Many Bayesian clinical trial methods are based on random inequalities. For some distribution families, these inequalities can be computed in closed form. For example,  gives closed-form solutions to computing
P(X > Y)
when X and Y are either independent normal or independent gamma random variables. However the case of beta random variables is very important and no closed form solution for this case is known. Such inequalities must be evaluated numerically. Simulation programs using these inequalities spend nearly all their time computing the inequalities. This report presents a close-form approximation for beta inequalities that is two orders of magnitude faster to evaluate.
Cook, John D., "Fast approximation of Beta inequalities" (October 2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 76.