  We construct new Ginsparg-Wilson fermions for QCD by inserting an
approximately chiral Dirac operator - which involves ingredients of a perfect
action - into the overlap formula. This accelerates the convergence of the
overlap Dirac operator by a factor of 5 compared to the standard construction,
which inserts the Wilson fermion as a point of departure. Taking into account
the effort for treating the improved fermion, we are left with an total
computational overhead of about a factor 3. This remaining factor is likely to
be compensated by other virtues; here we show that the level of locality is
clearly improved, so that the exponent of the correlation decay is doubled. We
also show that approximate rotation invariance is drastically improved, but a
careful scaling test has to be postponed.
