How to derive the following probability in a close form formula with respect to $x$? \begin{equation} p(\vert x\vert\le \vert a - f\vert\ , \vert x \vert\le \vert b - f\vert) \end{equation} where $f \sim \mathcal{N}(\mu, \sigma^2)$, $a$,$b$ are constant.
At first glance I thought it can be derived by from a MultiVariate Folded Normal distribution (MVFN). However, it seems the cummlative probability function(CDF) of MVFN is a non trivial thing and even the some recent work[1][2] on MVFN didn't mention about this(please correct me if I missed some significant part).
I am not a mathematical researcher but I wonder if I can leverage the fact that actually the probability is only governed by one distribution of $f$, but failed myself. What's more I really don't want to sample this probability.
Could Anyone give me some hint?
[1] Chakraborty, Ashis Kumar, and Moutushi Chatterjee. "On multivariate folded normal distribution." Sankhya B 75.1 (2013): 1-15.
[2] Murthy, G.S.R. A Note on Multivariate Folded Normal Distribution. Sankhya B 77, 108–113 (2015) doi:10.1007/s13571-014-0092-9