Approximate a float using a minimal fraction
The partial convergents of the continued fraction of $x$ consists of all the best rational approximations of $x$; see Wikipedia, for example. A best rational approximation of $x$ is a rational number $p/q$ such that $\left|x-\frac{p}{q}\right| \leq \left|x-\frac{p'}{q'}\right|$ for all $q' \leq q$. Your $p/q$ is in particular a best rational approximation, so it would be one of the convergents.