This shape is just the CDF of Wigner's semicircle law
Assuming[{-1 < x < 1},
f[x_] = CDF[WignerSemicircleDistribution[1], x] // Refine
];
g = InverseFunction[f[#] &];
LogPlot[g[1 - x], {x, 0, 0.5}, Filling -> Axis, PlotRange -> All]
Answered in https://math.stackexchange.com/a/4187243/998