  We present a method for solving impurity models with electron-phonon
coupling, which treats the phonons efficiently and without approximations. The
algorithm is applied to the Holstein-Hubbard model in the dynamical mean field
approximation, where it allows access to strong interactions, very low
temperatures and arbitrary fillings. We show that a renormalized
Migdal-Eliashberg theory provides a reasonlable description of the phonon
contribution to the electronic self energy in strongly doped systems, but fails
if the quasiparticle energy becomes of order of the phonon frequency.
