The correct equation is

${\displaystyle H=-\frac{1}{2}g{[\frac{D}{V}-\sqrt{\frac{2H}{g}}]}^2+\mathrm{eg}\sqrt{\frac{2H}{g}}[\frac{D}{V}-\sqrt{\frac{2H}{g}}]<T}$

The parts in the brackets and the power of g are obtained by dimensional analysis (the coefficient of restitution, being a ratio of velocities, is dimensionless)

The overall signs are obtained by noting that the first part of the motion is down and the second up.