If $ f(x)=1-\frac{4}{\pi}\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}e^{-\frac{\pi^2 (2k+1)^2}{8 x^2}}$, find $\min\{x:f(x)\geq 1-y \}$.

The function $f(x)$ is increasing and its output falls in $[0,1]$. Usually, we have to find the inverse of $f(x)$. I don't know how to do this. If we don't have an inverse, we should bound $f(x)$. Even if we consider the first three terms of $f(x)$ to bound it, the inverse of the sum of exponentials is not easy to find, and it is not a good bound for me. Any idea?

I tried $erf(x)$ but it is not always was not a valid bound on the whole region of $x \geq 0$. $erf(x)-1$ is somehow close, but still, some part of the x-axis is lower bound and other parts it is upper bound.

The function $f(x)$ is CDF of maximum of absolute value of a standard Weiner process in $t\in [0,1]$.


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