A Formalization of Floating-Point
Numeric Base Conversion · 1970
Theprocess ofconverting arbitrary realnumbersinto a floating-point formatisformalized as a mappingoftherealsintoa specified subsetofrealnumbers.Thestructureofthissubset, theset ofn significant digitbase floating-point numbers,isanalyzed and properties ofconversion mappingsare determined. Fora restricted conversion mappingofthen significant digitbase,Bnumberstothe m significant-digit base6 numbers,theone-to-one,onto,andorder- preserving proper.ties ofthemappingare summarized. Multiple con- versions consisting ofa composition ofindividual conversion map- pingsareinvestigated andsome results oftheinvariant pointsofsuch compoundconversions are presented. The hardwareandsoftware implications oftheseresults withregardto establishing goalsand standards forfloating-point formatsandconversion procedures are considered. IndexTerms-Accumulated error ofsuccessive conversions, base conversion, equivalent digitsformula,floating-point numbers, mixed-base computational environments, round-off error,significant digits.