Changes

<math>z \in [-\infty,z_m] \quad r^*=0</math>

<math>z \in (z_m,y] \quad r^*=\alpha (x-y)</math>

<math>z \in (y,\frac{x+y}{2}] \quad r^*=2 \alpha \left (\frac{x+y}{2} + z \right )</math>

<math>z \in (\frac{x+y}{2},\infty] \quad r^*=0</math>

In the case where <math>y > 0 \quad r^*=\frac{\alpha}{4}\left ( x+y \right )^2 - \alpha y^2</math>

In the case where <math>y \le 0 \quad r^*=\frac{\alpha}{4}\left ( x+y \right )^2</math>