Mayero, Jean-Michel Muller, Ioana Paşca, Laurence Rideau, Laurent Théry
2012 Lecture Notes in Computer Science  
We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation.  ...  We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C.  ...  In a previous version of our Coq development we had managed to prove correct Taylor models for some elementary functions and addition.  ... 
doi:10.1007/978-3-642-28891-3_9 fatcat:q5sgnftbmfdzhhimk4zmcwi5z4

Formally Verified Approximations of Definite Integrals [chapter]

Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote
2016 Lecture Notes in Computer Science  
This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system.  ...  Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting.  ...  Elementary real analysis in Coq Coq's standard library Reals 3 axiomatizes real arithmetic, with a classical flavor [9] .  ... 
doi:10.1007/978-3-319-43144-4_17 fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa

Formally Verified Approximations of Definite Integrals

Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote
2018 Journal of automated reasoning  
This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system.  ...  Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting.  ...  Acknowledgements We would like to thankÉrik Martin-Dorel for his improvements to the Coq framework for computing rigorous polynomial approximations and Philippe Dumas for stimulating discussions and suggestions  ... 
doi:10.1007/s10817-018-9463-7 fatcat:g3jumo65prehlovv7mwnoqjsse

Certified and Fast Computation of Supremum Norms of Approximation Errors

Sylvain Chevillard, Mioara Joldes, Christoph Lauter
2009 2009 19th IEEE Symposium on Computer Arithmetic  
The tight yet certain bounding of this error is an important step towards safe implementations.  ...  We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors.  ...  Tight bounding of the zeros of a function We have reduced the initial problem to the following one: let τ be a function defined on an interval [a, b].  ... 
doi:10.1109/arith.2009.18 dblp:conf/arith/ChevillardJL09 fatcat:zo6uwrpwvrhy7hdfl65gsm2eyi

A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3) [chapter]

Frédéric Chyzak, Assia Mahboubi, Thomas Sibut-Pinote, Enrico Tassi
2014 Lecture Notes in Computer Science  
This paper describes the formal verification of an irrationality proof of ζ(3), the evaluation of the Riemann zeta function, using the Coq proof assistant.  ...  This result was first proved by Apéry in 1978, and the proof we have formalized follows the path of his original presentation.  ...  A bound 3 n is however tight enough for our purpose and has been proved by several independent and elementary proofs, for instance by Hanson [14] and Feng [12] .  ... 
doi:10.1007/978-3-319-08970-6_11 fatcat:mlorqqxcxbb7dpjx5i53dfmudu

Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation [article]

Xavier Allamigeon, Stéphane Gaubert, Victor Magron, Benjamin Werner
2013 arXiv   pre-print
The max-plus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap.  ...  Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical  ...  In this way, starting from a transcendental univariate elementary function f ∈ T , such as arctan, exp, etc, defined on a real bounded interval I, we arrive at a semialgebraic lower bound of f , which  ... 
arXiv:1307.7002v1 fatcat:friwu32z2vbddhdtpiie72xf2u

A Simple Test Qualifying the Accuracy of Horner'S Rule for Polynomials

Sylvie Boldo, Marc Daumas
2004 Numerical Algorithms  
An example of use is given with the approximation of elementary functions. Résumé : Les polynômes sont utilisés dans de nombreuses applications et enfouis dans des librairies telles que libm.  ...  Whereas the accuracy of the functions used by linear algebra have long been studied, little is available to decide on one scheme to evaluate a polynomial.  ...  The Coq theorems proved in Section 2 can be applied to build faithful approximations to the elementary functions over the full range of the floating point input.  ... 
doi:10.1023/b:numa.0000049487.98618.61 fatcat:7s4k45iohffynkhaefrzgfyvwy

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm's and Tarski's Theorems

Anthony Narkawicz, César Muñoz, Aaron Dutle
2015 Journal of automated reasoning  
The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval  ...  Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval.  ...  Those algorithms are quite powerful and can prove tight bounds on complex polynomials with up to 16 variables and degree 4.  ... 
doi:10.1007/s10817-015-9320-x fatcat:l7jxvtfsxbhypbq3v2yje6dmbq

Polynomial function intervals for floating-point software verification

Jan Duracz, Michal Konečný
2014 Annals of Mathematics and Artificial Intelligence  
We also include a scalability study that explores the limits of PolyPaver in proving tight functional specifications of progressively larger randomly generated programs.  ...  In this paper we report on experiments using Poly-Paver that indicate that the additional expressivity does not come at a performance cost when comparing with other publicly available state-of-the-art  ...  Thus, (2) tells us that the total approximation error made when computing with a standardcompliant implementation fun of an elementary function f can be bounded in terms of an interval expression.  ... 
doi:10.1007/s10472-014-9409-7 fatcat:3y3qukebzvcsrpafjwgihilqqm

A Certificate-Based Approach to Formally Verified Approximations

Florent Bréhard, Assia Mahboubi, Damien Pous, Michael Wagner
2019 International Conference on Interactive Theorem Proving  
We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant.  ...  Based on interval arithmetic, this library also implements a technique of validation a posteriori based on the Banach fixed-point theorem.  ...  We fix abstract operations C: Ops1 for the remaining Coq snippets in this section, and we translate this routine into a recursive function with two accumulators: In the right-hand side, catrev is the function  ... 
doi:10.4230/lipics.itp.2019.8 dblp:conf/itp/BrehardMP19 fatcat:b2twcbftqrdihpid6s3fxlw4ha

Certification of real inequalities: templates and sums of squares

Victor Magron, Xavier Allamigeon, Stéphane Gaubert, Benjamin Werner
2014 Mathematical programming  
We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions.  ...  The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like cos, arctan, exp, etc.  ...  In this way, starting from a transcendental univariate elementary function f ∈ D, such as arctan, exp, etc , defined on a real bounded interval I, we arrive at a semialgebraic lower bound of f , which  ... 
doi:10.1007/s10107-014-0834-5 fatcat:envrtolapvafxoltclmof32ssu

Certification of Real Inequalities -- Templates and Sums of Squares [article]

Xavier Allamigeon, Stéphane Gaubert, Victor Magron, Benjamin Werner
2014 arXiv   pre-print
We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions.  ...  The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like cos, arctan, , etc.  ...  In this way, starting from a transcendental univariate elementary function f ∈ D, such as arctan, exp, etc , defined on a real bounded interval I, we arrive at a semialgebraic lower bound of f , which  ... 
arXiv:1403.5899v2 fatcat:df2jhxchzrbyjhht6a4atd6gqu

Validated and Numerically Efficient Chebyshev Spectral Methods for Linear Ordinary Differential Equations

Florent Bréhard, Nicolas Brisebarre, Mioara Joldeş
2018 ACM Transactions on Mathematical Software  
One notices that we obtain a quite tight error bound, even for the · ∞ norm.  ...  However, we can now prove the far better convergence result of the main Theorem 4. 1 : 1 Elementary operations on almost-banded (a.  ...  In the examples analyzed in this section, we particularly investigate their evolution in function of the parameters of the problems.  ... 
doi:10.1145/3208103 fatcat:jmuaqizi3vdixdhaautf5uszb4

Certification of inequalities involving transcendental functions: Combining SDP and max-plus approximation

Xavier Allamigeon, Stephane Gaubert, Victor Magron, Benjamin Werner
2013 2013 European Control Conference (ECC)   unpublished
The max-plus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap.  ...  Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical  ...  In this way, starting from a transcendental univariate elementary function f ∈ T , such as arctan, exp, etc, defined on a real bounded interval I, we arrive at a semialgebraic lower bound of f , which  ... 
doi:10.23919/ecc.2013.6669514 fatcat:lnt2qqt3jfdv5k4txtt64euk5e
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