Suppose I want to design an algorithm that, for an arbitrary polynomial $p$, returns YES iff there are two roots $z_1$ and $z_2$ of $p$ such that $\left|z_1 - z_2\right| = 1$.
How do I design such an algorithm, given that any computerized root-finding algorithm will only give the roots of arbitrary $p$ approximately? It would seem that some error tolerance would be required -- say, returning YES iff $1 - \epsilon < |z_1 - z_2| < 1 + \epsilon$. But even then, how to decide which $\epsilon$ is appropriate to avoid false positives? Is there a way of bounding the probability of an incorrect decision given an error tolerance of $\epsilon$? Etc.