This distribution is a special case of a stable distribution with shape parameter α = 1/2 and skewness parameter β = 0.
And it does not appear to be widely used.
This name "SαS point5 distribution" is not official but provisional.

Such a distribution with β = 0 is called symmetric alpha-stable distribution.

• α = 2: Normal distribution
• α = 1: Cauchy distribution

The SαS point5 distribution, like these distribution, has a closed-from expression, but it can't be expressed in terms of elementary function.

The SαS point5 distribution, generalized to a stable distribution by introducing position and scale parameters, is as follows:

Since scaling and translations allow for standardization, standard parameters are discussed here.

Using fresnel integral function, it's obtained as follows:
This equation becames difficult to obtain accurately when x is extremely small.

When x is extremely small, the following series expression can also be used:

When x is large, the following equation can be used as an asymptotic expression.
The coefficients decay rapidly, making them suitable numerical evaluation:

PDF Precision 150
CDF Precision 150
Quantile Precision 142

K.I.Hopcraft, E.Jakeman, R.M.J.Tanner, "Lévy random walks with fluctuating step number and multiscale behavior" (1999)

HoltsmarkDistribution