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Certification of bounds on expressions involving rounded operators [article]

Marc Daumas, Guillaume Melquiond (LIP, INRIA Rhône-Alpes)
2007 arXiv   pre-print
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.  ...  ÒÙÑ Ö Ø ÖÑ× Û Ö ÓÒ× Ö Ò Ð Ð Ò Û Ö ÓÔØ Ñ Þ ÓÙØ Ó Ø ÑÔÐ Ñ ÒØ Ø ÓÒ y¸Ø × Ø ÖÑ× Ö ÒØÖÓ Ù Ò Y º ËÓ Ø ÜÔÖ ×× ÓÒ y Ú × Ø Ø Ú ÐÝ ÓÑÔÙØ Ú ÐÙ Û Ð Ø ÜÔÖ ×× ÓÒ Y Ú × Ø Ð Ú ÐÙ y ØÖ × ØÓ ÔÔÖÓÜ Ñ Ø º ÁÒ ÓÖ Ö ØÓ ÖØ Ý  ... 
arXiv:cs/0701186v2 fatcat:mel4y5bsxbb35czmvk3obtlenq

Certification of bounds on expressions involving rounded operators

Marc Daumas, Guillaume Melquiond
2010 ACM Transactions on Mathematical Software  
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]

Mariano Moscato, Laura Titolo, Aaron Dutle, César A. Muñoz
2017 Lecture Notes in Computer Science  
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]

Victor Magron, George Constantinides, Alastair Donaldson
2016 arXiv   pre-print
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

Victor Magron, George Constantinides, Alastair Donaldson
2017 ACM Transactions on Mathematical Software  
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

Gilles Villard
2007 Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07  
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]

Gilles Villard
2007 arXiv   pre-print
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]

Laura Titolo, Marco A. Feliú, Mariano Moscato, César A. Muñoz
2017 Lecture Notes in Computer Science  
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]

Jiankai Jin, Olga Ohrimenko, Benjamin I. P. Rubinstein
2022 arXiv   pre-print
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

Alexey Solovyev, Marek S. Baranowski, Ian Briggs, Charles Jacobsen, Zvonimir Rakamarić, Ganesh Gopalakrishnan
2018 ACM Transactions on Programming Languages and Systems  
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]

Alexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, Ganesh Gopalakrishnan
2015 Lecture Notes in Computer Science  
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]

Victor Magron, Xavier Allamigeon, Stéphane Gaubert, Benjamin Werner
2015 arXiv   pre-print
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]

Heiko Becker, Nikita Zyuzin, Raphael Monat, Eva Darulova, Magnus O. Myreen, Anthony Fox
2018 arXiv   pre-print
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

Victor Magron, Xavier Allamigeon, Stéphane Gaubert, Benjamin Werner
2014 Journal of Formalized Reasoning  
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]

Florent De Dinechin, Guillaume Melquiond
2008 arXiv   pre-print
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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