Gradual and tapered overflow and underflow: A functional differential equation and its approximation

Abstract

A modified structure for floating-point representation, which can essentially eliminate overflow from floating-point calculations, is analyzed. The main part of the paper deals with a computational examination of a model for floating-point exponents and the probabilities of overflow and underflow. Earlier results in the absence of gradual underflow are presented for comparison but the primary focus is on the effect of gradual underflow, a version of gradual overflow, and of a proposal for an extended treatment of these exceptions which we call tapered overflow and underflow. The latter virtually eliminates the exceptions; its overflow threshold is 10 600 000 000 . This paper reviews several models for the distribution of exponents of floating-point numbers and their evolution in the presence of repeated multiplicative operations. A continuous model, which closely approximates the discrete case, is seen to satisfy a functional differential equation with both delay and advance terms.