Assurance of Accuracy in Floating-Point Calculations - A Software Model Study

Since the days of the earliest computers, floating-point arithmetic has been a workable solution to represent real numbers with a finite number of bits, though it cannot precisely represent all real numbers. This results in invisible, unknowable, and sometimes catastrophic error - particularly in complex calculations - when an insufficient number of significant digits have been retained. This paper describes a novel bounded floating-point extension of the current floating-point standard that makes this error visible and knowable. It ensures accuracy by bounding a calculation result between a lower bound and an upper bound on the real value represented. A bounded floating-point software model (emulating the hardware implementation of the bounded floating-point system) is presented that can identify the loss of significant digits and pinpoint failure points. By utilizing this model, the superiority of the Kahan thin triangle algorithm versus earlier Heron algorithm is shown.