u = y - x \in \left[-1, 1\right] \\
v = xy \in \left[ 1, 2 \right] \\
\psi (x,y) = (y - x, xy) \Rightarrow J(\psi) = -x - y \Rightarrow J(\phi) = \frac{1}{J(\psi)} = -\frac{1}{x + y}=-\frac{1}{\sqrt{u^2 + 4v}}