This refers to the period before a common apparent horizon is found. Once a common apparent horizon is found, then pretracking can be disabled as the apparent horizon finder can easily “track” the apparent horizon’s motion from one time step to the next. With a binary search the number of iterations depends only weakly (logarithmically) on the pretracking algorithm’s accuracy tolerance. It might be possible to replace the binary search by a more sophisticated 1-dimensional search algorithm (I discuss such algorithms in Appendix A), potentially cutting the number of iterations substantially. This might be a fruitful area for further research.