We propose a general methodology to speed up the test generation process for circuits with high-level primitives. Our search procedure is a variation of depth first search that tries to fully exploit the capabilities of a computer to execute complex arithmetic and logical operations. We present techniques for signal value justification, and fault propagation, which are used by our algorithm. We have implemented a dependency-directed backtracking method to speed up our algorithm. This methodology has been applied to six circuits and the results are found to be very encouraging.