Feynman's "little trick" with $\epsilon/2$ is actually called the Leapfrog method (or Leapfrog integration). Unlike the first method introduced (which is named Euler integration by the way), Leapfrog is stable for any repeating (oscillatory) motion (if the timestep $\epsilon$ satisfies some conditions, namely $\epsilon \lt \frac{2}{\omega}$, where $\omega$ is a property that is related to how much time the motion takes to "repeat" and return to its initial state; in the above simulation, $\omega = 1$, so $\epsilon$ has to be smaller than $\frac{2}{1} = 2$, otherwise the Leapfrog integration becomes unstable).