I have the below exponential function which I wish to solve it for $b$. Other than resorting to the Lambert W function, is there alternative way of representing the solution?
$$ \frac{(1+a)(1-b)}{ab +a -b +1} = \exp \left({\frac{2a(ab-a-b)}{ab +a -b +1}}\right)$$ where $a,b \in (0,1)$.
Thanks a lot.
I even do not think that you could get an explicit solution in terms of Lambert function and, as Alex S. commented, only numerical methods could be used to solve this equation.
There is one thing which is interesting : if you plot the contour level, you should notice that $b$ is a very smooth function of $a$. If fact, for the range you gave, $b$ varies from $\frac 12$ to $(1-\frac 1e)$. Making a linear interpolation, you can generate a very reasonable estimate of $b$ using $$b_{est}=\frac 12+\big(\frac{1}{2}-\frac{1}{e}\big)a$$ and Newton method would converge in very few iterations.
A quite better estimates seems to be $$b_{est}=\frac 12+\big(\frac{1}{2}-\frac{1}{e}\big)a^{3/4}$$