The one-third power law models how human speed of motion depends on the path's curvature. This paper studies the interest of using this law for humanoid robot walking control along a planar reference trajectory. We predicted that humanoid robots following a reference trajectory may benefit from one-third power law speed profiles by reducing closed-loop drift and energy consumption. To robustly execute reference trajectories, we use contracting morphed Andronov-Hopf oscillators, regularized to follow a power law while converging to a planned cyclic trajectory. The walking pattern generator of HRP-2 uses these guiding dynamics to walk along elliptic trajectories. In dynamic simulation, we observe minimal geometric drift with the one-third power law, demonstrating increased precision compared with constant speed and other power laws. Closed-loop experiments on HRP-2 result in a small drift of all power law motions from the reference trajectory, showing the efficiency of the control architecture. We observe that the one-third power law controller demands less compensatory action, and therefore lowers the burden on the hardware. Slowing in curved movement regions also unexpectedly allows for faster overall movement.