  The Fixed Point Dirac Operator and Chirally Improved Fermions both use large
numbers of gauge paths and the full Dirac structure to approximate a solution
of the Ginsparg-Wilson equation. After a brief review of the two approaches we
present recent results for quenched QCD with pion masses down to 210 MeV. We
discuss the limits and advantages of approximate parameterizations and outline
future perspectives.
