An Overview of Semantics for the Validation of Numerical Programs

Martel, Matthieu

doi:10.1007/978-3-540-30579-8_4

Cited by 19 publications

(17 citation statements)

References 25 publications

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“…There are also a large body of work on abstract interpretation, SMT solving, model checking, and code perturbation to tackle the representation error problem [5,9,12,14,16,21,27,28,30,35]. Particularly, robustness analysis [7] tries to statically prove that a floating point program is free from instability problems.…”

Section: Related Workmentioning

confidence: 99%

On-the-fly detection of instability problems in floating-point program execution

BaoTao

ZhangXiangyu

2013

SIGPLAN Not.

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The machine representation of floating point values has limited precision such that errors may be introduced during execution. These errors may get propagated and magnified by the following operations, leading to instability problems, e.g., control flow path may be undesirably altered and faulty output may be emitted. In this paper, we develop an onthe-fly efficient monitoring technique that can predict if an execution is stable. The technique does not explicitly compute errors as doing so incurs high overhead. Instead, it detects possible places where an error becomes substantially inflated regarding the corresponding value, and then tags the value with one bit to denote that it has an inflated error. It then tracks inflation bit propagation, taking care of operations that may cut off such propagation. It reports instability if any inflation bit reaches a critical execution point, such as a predicate, where the inflated error may induce substantial execution difference, such as different execution paths. Our experiment shows that with appropriate thresholds, the technique can correctly detect that over 99.999996% of the inputs of all the programs we studied are stable while a traditional technique relying solely on inflation detection mistakenly classifies majority of the inputs as unstable for some of the programs. Compared to the state of the art technique that is based on high precision computation and causes several hundred times slowdown, our technique only causes 7.91 times slowdown on average and can report all the true unstable executions with the appropriate thresholds.

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Section: Related Workmentioning

confidence: 99%

On-the-fly detection of instability problems in floating-point program execution

BaoTao

ZhangXiangyu

2013

SIGPLAN Not.

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“…The semantics of the other elementary operations (division and square root) relies on a power series development and is more complicated. It is fully explicited in [16,17].…”

Section: Numerical Domainsmentioning

confidence: 99%

“…This latter information is of great interest to improve the accuracy of the implementation. More generally, program transformations to detect numerical errors are runtime are discussed in [3] and other methods, not based on static analysis, are compared in [16]. Concerning the transformation of programs in order to enhance their numerical accuracy, there only exists non-automatic methods dedicated to specific classes of formulas, for example to improve the evaluation of polynomial expressions [4,13].…”

Section: Introductionmentioning

confidence: 99%

Program transformation for numerical precision

Martel

2009

Proceedings of the 2009 ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation

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This article introduces a new program transformation in order to enhance the numerical accuracy of floating-point computations. We consider that a program would return an exact result if the computations were carried out using real numbers. In practice, roundoff errors due to the finite representation of values arise during the execution. These errors are closely related to the way formulas are evaluated. Indeed, mathematically equivalent formulas, obtained using laws like associativity, distributivity, etc., may lead to very different numerical results in the computer arithmetic. We propose a semantics-based transformation in order to optimize the numerical accuracy of programs. This transformation is expressed in the abstract interpretation framework and it aims at rewriting pieces of numerical codes in order to obtain results closer to what the computer would output if it used the exact arithmetic.

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“…So ASTRÉE will detect catastrophic losses of precision leading to overflows, and their significance and source. Static analyzers like FLUCTUAT [55] are more specifically designed to analyze the relative contributions of the rounding errors at all stages of a floating point computation.…”

Section: Floatsmentioning

confidence: 99%