The answer to What is the probability P(X > Y) given X ~ Be(a1, b1), and Y ~ Be(a2, b2), and X and Y are independent?What is the probability P(X > Y) given X ~ Be(a1, b1), and Y ~ Be(a2, b2), and X and Y are independent? provides an analytical solution for this, but is there a less computationally intensive expression if one is willing to lose some precision?
Assume integer a and b, if that helps.