We consider the reachability problem on semi-algebraic hybrid automata. In particular, we deal with the effective cost that has to be afforded to solve reachability through first-order satisfiability. The analysis we perform with some existing tools shows that even simple examples cannot be efficiently solved. We need approximations to reduce the number of variables in our formulae: this is the main source of time computation growth. We study standard approximation methods based on Taylor polynomials and ad-hoc strategies to solve the problem and we show their effectiveness on the repressilator case study.