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Verifying the accuracy of polynomial approximations in HOL [chapter]

John Harrison
1997 Lecture Notes in Computer Science  
In verifying such an algorithm, one is faced with the problem of bounding the error in this polynomial approximation.  ...  We discuss a technique for proving such results formally in HOL, via the formalization of a number of results in polynomial theory, e.g. squarefree decomposition and Sturm's theorem, and the use of a computer  ...  This is all the more true in HOL, where a single evaluation of a transcendental function to moderate accuracy can take many seconds (Harrison 1996) .  ... 
doi:10.1007/bfb0028391 fatcat:yt6aregsfnfqfdnz6xjaq6fbca

Dandelion: Certified Approximations of Elementary Functions [article]

Heiko Becker, Mohit Tekriwal, Eva Darulova, Anastasia Volkova, Jean-Baptiste Jeannin
2022 arXiv   pre-print
In fact, the Remez algorithm computes the best possible approximation for a given polynomial degree, but has so far not been formally verified.  ...  This paper presents Dandelion, an automated certificate checker for polynomial approximations of elementary functions computed with Remez-like algorithms that is fully verified in the HOL4 theorem prover  ...  Further, we thank Magnus Myreen and Michael Norrish for their help with improving the HOL4 implementation of Dandelion. We also thank Samuel Coward for helping us with the MetiTarski evaluation.  ... 
arXiv:2202.05472v2 fatcat:6uo35kna3vcdvdulmhvrfimymu

Floating-Point Verification [chapter]

John Harrison
2005 Lecture Notes in Computer Science  
This proof, for example, involves verifying the accuracy of polynomial approximations to transcendental functions (optimal Remez polynomials rather than simply truncated Taylor series), precisely bounding  ...  To emphasize the last point, an error in the FDIV (floating-point division) instruction of some early Intel® Pentium® processors in 1994 resulted in a charge to Intel of approximately $475M.  ... 
doi:10.1007/11526841_35 fatcat:lg5ossoiijbblf2ojgwlkqm3p4

Floating Point Verification [chapter]

John Harrison
1998 Theorem Proving with the Real Numbers  
This proof, for example, involves verifying the accuracy of polynomial approximations to transcendental functions (optimal Remez polynomials rather than simply truncated Taylor series), precisely bounding  ...  To emphasize the last point, an error in the FDIV (floating-point division) instruction of some early Intel® Pentium® processors in 1994 resulted in a charge to Intel of approximately $475M.  ... 
doi:10.1007/978-1-4471-1591-5_7 fatcat:6owbmyxx6zevjju47bqj72pl3i

Formal Verification of Floating Point Trigonometric Functions [chapter]

John Harrison
2000 Lecture Notes in Computer Science  
We have formal verified a number of algorithms for evaluating transcendental functions in double-extended precision floating point arithmetic in the Intel® IA-64 architecture.  ...  In this paper we describe in some depth the formal verification of the sin and cos functions, including the initial range reduction step.  ...  In order to provably bound the accuracy of a polynomial approximation to any mathematical function, the only work required of the user is to provide these functions.  ... 
doi:10.1007/3-540-40922-x_14 fatcat:e4fumqurobcsrmgjjgqygwko6m

Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations [chapter]

Alexey Solovyev, Thomas C. Hales
2013 Lecture Notes in Computer Science  
Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains.  ...  One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds.  ...  John Harrison, the developer of HOL Light, contributed a lot to the Flyspeck project by proving many important foundational theorems in HOL Light.  ... 
doi:10.1007/978-3-642-38088-4_26 fatcat:wqfua6ytfjdklayyn7h5rw6mpa

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

Érik Martin-Dorel, Guillaume Melquiond
2015 Journal of automated reasoning  
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.  ...  Acknowledgements We would like to thank the people from the ANR TaMaDi project for initiating and greatly contributing to the CoqApprox project.  ... 
doi:10.1007/s10817-015-9350-4 fatcat:4ges4bruovhqbphmqstyfhqkz4

Computer-assisted proofs for Lyapunov stability via Sums of Squares certificates and Constructive Analysis [article]

Grigory Devadze and Victor Magron and Stefan Streif
2020 arXiv   pre-print
Our framework relies on constructive analysis together with formally certified sums of squares Lyapunov functions. The crucial steps are formalized within of the proof assistant Minlog.  ...  We provide a computer-assisted approach to ensure that a given continuous or discrete-time polynomial system is (asymptotically) stable.  ...  Then, we obtain an approximate Cholesky's decomposition LL T ofG with accuracy δ c , and for all i = 1, . . . , r, we compute the inner product of the i-th row of L by v d (x) to obtain a polynomial s  ... 
arXiv:2006.09884v1 fatcat:224lxtqlv5aivo6acnwnsaukou

Page 5864 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
of CSP to verify authentication proto- cols (121-136); John Harrison, Verifying the accuracy of poly- nomial approximations in HOL (137-152); Daniel Hirschkoff, A full formalisation of z-calculus theory  ...  Stearns, The polynomial time decidability of simulation relations for finite state processes: a HORNSAT based approach (603-641); Lefteris M.  ... 

Verifying a Synthesized Implementation of IEEE-754 Floating-Point Exponential Function using HOL

B. Akbarpour, A. T. Abdel-Hamid, S. Tahar, J. Harrison
2009 Computer journal  
In this paper, we have hierarchically formalized and verified a hardware implementation of the IEEE-754 table-driven floating-point exponential function algorithm using the HOL theorem prover.  ...  The high ability of abstraction in the HOL verification system allows its use for the verification task over the whole design path of the circuit, starting from gate level implementation of the circuit  ...  Our method for bounding the approximation error in this polynomial [7] is posthoc, and works equally well if the polynomial is derived in other ways, e.g., via Chebyshev expansions [38] or more delicate  ... 
doi:10.1093/comjnl/bxp023 fatcat:wvf6ehqqefaqtj6yhois52grmy

Formal Verification of Square Root Algorithms

John Harrison
2003 Formal methods in system design  
After briefly surveying some of our formal verification work, we discuss in more detail the verification of a square root algorithm, which helps to illustrate why some features of HOL Light, in particular  ...  A number of important algorithms have been proven correct using the HOL Light theorem prover.  ...  Similar algorithms for the IBM Power 9 series are considered by Sawada and Gamboa (2002) , who formally verifiy the accuracy of the polynomial approximations, but not the final rounding method (the rounding  ... 
doi:10.1023/a:1022973506233 fatcat:d6wrk2negrakli464topcmrlpi

Floating-Point Verification Using Theorem Proving [chapter]

John Harrison
2006 Lecture Notes in Computer Science  
This chapter describes our work on formal verification of floating-point algorithms using the HOL Light theorem prover. let th6 = REAL_ARITH 'abs(c -a) < e ∧ abs(b) <= d =⇒ abs((a + b) -c) < d + e';;  ...  In order to get a still better approximation, one can either use a longer polynomial in e, or repeat the Newton-Raphson linear correction several times.  ...  The core approximation is then a polynomial approximation to sin(r) or cos(r) as appropriate, similar to a truncation of the familiar Taylor series: important: (i) available library of formalized real  ... 
doi:10.1007/11757283_8 fatcat:kz7ckh7iyrglbm4yw3mv4zzd3a

Sobolev gradient preconditioning for the electrostatic potential equation

J. Karátson, L. Lóczi
2005 Computers and Mathematics with Applications  
Convergence is then verified for the corresponding sequence in Sobolev space, implying mesh independent convergence results for the discretized problems.  ...  The simplicity of the algorithm and the fast linear convergence are finally illustrated in a numerical test example. ~)  ...  In each step of (13), we approximate e =-by a suitable Taylor polynomial and define the subsequent iterate by v'% p(u,) = A.~ j! ' 2t~ + 1 (u,~ --2~wn), where -Aw. = p(u.), w,,io B = O.  ... 
doi:10.1016/j.camwa.2005.08.011 fatcat:uskglbrkmzbarlo5dy4ascvgpu

Formalization of Lerch's Theorem using HOL Light [article]

Adnan Rashid, Osman Hasan
2018 arXiv   pre-print
Next, the uniqueness of the Laplace transform provides the solution of these differential equations in the time domain.  ...  One of the main contributions of the paper is the formalization of Lerch's theorem, which describes the uniqueness of the Laplace transform and thus plays a vital role in solving linear differential equations  ...  On the other hand, Orloff [22] provides the proof of the lemma by approximating the function φ(x) with a polynomial p(x) in the interval [0, 1].  ... 
arXiv:1806.03049v1 fatcat:dxqnz5t2pvhvzaueddrd2hxq7a

Formally Verified Approximations of Definite Integrals [chapter]

Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote
2016 Lecture Notes in Computer Science  
Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting.  ...  Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy.  ...  Now that we have polynomial approximations, we can make use of the following lemma. Lemma 3 (Polynomial approximation).  ... 
doi:10.1007/978-3-319-43144-4_17 fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa
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