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Proving bounds on realvalued functions with computations
 4th International Joint Conference on Automated Reasoning. Volume 5195 of Lecture Notes in Artificial Intelligence
, 2008
"... Abstract. Intervalbased methods are commonly used for computing numerical bounds on expressions and proving inequalities on real numbers. Yet they are hardly used in proof assistants, as the large amount of numerical computations they require keeps them out of reach from deductive proof processes. ..."
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Abstract. Intervalbased methods are commonly used for computing numerical bounds on expressions and proving inequalities on real numbers. Yet they are hardly used in proof assistants, as the large amount of numerical computations they require keeps them out of reach from deductive proof processes. However, evaluating programs inside proofs is an efficient way for reducing the size of proof terms while performing numerous computations. This work shows how programs combining automatic differentiation with floatingpoint and interval arithmetic can be used as efficient yet certified solvers. They have been implemented in a library for the Coq proof system. This library provides tactics for proving inequalities on realvalued expressions. 1
Emulation of a FMA and CorrectlyRounded Sums: Proved Algorithms Using Rounding to Odd
 IEEE Trans. Computers
, 2008
"... Rounding to odd is a nonstandard rounding on floatingpoint numbers. By using it for some intermediate values instead of rounding to nearest, correctly rounded results can be obtained at the end of computations. We present an algorithm to emulate the fused multiplyandadd operator. We also present ..."
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Rounding to odd is a nonstandard rounding on floatingpoint numbers. By using it for some intermediate values instead of rounding to nearest, correctly rounded results can be obtained at the end of computations. We present an algorithm to emulate the fused multiplyandadd operator. We also present an iterative algorithm for computing the correctly rounded sum of a set floatingpoint numbers under mild assumptions. A variation on both previous algorithms is the correctly rounded sum of any three floatingpoint numbers. This leads to efficient implementations, even when this rounding is not available. In order to guarantee the correctness of these properties and algorithms, we formally proved them using the Coq proof checker.
Floatingpoint arithmetic in the Coq system
"... The process of proving some mathematical theorems can be greatly reduced by relying on numericallyintensive computations with a certified arithmetic. This article presents a formalization of floatingpoint arithmetic that makes it possible to efficiently compute inside the proofs of the Coq system. T ..."
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The process of proving some mathematical theorems can be greatly reduced by relying on numericallyintensive computations with a certified arithmetic. This article presents a formalization of floatingpoint arithmetic that makes it possible to efficiently compute inside the proofs of the Coq system. This certified library is a multiradix and multiprecision implementation free from underflow and overflow. It provides the basic arithmetic operators and a few elementary functions. 1
Multiplications of Floating Point Expansions
 IN PROCEEDINGS OF THE 14TH SYMPOSIUM ON COMPUTER ARITHMETIC, I. KOREN AND P. KORNERUP (EDS
, 1999
"... In modern computers, the floating point unit is the part of the processor delivering the highest computing power and getting most attention from the design team. Performance of any multiple precision application will be dramatically enhanced by adequate use of floating point expansions. We present i ..."
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In modern computers, the floating point unit is the part of the processor delivering the highest computing power and getting most attention from the design team. Performance of any multiple precision application will be dramatically enhanced by adequate use of floating point expansions. We present in this work three multiplication algorithms faster and more integrated than the stepwise algorithm proposed earlier. We have tested these new algorithms on an application that computes the determinant of a matrix. In the absence of overflow or underflow, the process is error free and possibly more efficient than its integer based counterpart.
Stochastic Formal Methods: An application to accuracy of numeric software
, 2006
"... Abstract — This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safetycritical software, as these systems are now running fast enough and long eno ..."
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Abstract — This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safetycritical software, as these systems are now running fast enough and long enough for their errors to impact on their functionality. Furthermore, worstcase analysis would blindly advise the replacement of existing systems that have been successfully running for years. We present here a set of formal theorems validated by the PVS proof assistant. These theorems will allow code analyzing tools to produce formal certificates of accurate behavior. For example, FAA regulations for aircraft require that the probability of an error be below 10 −9 for a 10 hour flight [1]. I.
Provably faithful evaluation of polynomials
 In Proceedings of the 21st Annual ACM Symposium on Applied Computing
, 2006
"... We provide sufficient conditions that formally guarantee that the floatingpoint computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers and rounding modes in the Program Verification System (PVS). Our work is based on a wellknown formali ..."
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We provide sufficient conditions that formally guarantee that the floatingpoint computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers and rounding modes in the Program Verification System (PVS). Our work is based on a wellknown formalization of floatingpoint arithmetic in the proof assistant Coq, where polynomial evaluation has been already studied. However, thanks to the powerful proof automation provided by PVS, the sufficient conditions proposed in our work are more general than the original ones.
Bridging the gap between formal specification and bitlevel floatingpoint arithmetic
"... Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defin ..."
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Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defined functions for any rounding mode described by the standard. Our library can now be applied to certify
both software and hardware. Developing results in those two dramatically different
directions, like no other formal development so far, guarantees that nothing was
forgotten or poorly specified in our formalization. It also lets us compare our work
with the existing bitlevel formalizations developed with other proof assistants.
Formally Verified Argument Reduction with a FusedMultiplyAdd, in
 n o 8, 2009, p. 11391145, http://arxiv.org/abs/0708.3722 US . Activity Report INRIA 2009
"... Abstract — Cody & Waite argument reduction technique works perfectly for reasonably large arguments but as the input grows there are no bit left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fusedmultiplyadd provides a fully a ..."
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Abstract — Cody & Waite argument reduction technique works perfectly for reasonably large arguments but as the input grows there are no bit left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fusedmultiplyadd provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach double full accuracy (all the significand bits of two numbers are significant) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker. Index Terms — Argument reduction, fma, formal proof, Coq. I.
When double rounding is odd
, 2004
"... Double rounding consists in a first rounding in an intermediate extended precision and then a second rounding in the working precision. The natural question is then of the precision and correctness of the final result. Unfortunately, the used double rounding algorithms do not obtain a correct roundi ..."
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Double rounding consists in a first rounding in an intermediate extended precision and then a second rounding in the working precision. The natural question is then of the precision and correctness of the final result. Unfortunately, the used double rounding algorithms do not obtain a correct rounding of the initial value. We prove an efficient algorithm for the double rounding to give the correct rounding to the nearest value assuming the first rounding is to odd. As this rounding is unusual and this property is surprising, we formally proved this property using the Coq automatic proof checker. Keywords: Floatingpoint, double rounding, formal proof, Coq. Résumé Le double arrondi consiste en un premier arrondi dans une précision étendue suivi d’un second arrondi dans la précision de travail. La question naturelle qui se pose est celle de la précision et de la correction du résultat final ainsi obtenu. Malheureusement, les algorithmes de double arrondi utilisés ne produisent pas un arrondi correct de la valeur initiale. Nous décrivons ici un algorithme efficace de double arrondi permettant de fournir l’arrondi correct au plus proche à condition que le premier arrondi soit impair. Comme cet arrondi est inhabituel et que cette propriété est surprenante, nous avons prouvé formellement ce résultat en utilisant l’assistant de preuves Coq. Motsclés: Virgule flottante, double arrondi, preuve formelle, Coq. When double rounding is odd 1