Overflow and underflow in multiplication and division Original Research Article

In this paper the distribution of floating-point exponents and its effect on the frequency of underflow and overflow is studied. Previous work suggested that the uniform distribution of exponents was the only continuous one consistent with the logarithmic distribution of floating-point fractions. This paper takes that as a starting point and analyzes, both mathematically and computationally, how the distribution evolves as a result of repeated multiplications and divisions. We see that continuous and discrete models exhibit very similar behavior, generating subsequent distributions which are splines of increasing degree with characteristics similar to those of a normal distribution. These distributions are related to a system of functional differential equations. The analysis leads to a simple computational model which is used as a basis for the experimental results. These confirm that the likelihood of exponent spill is substantially reduced if the initial distribution is restricted to a small range. Even better results are obtained if the initial range is symmetric.