  We show that the most general scalar-tensor theory of gravity up to four
derivatives in $3+1$ dimensions is well-posed in a modified version of the CCZ4
formulation of the Einstein equations in singularity-avoiding coordinates. We
demonstrate the robustness of our new formulation in practise by studying equal
mass black hole binary mergers for different values of the coupling constants.
Although our analysis of well-posedness is restricted to cases in which the
couplings are small, we find that in simulations we are able to push the
couplings to larger values, so that a certain weak coupling condition is order
one, without instabilities developing. Our work provides the means for such
simulations to be undertaken by the many numerical relativity codes that rely
on the moving puncture gauge to evolve black hole singularities.
