Mathematica's NIntegrate function is extremely fast when solving discontinuous, piecewise functions. Presumably, this is because of the SymbolicPiecewiseSubdivision routine it calls in order to separate the integral into disjoint, smooth, domains. How does this routine work in higher dimensions? All of the examples I've seen only use one-dimensional integrals, but the difference is still there for 3-dimensional integrals.

To be clear, I'm interested in the underlying transformation/mapping that SymbolicPiecewiseSubdivision uses; actually using the routine is mostly automatic.