This paper is concerned with separation bounds--how small can a nonzero number defined by a certain kind of expression be? Put another way, how accurate do we need to be to answer with positive, negative, or zero? Such questions are fundamental to computational geometry, since getting them wrong can lead to structures that are topologically incoherent. The starting point is a bound [1] for expressions involving the four arithmetic operations and radicals.

This bound is extended to include “the k-th real root of a polynomial,” and is made linear even if division is present. The key to this improvement lies in, for example, writing √a/b as a√ab rather than √a/√b. The authors compare their bound with the previous one [1] and one from Li and Yap’s paper [2]; however, their new bound is both theoretically and practically incomparable with the latter, since Li and Yap’s bound is sometimes better and sometimes worse.

It should be noted: although the paper’s title refers to real expressions, the methodology is equally applicable to complex ones. Also, “log” means “log₂” throughout the paper, and the authors sometimes optimize out divisions for the older work [1].