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Assisted verification of elementary functions using Gappa
 In Proceedings of the 2006 ACM symposium on Applied computing
, 2006
"... The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lin ..."
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Cited by 17 (6 self)
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The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lines of code for each function, and will usually be broken by the smallest change to the code (e.g. for maintenance or optimization purpose). Therefore, it is very tedious and errorprone if done by hand. This article discusses the use of the Gappa proof assistant in this context. Gappa has two main advantages over previous approaches: Its input format is very close to the actual C code to validate, and it automates error evaluation and propagation using interval arithmetic. Besides, it can be used to incrementally prove complex mathematical properties pertaining to the C code. Yet it does not require any specific knowledge about automatic theorem proving, and thus is accessible to a wider community. Moreover, Gappa may generate a formal proof of the results that can be checked independently by a lowerlevel proof assistant like Coq, hence providing an even higher confidence in the certification of the numerical code. 1.
Certifying the floatingpoint implementation of an elementary function using Gappa
 IEEE TRANSACTIONS ON COMPUTERS, 2010. 9 HTTP://DX.DOI.ORG/10.1145/1772954.1772987 10 HTTP://DX.DOI.ORG/10.1145/1838599.1838622 11 HTTP://SHEMESH.LARC.NASA.GOV/NFM2010/PAPERS/NFM2010_14_23.PDF 12 HTTP://DX.DOI.ORG/10.1007/9783642142031_11 13 HTTP://DX.
, 2011
"... High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a timeconsuming proof fo ..."
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Cited by 8 (3 self)
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High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a timeconsuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for maintenance or optimization purpose. Certifying floatingpoint programs by hand is, therefore, very tedious and errorprone. The Gappa proof assistant is designed to make this task both easier and more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of this tool for a widely used class of floatingpoint programs: implementations of elementary functions in a mathematical library.
Software techniques for perfect elementary functions in floatingpoint interval arithmetic
 IN REAL NUMBERS AND COMPUTERS
, 2006
"... ..."
Certifying floatingpoint implementations using Gappa
, 2008
"... High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. Such work may require several lines of proof for each lin ..."
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Cited by 1 (0 self)
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High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. Such work may require several lines of proof for each line of code, and will usually be broken by the smallest change to the code (e.g. for maintenance or optimization purpose). Certifying these programs by hand is therefore very tedious and errorprone. This article discusses the use of the Gappa proof assistant in this context. Gappa has two main advantages over previous approaches: Its input format is very close to the actual C code to validate, and it automates error evaluation and propagation using interval arithmetic. Besides, it can be used to incrementally prove complex mathematical properties pertaining to the C code. Yet it does not require any specific knowledge about automatic theorem proving, and thus is accessible to a wide community. Moreover, Gappa may generate a formal proof of the results that can be checked independently by a lowerlevel proof assistant like Coq, hence providing an even higher confidence in the certification of the numerical code. The article demonstrates the use of this tool on a realsize example, an elementary function with correctly rounded output. 1
ONE METHOD FOR PROVING INEQUALITIES BY COMPUTER
, 2006
"... In this paper we consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program. 1. Some particular inequalities In this section we prove two new inequalities given in Theore ..."
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Cited by 1 (0 self)
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In this paper we consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program. 1. Some particular inequalities In this section we prove two new inequalities given in Theorem 1.2 and Theorem 1.10. While proving these theorems we use a method for inequalities of the following form f(x) ≥ 0, for the continues function f: [a, b] − → R. 1.1. Let us consider some inequalities for the gamma function which is defined by the integral: Γ(z) = e −t t z−1 (1) dt 0 which converges for Re(z)> 0. In the paper [10] the following statement is proved. Lemma 1.1 For x ∈ [0, 1] the following inequalities are true: