We study the problem of computing the reachable principals of the simulation preorder and the reachable blocks of simulation equivalence. Following a theoretical investigation of this problem, which %, in particular, highlights a sharp contrast with the already settled case of bisimulation, we design algorithms to solve this problem by leveraging the idea of interleaving reachability and simulation computation while possibly avoiding the computation of all the reachable states or the whole simulation preorder. In particular, we put forward a symbolic algorithm processing state partitions and, in turn, relations between their blocks, which is suited for processing infinite-state systems.