Results 1  10
of
22
Why: A MultiLanguage MultiProver Verification Tool
"... This article introduces the verification tool Why. This tool produces verification conditions from annotated programs given as input. ..."
Abstract

Cited by 39 (0 self)
 Add to MetaCart
This article introduces the verification tool Why. This tool produces verification conditions from annotated programs given as input.
An overview of semantics for the validation of numerical programs
, 2005
"... Abstract. In this article, we introduce a simple formal semantics for floatingpoint numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enable ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
Abstract. In this article, we introduce a simple formal semantics for floatingpoint numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enables us to formally compare the properties calculated in each semantics to our reference, simple semantics. Most of these methods having been developed to verify numerical intensive codes, we also discuss their adequacy to the formal validation of softwares and to static analysis. Finally, this study is completed by experimental results. 1
MultiProver Verification of FloatingPoint Programs ⋆
"... Abstract. In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a firstorder axiomatization of floatingpoint operations which allows to reduce verifica ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
Abstract. In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a firstorder axiomatization of floatingpoint operations which allows to reduce verification to checking the validity of logic formulas, in a suitable form for a large class of provers including SMT solvers and interactive proof assistants. Experiments using the FramaC platform for static analysis of C code are presented. 1
Combining Coq and Gappa for Certifying FloatingPoint Programs ⋆
"... Abstract. Formal verification of numerical programs is notoriously difficult. On the one hand, there exist automatic tools specialized in floatingpoint arithmetic, such as Gappa, but they target very restrictive logics. On the other hand, there are interactive theorem provers based on the LCF approa ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Abstract. Formal verification of numerical programs is notoriously difficult. On the one hand, there exist automatic tools specialized in floatingpoint arithmetic, such as Gappa, but they target very restrictive logics. On the other hand, there are interactive theorem provers based on the LCF approach, such as Coq, that handle a generalpurpose logic but that lack proof automation for floatingpoint properties. To alleviate these issues, we have implemented a mechanism for calling Gappa from a Coq interactive proof. This paper presents this combination and shows on several examples how this approach offers a significant speedup in the process of verifying floatingpoint programs. 1
Formal Verification of the VAMP Floating Point Unit
 In CHARME 2001, volume 2144 of LNCS
, 2001
"... We report on the formal verification of the floating point unit used in the VAMP processor. The FPU is fully IEEE compliant, and supports denormals and exceptions in hardware. The supported operations are addition, subtraction, multiplication, division, comparison, and conversions. The hardware is v ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
We report on the formal verification of the floating point unit used in the VAMP processor. The FPU is fully IEEE compliant, and supports denormals and exceptions in hardware. The supported operations are addition, subtraction, multiplication, division, comparison, and conversions. The hardware is verified on the gate level against a formal description of the IEEE standard by means of the theorem prover PVS.
Semantics of roundoff error propagation in finite precision computations
 Journal of Higher Order and Symbolic Computation
, 2006
"... Abstract. We introduce a concrete semantics for floatingpoint operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it. In our model, every elementary ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
Abstract. We introduce a concrete semantics for floatingpoint operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it. In our model, every elementary operation introduces a new first order error term, which is later propagated and combined with other error terms, yielding higher order error terms. The semantics is parameterized by the maximal order of error to be examined and verifies whether higher order errors actually are negligible. We consider also coarser semantics computing the contribution, to the final error, of the errors due to some intermediate computations. As a result, we obtain a family of semantics and we show that the less precise ones are abstractions of the more precise ones.
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. ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
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
Theorems on efficient argument reductions
 Proceedings of the 16th IEEE Symposium on Computer Arithmetic (ARITH16
, 2003
"... A commonly used argument reduction technique in elementary function computations begins with two positive floating point numbers α and γ that approximate (usually irrational but not necessarily) numbers 1/C and C, e.g., C = 2π for trigonometric functions and ln 2 for e x. Given an argument to the fu ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
A commonly used argument reduction technique in elementary function computations begins with two positive floating point numbers α and γ that approximate (usually irrational but not necessarily) numbers 1/C and C, e.g., C = 2π for trigonometric functions and ln 2 for e x. Given an argument to the function of interest it extracts z as defined by xα = z + ς with z = k2 −N and ς  ≤ 2 −N−1, where k, N are integers and N ≥ 0 is preselected, and then computes u = x − zγ. Usually zγ takes more bits than the working precision provides for storing its significand, and thus exact x − zγ may not be represented exactly by a floating point number of the same precision. This will cause performance penalty when the working precision is the highest available on the underlying hardware and thus considerable extra work is needed to get all the bits of x − zγ right. This paper presents theorems that show under mild conditions that can be easily met on today’s computer hardware and still allow α ≈ 1/C and γ ≈ C to almost the full working precision, x − zγ is a floating point number of the same precision. An algorithmic procedure based on the theorems is obtained. The results will enhance performance, in particular on machines that has hardware support for fusedmultiplyadd (fma) instruction(s). 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 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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.