  We present an enhanced algorithm for exploring mirror symmetry for elliptic
curves through the correspondence of algebraic and tropical geometry, focusing
on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz
numbers. We present a new highly efficient algorithm for computing generating
series for these numbers. We have implemented the algorithm both using Singular
and OSCAR. The implementations outperform by far the current method provided in
Singular. The OSCAR implementation, benefiting in particular from just-in-time
compilation, again by far outperforms the implementation of the new algorithm
in Singular. This advancement in computing the Gromov-Witten invariants
facilitates a study of number theoretic and geometric properties of the
generating series, including quasi-modularity and homogeneity.
