In searching for various choices for the interpolation of exponential-towers to fractional heights (aka tetration) I came to the following type of function: $$ f_b(x) =\left[ \frac {t_0}{2} + \sum_{k=1}^{\infty} v_k \cdot (x-t_0)^k \right] + \left[ \frac{t_1}{2} + \sum_{k=1}^{\infty} w_k \cdot (x-t_1)^k \right]$$ which is an analytical formula for the doubly-infinite alternating series of the exponential-towers of consecutive heights (for $ ...+\log_b(\log_b(x))-\log_b(x) + x -b^x+b^{b^x}-...$ where $b$ is inside the range $]1 \ldots e^{\frac 1e}[$ and $t_0,t_1$ are the fixpoints, and the $x$ is in the range between the fixpoints - for what this information might be worth).
To find some smooth translation depending on another parameter I would like to have an inverse for that function, say I have some value for $f(x)$ and want to determine $x$; until now I apply Newton-iteration for this, but that is not much satisfying.
So perhaps there might be one clever idea how one could proceed to determine an inverse by some analytical formula.
[update]: After @Helge's comment I should perhaps add, that $f_b(x)$ seems to have no real fixpoint in the admissible range for $x$ and so the path, which is otherways the most straightforward one in more general problems, seems to be not viable here [/update]
