  The search for feebly-interacting new-physics particles in the MeV-GeV mass
range often involves high-intensity beams dumped into thick heavy targets. The
challenge of evaluating the expected backgrounds for these searches from first
principles is limited by the CPU time needed to generate the shower induced by
the primary beam. We present a Monte Carlo biasing method allowing a three
orders of magnitude increase in the efficiency for the simulation of the muon
production in a 400 GeV$/c$ proton beam-dump setup. At the same time, this
biasing method is maintaining nearly every feature of a simulation from first
principles.
