An important criteria to assert the security of a cryptographic primitive is its resistance against differential cryptanalysis. For word-oriented primitives, a common technique to determine the number of rounds required to ensure the immunity against differential distinguishers is to consider truncated differential characteristics and to count the number of active S-boxes. Doing so allows one to provide an upper bound on the probability of the best differential characteristic with a reduced computational cost. However, in order to design very efficient primitives, it might be needed to evaluate the probability more accurately. This is usually done in a second step, during which one tries to instantiate truncated differential characteristics with actual values and computes its corresponding probability. This step is usually done either with ad-hoc algorithms or with CP, SAT or MILP models that are solved by generic solvers. In this paper, we present a generic tool for automatically generating these models to handle all word-oriented ciphers. Furthermore the running times to solve these models are very competitive with all the previous dedicated approaches.