A New Abstract Domain for the Representation of Mathematically Equivalent Expressions

Arnault Ioualalen (1, 2) , Matthieu Martel (2, 1)

1 UPVD - Université de Perpignan Via Domitia
2 DALI - Digits, Architectures et Logiciels Informatiques

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 floating-point arithmetic is not intuitive, these errors are very di cult 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, roundo 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 di erent shapes and accuracies. Then, we extract optimized expressions from APEGs by searching the most accurate concrete expressions among the set of represented expressions.

Mots clés: numerical accuracy; ation; Static analysis; abstract interpretation; floating-point arithmetic; code synthesis; program transfor

Domaines: Arithmétique des ordinateurs

Arnault Ioualalen, Matthieu Martel. A New Abstract Domain for the Representation of Mathematically Equivalent Expressions. SAS: Static Analysis Symposium, Sep 2012, Deauville, France. pp.75-93. ⟨hal-00698618⟩