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Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

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] . ...

##
###
Rigorous Polynomial Approximation Using Taylor Models in Coq
[chapter]

2012
*
Lecture Notes in Computer Science
*

We carry out this work

doi:10.1007/978-3-642-28891-3_9
fatcat:q5sgnftbmfdzhhimk4zmcwi5z4
*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. ...##
###
Formally Verified Approximations of Definite Integrals
[chapter]

2016
*
Lecture Notes in Computer Science
*

This paper presents an efficient method for automatically computing and

doi:10.1007/978-3-319-43144-4_17
fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa
*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] . ...##
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Formally Verified Approximations of Definite Integrals

2018
*
Journal of automated reasoning
*

This paper presents an efficient method for automatically computing and

doi:10.1007/s10817-018-9463-7
fatcat:g3jumo65prehlovv7mwnoqjsse
*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 ...##
###
Certified and Fast Computation of Supremum Norms of Approximation Errors

2009
*
2009 19th IEEE Symposium on Computer Arithmetic
*

The

doi:10.1109/arith.2009.18
dblp:conf/arith/ChevillardJL09
fatcat:zo6uwrpwvrhy7hdfl65gsm2eyi
*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]. ...##
###
A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3)
[chapter]

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

doi:10.1007/978-3-319-08970-6_11
fatcat:mlorqqxcxbb7dpjx5i53dfmudu
*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] . ...##
###
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
[article]

2013
*
arXiv
*
pre-print

The max-plus approximation is iteratively refined and combined

arXiv:1307.7002v1
fatcat:friwu32z2vbddhdtpiie72xf2u
*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 ...##
###
A Simple Test Qualifying the Accuracy of Horner'S Rule for Polynomials

2004
*
Numerical Algorithms
*

An example of use is given

doi:10.1023/b:numa.0000049487.98618.61
fatcat:7s4k45iohffynkhaefrzgfyvwy
*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. ...##
###
Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm's and Tarski's Theorems

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

doi:10.1007/s10817-015-9320-x
fatcat:l7jxvtfsxbhypbq3v2yje6dmbq
*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. ...##
###
Polynomial function intervals for floating-point software verification

2014
*
Annals of Mathematics and Artificial Intelligence
*

We also include a scalability study that explores the limits of PolyPaver

doi:10.1007/s10472-014-9409-7
fatcat:3y3qukebzvcsrpafjwgihilqqm
*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*. ...##
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A Certificate-Based Approach to Formally Verified Approximations

2019
*
International Conference on Interactive Theorem Proving
*

We present a library to verify rigorous approximations of

doi:10.4230/lipics.itp.2019.8
dblp:conf/itp/BrehardMP19
fatcat:b2twcbftqrdihpid6s3fxlw4ha
*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*...##
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Certification of real inequalities: templates and sums of squares

2014
*
Mathematical programming
*

We consider the problem of certifying lower

doi:10.1007/s10107-014-0834-5
fatcat:envrtolapvafxoltclmof32ssu
*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 ...##
###
Certification of Real Inequalities -- Templates and Sums of Squares
[article]

2014
*
arXiv
*
pre-print

We consider the problem of certifying lower

arXiv:1403.5899v2
fatcat:df2jhxchzrbyjhht6a4atd6gqu
*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 ...##
###
Validated and Numerically Efficient Chebyshev Spectral Methods for Linear Ordinary Differential Equations

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. ...

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

2013
*
2013 European Control Conference (ECC)
*
unpublished

The max-plus approximation is iteratively refined and combined

doi:10.23919/ecc.2013.6669514
fatcat:lnt2qqt3jfdv5k4txtt64euk5e
*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 ...
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