  Nearly Euclidean Thurston (NET) maps are described by simple diagrams which
admit a natural notion of size. Given a size bound $C$, there are finitely many
diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of
size at most $C$, the problem of determining whether $F$ is equivalent to a
rational function is, in theory, a finite computation. We give bounds for the
size of this computation in terms of $C$ and one other natural geometric
quantity. This result partially explains the observed effectiveness of the
computer program NETmap in deciding rationality.
