I apologize for the very specific question I am asking. Define the relative entropy $D:[0,1]\times[0,1]\mapsto[0,\infty]$ by
$$D(x,y) = x\log_e\frac{x}{y}+(1-x)\log_e\frac{1-x}{1-y}.$$
Note $D(x,y)\geq 0$ with equality iff $x=y.$
It is possible to show that there exists a continuous decreasing bijection $f:[0,1]\mapsto [0,1]$ such that for $r=f(s),$ we have the equality
$$\frac{D(0.2+0.1r,0.2+0.1s)}{D(r,s)} = \frac{D(0.2+0.1s,0.2+0.1r)}{D(s,r)},$$
for all $s\in (0,1)$ except $s=s_0$ which satisfies $f(s_0)=s_0.$ [The constants 0.2,0.1 chosen here are generic. I am interested in the general case with two arbitrary constants a,b.]
One can observe that $f$ is an involution.
I would like to know
(a) if there is a systematic way to guess an explicit formula for a general involution;
(b) if there is a software package that can generate an explicit formula (when possible and not too hard) from an implicit one.
Thank you very much for your time.