We present an accelerated algorithm to forward-simulate origin--fixation models. Our algorithm requires on average only about two fitness evaluations per fixed mutation, whereas traditional algorithms require, per one fixed mutation, a number of fitness evaluations on the order of the effective population size Ne. Our accelerated algorithm yields the exact same steady state as the original algorithm but produces a different order of fixed mutations. By comparing several relevant evolutionary metrics, such as the distribution of fixed selection coefficients and the probability of reversion, we find that the two algorithms behave equivalently in many respects. However, the accelerated algorithm yields less variance in fixed selection coefficients. Notably, we are able to recover the expected amount of variance by rescaling population size, and we find a linear relationship between the rescaled population size and the population size used by the original algorithm. Considering the widespread usage of origin--fixation simulations across many areas of evolutionary biology, we introduce our accelerated algorithm as a useful tool for increasing the computational complexity of fitness functions without sacrificing much in terms of accuracy of the evolutionary simulation.