The inverse kinematics of five-axis milling machines produces large errors near stationary points of the required surface. When the tool travels across or around the point the rotation angles may jump considerably leading to unexpected deviations from the prescribed trajectories. We propose two new algorithms to repair the trajectories by adjusting the rotation angles in such a way that the kinematics error is minimized. The angle switching algorithm computes all feasible rotations and identifies an optimized sequence of rotations by the shortest path scheme. Further error reduction is accomplished by the angle insertion algorithm based on the equi distribution principle applied to rotations near the singularity. We have verified the algorithms by two five-axis milling machines, namely, MAHO600E at the CIM Lab of Asian Institute of Technology and HERMLE UWF920H at the CIM Lab of Kasetsart University.