We present a new heuristic point-to-point routing algorithm based on contraction hierarchies (CH). Given an epsilon >= 0, we can prove that the length of the path computed by our algorithm is at most (1+epsilon) times the length of the optimal (shortest) path. CH is based on node contraction: removing nodes from a network and adding shortcut edges to preserve shortest path distances. Our algorithm tries to avoid shortcuts even when a replacement path is epsilon times longer.