I read this post about math for Shapiro-Wilk test. And there is an expression for inverse normal distribution $\large m(n)=F^{-1}\left(\frac{n-\frac{3}{8}}{N+0.25}\right)$ for $n\in {1,2,..,N}.$
I suppose $F^{-1}$ stands for ${\displaystyle f(x;\mu ,\lambda )={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp \left[-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}\right]}$
But where did the argument $\frac{n-\frac{3}{8}}{N+0.25}$ come from?
$F^{-1}$ is the inverse-cdf (or quantile function).
What the Shapiro-Wilk does is use the inverse of the normal cdf (the normal quantile function) to transform adjusted uniform quantiles as a way of approximating expected normal quantiles.
You're looking at the density for the inverse Gaussian distribution which another distribution not relevant to the Shapiro-Wilk, and the density is not the thing being discussed.
Adjusted uniform quantiles of the form $u_i=\frac{i-\alpha}{n+1-2\alpha}$ $-$ of which this is an example (with $\alpha=\frac38$) $-$ come from Blom, 1958 [1].
It's a way of attempting to approximate expected values of order statistics that's suitable for symmetric distributions. That particular value of $\alpha$ is often used for the normal case.
[1] Blom, G. (1958), Statistical estimates and transformed beta variables, New York: John Wiley and Sons
