Provably faithful evaluation of polynomials

Boldo, Sylvie; Muñoz, César

doi:10.1145/1141277.1141586

Cited by 6 publications

(5 citation statements)

References 24 publications

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“…Using similar techniques, a number of algorithms with faithfully rounded result have been developed for several standard problems in numerical analysis. For example, Graillat gave in [10] Boldo and Muñoz showed in [3] a compensated polynomial evaluation to be faithful provided that k :"…”

Section: Faithful Results By a Simplified Pair Arithmeticmentioning

confidence: 99%

“…a method to evaluate t is available satisfying the estimate in (43) with appropriate interpretation. For the special case of IEEE-754 binary64 floating-point arithmetic with rounding to nearest, Assumption 5.7 is satisfied if the real resultĉ does not cause over-or underflow by setting v :" u{p1`uq for u :" 2´5 3 , by replacing c Ð a˝b by c " flpa˝bq and evaluating the expressions in (44) by appropriate error-free transformations 3 , possibly using the fused multiplyadd operation FMA.…”

Section: Faithful Results By a Simplified Pair Arithmeticmentioning

confidence: 99%

See 1 more Smart Citation

Error Bounds for Computer Arithmetics

Rump

2019

2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)

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This note summarizes recent progress in error bounds for compound operations performed in some computer arithmetic. Given a general set of real numbers together with some operations satisfying the first standard model, we identify three types A, B, and C of weak sufficient assumptions implying new results and sharper error estimates. Those include linearized error estimates in the number of operations, faithfully rounded and reproducible results. All types of assumptions are satisfied for an IEEE-754 p-digit base-β floating-point arithmetic.

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Section: Faithful Results By a Simplified Pair Arithmeticmentioning

confidence: 99%

Section: Faithful Results By a Simplified Pair Arithmeticmentioning

confidence: 99%

Error Bounds for Computer Arithmetics

Rump

2019

2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)

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“…Automatic treatment of all the following applications will use interval arithmetic on floating point arithmetic that has been presented in previous publications and is now available in formal tools [4,9,10]. Lévy's inequality works with independent symmetric random variables, as we safely assume in Sects.…”

Section: Applicationsmentioning

confidence: 99%

Improved bound for stochastic formal correctness of numerical algorithms

Daumas

Lester

Martin-Dorel

et al. 2010

Innovations Syst Softw Eng

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We provide bounds on the probability that accumulated errors were never above a given threshold on numerical algorithms. Such algorithms are used, for example, in aircraft and nuclear power plants. This report contains simple formulas based on Lévy's, Markov's and Hoeffding's inequalities and it presents a formal theory of random variables with a special focus on producing concrete results. We select three very common applications that cover the common practices of systems that evolve for a long time. We compute the number of bits that remain continuously significant in the first two applications with a probability of failure around one out of a billion, where worst case analysis considers that no significant bit remains. We are using PVS as such formal tools force explicit statement of all hypotheses and prevent incorrect uses of theorems.

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“…• Development of a fully functional floating point arithmetic library [33] in order to generate guaranteed proofs of round-off-errors [32]. • Integration of this library and an exact arithmetic formalization in PVS developed by one of the authors [34].…”

Section: Conclusion and Limits Of Tractabilitymentioning

confidence: 99%

Verified Real Number Calculations: A Library for Interval Arithmetic

Daumas

Lester

Muoz

2009

IEEE Trans. Comput.

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Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

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Provably faithful evaluation of polynomials

Cited by 6 publications

References 24 publications

Error Bounds for Computer Arithmetics

Error Bounds for Computer Arithmetics

Improved bound for stochastic formal correctness of numerical algorithms

Verified Real Number Calculations: A Library for Interval Arithmetic

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