A Quest for Exactness: Program Transformation for Reliable Real Numbers. (Transformation de Programmes pour des Nombres Réels Fiables)

This thesis presents an algorithm that eliminates square root and division op-erations in some straight-line programs used in embedded systems while pre-serving the semantics. Eliminating these two operations allows to avoid errorsat runtime due to rounding. These errors can lead to a completely unexpectedbehavior from the program. This transformation respects the constraints of em-bedded systems, such as the need for the program to be executed in a fixed sizememory. The transformation uses two fundamental algorithms developed in thisthesis. The first one allows to eliminate square roots and divisions from Booleanexpressions built with comparisons of arithmetic expressions. The second one isan algorithm that solves a particular anti-unification problem, that we call con-strained anti-unification. This program transformation is defined and proven inthe PVS proof assistant. It is also implemented as a strategy for this system. Con-strained anti-unification is also used to extend this transformation to programscontaining functions. It allows to eliminate square roots and divisions from PVSspecifications. Robustness of this method is highlighted by a major example: theelimination of square roots and divisions in a conflict detection algorithm used inaeronautics.tel-00924379,version1-6Jan2014