We present an algorithm for the bounded unification of higher-order terms.The algorithm extends G. P. Huet's pre-unification algorithm with rules for the generation and folding of regular terms.The concise form of the algorithm allows the reuse of the pre-unification correctness proof. Furthermore, the regular termscan be restricted in order to decide the unifiability problem.Finally, the algorithm avoids re-computation of terms in a non-deterministic search which leads to a better performance in practice when compared to other boundedunification algorithms.