  Numerical relativity simulations of compact binaries with the Z4c and BSSNOK
formulations are compared. The Z4c formulation is advantageous in every case
considered. In simulations of non-vacuum spacetimes the constraint violations
due to truncation errors are between one and three orders of magnitude lower in
the Z4c evolutions. Improvements are also found in the accuracy of the computed
gravitational radiation. For equal-mass irrotational binary neutron star
evolutions we find that the absolute errors in phase and amplitude of the
waveforms can be up to a factor of four smaller. The quality of the Z4c
numerical data is also demonstrated by a remarkably accurate computation of the
ADM mass from surface integrals. For equal-mass non-spinning binary puncture
black hole evolutions we find that the absolute errors in phase and amplitude
of the waveforms can be up to a factor of two smaller. In the same evolutions
we find that away from the punctures the Hamiltonian constraint violation is
reduced by between one and two orders of magnitude. Furthermore, the utility of
gravitational radiation controlling, constraint preserving boundary conditions
for the Z4c formulation is demonstrated. The evolution of spacetimes containing
a single compact object confirm earlier results in spherical symmetry. The
boundary conditions avoid spurious and non-convergent effects present in high
resolution runs with either formulation with a more naive boundary treatment.
We conclude that Z4c is preferable to BSSNOK for the numerical solution of the
3+1 Einstein equations with the puncture gauge.
