For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$.
In particular, I'd like to generate approximations where $r$, $s$ and $t$ are moderately sized but for which the sign of $r + s\sqrt p + t\sqrt q$ cannot be resolved correctly using double precision arithmetic.
I'd also be interested in specific examples, e.g. for $p = 2$, $q = 3$.