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Certification of bounds on expressions involving rounded operators
[article]
2007
arXiv
pre-print
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Gappa uses interval arithmetic to certify bounds on mathematical expressions that involve rounded as well as exact operators. Gappa generates a theorem with its proof for each bound treated. ...
Gappa uses multiple-precision dyadic fractions for the endpoints of intervals and performs forward error analysis on rounded operators when necessary. ...
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arXiv:cs/0701186v2
fatcat:mel4y5bsxbb35czmvk3obtlenq
Certification of bounds on expressions involving rounded operators
2010
ACM Transactions on Mathematical Software
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It uses interval arithmetic and forward error analysis to bound mathematical expressions that involve rounded as well as exact operators. ...
Gappa uses multiple-precision dyadic fractions for the endpoints of intervals and performs forward error analysis on rounded operators when necessary. ...
N, March 2009.Certification of bounds on expressions involving rounded operators · 17Listing 1. ...
doi:10.1145/1644001.1644003
fatcat:nb6win5sfze4xoz75loxdj4z4i
Automatic Estimation of Verified Floating-Point Round-Off Errors via Static Analysis
[chapter]
2017
Lecture Notes in Computer Science
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This paper introduces a static analysis technique for computing formally verified round-off error bounds of floating-point functional expressions. ...
The technique is based on a denotational semantics that computes a symbolic estimation of floating-point round-off errors along with a proof certificate that ensures its correctness. ...
Let R B ∶B → B be the function converting a floatingpoint expression to a real one, by simply replacing each operation on floating-point with the corresponding operation on reals and by applying R to the ...
doi:10.1007/978-3-319-66266-4_14
fatcat:aeouavpsinfkhdu45gjn7je3ny
Certified Roundoff Error Bounds Using Semidefinite Programming
[article]
2016
arXiv
pre-print
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We present a framework to provide upper bounds on absolute roundoff errors of floating-point nonlinear programs. ...
The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementations. ...
Acknowledgment The authors would like to specially acknowledge the precious help of Alexey Solovyev, his excellent feedback and suggestions. ...
arXiv:1507.03331v7
fatcat:5h7nzr6fubfavblpseyj5r4xve
Certified Roundoff Error Bounds Using Semidefinite Programming
2017
ACM Transactions on Mathematical Software
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We present a framework to provide upper bounds on absolute roundoff errors of floating-point nonlinear programs. ...
The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementations. ...
Acknowledgment The authors would like to specially acknowledge the precious help of Alexey Solovyev, his excellent feedback and suggestions. ...
doi:10.1145/3015465
fatcat:ouzr67bwmjafxiwqhpefklphpy
Certification of the QR factor R and of lattice basis reducedness
2007
Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07
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For implementing the certificate itself, we propose a floating point algorithm for computing (certified) error bounds for the entries of the R factor of the QR matrix factorization. ...
Given a lattice basis of n vectors in Z n , we propose an algorithm using 12n 3 + O(n 2 ) floating point operations for checking whether the basis is LLL-reduced. ...
With no subtraction involved, certified bounds can be computed using directed rounding. ...
doi:10.1145/1277548.1277597
dblp:conf/issac/Villard07
fatcat:6mowbwxssffgznda3zali7auti
Certification of the QR factor R, and of lattice basis reducedness
[article]
2007
arXiv
pre-print
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For implementing the certificate itself, we propose a floating point algorithm for computing (certified) error bounds for the entries of the R factor of the QR matrix factorization. ...
Given a lattice basis of n vectors in Z^n, we propose an algorithm using 12n^3+O(n^2) floating point operations for checking whether the basis is LLL-reduced. ...
With no subtraction involved, certified bounds can be computed using directed rounding. ...
arXiv:cs/0701183v1
fatcat:cfmuh472wje45b7ymthlc2ag2e
An Abstract Interpretation Framework for the Round-Off Error Analysis of Floating-Point Programs
[chapter]
2017
Lecture Notes in Computer Science
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An instantiation of this framework is implemented in the prototype tool PRECiSA that generates formal proof certificates stating the correctness of the computed round-off errors. ...
An abstraction on the control flow of the program is proposed to mitigate the explosion of the number of elements generated by the analysis. ...
Times in seconds for the generation of round-off error bounds and certificates (bold indicates the best time, italic indicates the second best.) ...
doi:10.1007/978-3-319-73721-8_24
fatcat:xo3tbkgiljg6zj4p4i45rh6qfe
Getting a-Round Guarantees: Floating-Point Attacks on Certified Robustness
[article]
2022
arXiv
pre-print
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Finally, as a mitigation, we advocate the use of rounded interval arithmetic to account for rounding errors. ...
We show that the attack can be carried out against several linear classifiers that have exact certifiable guarantees and against neural network verifiers that return a certified lower bound on a robust ...
The first author is supported by the University of Melbourne research scholarship (MRS) scheme. ...
arXiv:2205.10159v1
fatcat:t34aqd4h4nh7pljx4orinnmzou
Rigorous Estimation of Floating-Point Round-Off Errors with Symbolic Taylor Expansions
2018
ACM Transactions on Programming Languages and Systems
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It can be shown that one of the primary sources of errors in this expression is attributable to the propagation of error in t ⊕ 1 into the division operator. ...
In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. ...
Values of e and d depend on the rounding mode and the operation itself. Special care must be taken in case of exceptions (overflows or invalid operations). ...
doi:10.1145/3230733
fatcat:mast7t5wsffajksjqdti5dvkke
Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions
[chapter]
2015
Lecture Notes in Computer Science
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It can be shown that one of the primary sources of errors in this expression is attributable to the propagation of error in t ⊕ 1 into the division operator. ...
In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. ...
Values of e and d depend on the rounding mode and the operation itself. Special care must be taken in case of exceptions (overflows or invalid operations). ...
doi:10.1007/978-3-319-19249-9_33
fatcat:ujzjmfvwqvawlkh2yyfji2efqi
Formal Proofs for Nonlinear Optimization
[article]
2015
arXiv
pre-print
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We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature. ...
This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. ...
The tactic interval [Mel12] , built on top of Flocq, can simplify inequalities on expressions of real numbers. ...
arXiv:1404.7282v4
fatcat:rbvkjx7wbfbdzakpxwaqs7v4dm
A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4
[article]
2018
arXiv
pre-print
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We present implementations of certificate generation and checking for both Coq and HOL4 and evaluate it on a number of examples from the literature. ...
This paper presents a formally verified and modular tool which fully automatically checks the correctness of finite-precision roundoff error bounds encoded in a certificate. ...
Similar bounds can be derived for the other arithmetic operations. However, for multiplication and division, the propagation of errors is more involved. ...
arXiv:1707.02115v2
fatcat:2c2jwcvlpfcsvnu4ncod2wuqdm
Formal Proofs for Nonlinear Optimization
2014
Journal of Formalized Reasoning
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The file type is application/pdf.
We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature. ...
This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. ...
The tactic interval [Mel12] , built on top of Flocq, can simplify inequalities on expressions of real numbers. ...
doi:10.6092/issn.1972-5787/4319
dblp:journals/jfrea/MagronAGW15
fatcat:jpaovl3hdrbptkswa5od7w26am
Certifying floating-point implementations using Gappa
[article]
2008
arXiv
pre-print
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The article demonstrates the use of this tool on a real-size example, an elementary function with correctly rounded output. ...
One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. ...
Given the description of a logical property involving the bounds of mathematical expressions, the tool tries to prove the validity of this property. ...
arXiv:0801.0523v1
fatcat:ckxqfuzb6jgftdoywe5m3i6dva
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