In this article, we investigate both over- and under-approximations of reachable sets for analytic autonomous dynamical systems beyond polynomial dynamics. We start with the concept of evolution function, whose subzero-level set can be used to describe reachable set, and find a series representation of the evolution function with its Lie derivatives. Afterwards, based on the partial sums of this series, two different methodologies are introduced to compute over- and under-approximations of reachable sets, using numerical quantifier elimination for the semi-algebraic constraints and remainder estimation of the partial sum, respectively. Some benchmarks are given, including an eight-dimensional nonpolynomial quadrotor model, to show the advantages of our computational methods over some existing methods in the literature. Especially, our methods also work for nonconvex initial sets.