Improving the numerical accuracy of programs by automatic transformation

Abstract

International audiencehe dangers of programs performing floating- point computations are well known. This is due to the sensitivity of the results to the way formulæ are written. These last years, several techniques have been proposed concerning the transformation of arithmetic expressions in order to improve their numerical accuracy and, in this arti- cle, we go one step further by automatically transforming larger pieces of code containing assignments and control structures. We define a set of transformation rules allowing the generation, under certain conditions and in polynomial time, of larger expressions by performing limited formal computations, possibly among several iterations of a loop. These larger expressions are better suited to improve, by reparsing, the numerical accuracy of the program results. We use abstract interpretation-based static analysis techniques to over-approximate the round-off errors in programs and during the transformation of expressions. A tool has been implemented and experimental results are presented con- cerning classical numerical algorithms and algorithms for embedded systems

Eliminating Unstable Tests in Floating-Point Programs

“…Then, it selects the ones for which a better accuracy bound can be proved. Salsa [4] combines Sardana with techniques for intra-procedure [5] and inter-procedure [6,7] program transformation in order to improve the accuracy of a target variable in larger pieces of code containing assignments and control structures. To the best of the authors' knowledge, the program transformation proposed in this work is the only approach that addresses the problem of conditional instability for floating-point programs.…”

LCV: A Verification Tool for Linear Controller Software

“…Code generation tools such as Embedded Coder [30] and Simulink Coder [32] are widely used. The generated controller implementation is either used as it is in the control system, or sometimes transformed into another code before used for various reasons such as numerical accuracy improvement [8,9] and code protection [2,4,5]. For simplicity's sake herein, we will call code generation even when code generation is potentially followed by code transformation.…”

Improving the Numerical Accuracy of High Performance Computing Programs by Process Specialization

“…It reduces partly the round-off errors by automatically transforming C programs in a source-to-source manner. We have defined a set of intraprocedural transformation rules [9] like assignments, conditionals, loops, etc., and interprocedural transformation rules [10] for functions and other rules which deal with arrays. These rules have been implemented in the Salsa tool.…”

“…To cope with this limitation, we have proposed an automatic tool, Salsa, which takes programs made of assignments, conditionals, loops, functions, arrays, etc., and generates another program numerically more accurate using a source-to-source transformation. This is possible thanks to the set of intraprocedural and interprocedural transformation rules defined in previous work [10,9]. Salsa relies on static analysis by abstract interpretation [4] to compute variable ranges and round-off error bounds.…”

“…An extended description is given in [10]. We give an overview of the formal intraprocedural [9] and interprocedural [10] rules used in our transformation as well as on how the transformation of expressions is done [20].…”