A Formalization of Floating-Point Numeric Base Conversion

The process of converting arbitrary real numbers into a floating-point format is formalized as a mapping of the reals into a specified subset of real numbers. The structure of this subset, the set of n significant digit base β floating-point numbers, is analyzed and properties of conversion mappings are determined. For a restricted conversion mapping of the n significant digit base δ numbers to the m significant-digit base δ numbers, the one-to-one, onto, and order- preserving properties of the mapping are summarized. Multiple conversions consisting of a composition of individual conversion mappings are investigated and some results of the invariant points of such compound conversions are presented. The hardware and software implications of these results with regard to establishing goals and standards for floating-point formats and conversion procedures are considered.