A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

##
###
Verifying the accuracy of polynomial approximations in HOL
[chapter]

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) . ...

##
###
Dandelion: Certified Approximations of Elementary Functions
[article]

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. ...

##
###
Floating-Point Verification
[chapter]

2005
*
Lecture Notes in Computer Science
*

This proof, for example, involves

doi:10.1007/11526841_35
fatcat:lg5ossoiijbblf2ojgwlkqm3p4
*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. ...##
###
Floating Point Verification
[chapter]

1998
*
Theorem Proving with the Real Numbers
*

This proof, for example, involves

doi:10.1007/978-1-4471-1591-5_7
fatcat:6owbmyxx6zevjju47bqj72pl3i
*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. ...##
###
Formal Verification of Floating Point Trigonometric Functions
[chapter]

2000
*
Lecture Notes in Computer Science
*

We have formal

doi:10.1007/3-540-40922-x_14
fatcat:e4fumqurobcsrmgjjgqygwko6m
*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. ...##
###
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
[chapter]

2013
*
Lecture Notes in Computer Science
*

Our tool is implemented

doi:10.1007/978-3-642-38088-4_26
fatcat:wqfua6ytfjdklayyn7h5rw6mpa
*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. ...##
###
Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

2015
*
Journal of automated reasoning
*

Due to

doi:10.1007/s10817-015-9350-4
fatcat:4ges4bruovhqbphmqstyfhqkz4
*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. ...##
###
Computer-assisted proofs for Lyapunov stability via Sums of Squares certificates and Constructive Analysis
[article]

2020
*
arXiv
*
pre-print

Our framework relies on constructive analysis together with formally certified sums

arXiv:2006.09884v1
fatcat:224lxtqlv5aivo6acnwnsaukou
*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 ...##
###
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

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 ...

##
###
Formal Verification of Square Root Algorithms

2003
*
Formal methods in system design
*

After briefly surveying some

doi:10.1023/a:1022973506233
fatcat:d6wrk2negrakli464topcmrlpi
*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 ...##
###
Floating-Point Verification Using Theorem Proving
[chapter]

2006
*
Lecture Notes in Computer Science
*

This chapter describes our work on formal verification

doi:10.1007/11757283_8
fatcat:kz7ckh7iyrglbm4yw3mv4zzd3a
*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 ...##
###
Sobolev gradient preconditioning for the electrostatic potential equation

2005
*
Computers and Mathematics with Applications
*

Convergence is then

doi:10.1016/j.camwa.2005.08.011
fatcat:uskglbrkmzbarlo5dy4ascvgpu
*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. ...##
###
Formalization of Lerch's Theorem using HOL Light
[article]

2018
*
arXiv
*
pre-print

Next,

arXiv:1806.03049v1
fatcat:dxqnz5t2pvhvzaueddrd2hxq7a
*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]. ...##
###
Formally Verified Approximations of Definite Integrals
[chapter]

2016
*
Lecture Notes in Computer Science
*

Instead, it relies on computing and evaluating antiderivatives

doi:10.1007/978-3-319-43144-4_17
fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa
*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*). ...
« Previous

*Showing results 1 — 15 out of 440 results*