Enhancing the implementation of mathematical formulas for fixed-point and floating-point arithmetics

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

2009

Self Cite

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.

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Program transformation for numerical precision

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

2009

Self Cite

“…Our work concerns the synthesis at compile-time of accurate formulas, for given input ranges, to replace the expressions written by the programmers in source codes [10]. We consider that a program would return an exact result if the computations were carried out using real numbers.…”

Section: Introductionmentioning

confidence: 99%

“…An APEG is an abstraction of an exponential number of expressions and the profitability has to extract an accurate formula. We compute safe error bounds using established static analysis techniques for numerical accuracy [10] and we use a limited depth search algorithm to explore the APEG structure. In addition, APEGs contain abstraction boxes representing any parsing of a sequence of operations defined by a set of operands and one commutative operator.…”

Section: Introductionmentioning

confidence: 99%

Synthesizing accurate floating-point formulas

Ioualalen

2013 IEEE 24th International Conference on Application-Specific Systems, Architectures and Processors

2013

Self Cite

Abstract-Many critical embedded systems perform floatingpoint computations yet their accuracy is difficult to assert and strongly depends on how formulas are written in programs. In this article, we focus on the synthesis of accurate formulas mathematically equal to the original formulas occurring in source codes. In general, an expression may be rewritten in many ways. To avoid any combinatorial explosion, we use an intermediate representation, called APEG, enabling us to represent many equivalent expressions in the same structure. In this article, we specifically address the problem of selecting an accurate formula among all the expressions of an APEG. To validate our approach, we present experimental results showing how APEGs, combined with profitability analysis, make it possible to significantly improve the accuracy of floating-point computations.

A New Abstract Domain for the Representation of Mathematically Equivalent Expressions

Ioualalen

2012

Static Analysis

Self Cite

Abstract. Exact computations being in general not tractable for computers, they are approximated by floating-point computations. This is the source of many errors in numerical programs. Because the floatingpoint arithmetic is not intuitive, these errors are very difficult to detect and to correct by hand and we consider the problem of automatically synthesizing accurate formulas. We consider that a program would return an exact result if the computations were carried out using real numbers. In practice, roundoff errors arise during the execution and these errors are closely related to the way formulas are written. Our approach is based on abstract interpretation. We introduce Abstract Program Equivalence Graphs (APEGs) to represent in polynomial size an exponential number of mathematically equivalent expressions. The concretization of an APEG yields expressions of very different shapes and accuracies. Then, we extract optimized expressions from APEGs by searching the most accurate concrete expressions among the set of represented expressions.