$x=-\frac{1-fy}{yf^{2}-y-f}$
$\left(y\neq 0\text{ or }f\neq 0\right)\text{ and }\left(y=0\text{ or }f\neq \frac{\sqrt{4y^{2}+1}+1}{2y}\right)\text{ and }\left(y=0\text{ or }f\neq \frac{-\sqrt{4y^{2}+1}+1}{2y}\right)\text{ and }\left(\frac{\sqrt{4y^{2}+1}+1}{2}\neq 0\text{ or }f\neq 0\right)\text{ and }\left(\frac{-\sqrt{4y^{2}+1}+1}{2}\neq 0\text{ or }f\neq 0\right)$

$x=-\frac{1-fy}{yf^{2}-y-f}$
$\left(y\neq 0\text{ or }f\neq 0\right)\text{ and }\left(y=0\text{ or }f\neq \frac{\sqrt{4y^{2}+1}+1}{2y}\right)\text{ and }\left(y=0\text{ or }f\neq \frac{-\sqrt{4y^{2}+1}+1}{2y}\right)\text{ and }\left(\frac{-\sqrt{4y^{2}+1}+1}{2}\neq 0\text{ or }f\neq 0\right)$

$\left\{\begin{matrix}f=\frac{\sqrt{x^{2}-2xy+y^{2}+4\left(xy\right)^{2}}+x+y}{2xy}\text{; }f=\frac{-\sqrt{x^{2}-2xy+y^{2}+4\left(xy\right)^{2}}+x+y}{2xy}\text{, }&y\neq 0\text{ and }x\neq 0\\f=\frac{1-xy}{x+y}\text{, }&\left(x=0\text{ and }y\neq 0\right)\text{ or }\left(y=0\text{ and }x\neq 0\right)\end{matrix}\right.$