Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error Bound

Abstract

The evaluation of special functions often involves the evaluation of numerical constants. When the precision of the evaluation is known in advance (e.g., when developing libms) these constants are simply precomputed once and for all. In contrast, when the precision is dynamically chosen by the user (e.g., in multiple precision libraries), the constants must be evaluated on the fly at the required precision and with a rigorous error bound. Often, such constants are easily obtained by means of formulas involving simple numbers and functions. In principle, it is not a difficult work to write multiple precision code for evaluating such formulas with a rigorous round off analysis: one only has to study how round off errors propagate through sub expressions. However, this work is painful and error-prone and it is difficult for a human being to be perfectly rigorous in this process. Moreover, the task quickly becomes impractical when the size of the formula grows. In this article, we present an algorithm that takes as input a constant formula and that automatically produces code for evaluating it in arbitrary precision with a rigorous error bound. It has been implemented in the Solly a free software tool and its behavior is illustrated on several examples.