Exact Short Products From Truncated Multipliers

Abstract

Abstract We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage, e.g. a large integer (e.g. $5^{100}$) or an irrational number ($\pi $). We only need to access the most significant digits of the multiplier—as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.