When working in classical finite precision arithmetic, one is often faced with the following problem: the result of a computation cannot be represented exactly. In such cases, one usually has the goal of constructing the computation algorithm so that the output is the floating-point number nearest to the exact result, where “nearest” is determined according to the rounding mode selected by the user. Ziv developed a now-classical test that can be used to find out whether or not a given approximate solution to a calculation has this desirable property [1]. An interesting extension of Ziv’s approach arises when the algorithm’s input values are assumed to be of a higher precision than the output data.

A crucial parameter in all variants of Ziv’s test is a so-called “magic constant” that needs to be chosen by the user in a very careful way. If it is too small, then the test will be worthless because it can produce falsely positive results. In other words, incorrectly rounded values will not be recognized as wrong and thus will be accepted. This is an outcome that should certainly be avoided. On the other hand, a too-large value of the magic constant will lead to a high probability of false negatives, meaning correct results will not be recognized as such. In those cases, the user will be unnecessarily forced to repeat his or her computation with smaller tolerances, thus increasing the computational cost.

In this paper, de Dinechin et al. provide a detailed investigation of the properties of the magic constant and derive a method for choosing this parameter in a suitable way. Moreover, they demonstrate that their ideas lead to an almost optimal choice of the magic constant.