I have calculated the following PDF, $$ f(y) = \begin{cases} \frac{\sqrt{-y}}{2} e^{-\frac{y^2}{2}-y - \frac{1}{4}} \left( I_{-\frac{1}{4}}\left(\frac{y^2}{2}\right) - I_{\frac{1}{4}}\left(\frac{y^2}{2}\right) \right) ,& y<0 \\ \frac{\sqrt{2}}{\pi} e^{ - 1/4} \Gamma \left(\frac{5}{4}\right), &y=0\\ \frac{\sqrt{y}}{2} e^{-\frac{y^2}{2} -y-\frac{1}{4}} \left(I_{-\frac{1}{4}}\left(\frac{y^2}{2}\right) + I_{\frac{1}{4}}\left(\frac{y^2}{2}\right)\right), &y>0 \end{cases} $$ where $I_\nu$ are the modified Bessel functions of the first kind.
How do I find to which (if any) known distribution it belongs ?
I have searched for generalized Bessel distribution, and other similar things, but could not find anything that matches.