Russ Cox, January 2016
The blue line is ideal, Pr(rand.Float64() < x) = x. The red line is the analytically exact probability.
Make your browser wide enough to see four graphs across. Then the four columns show behavior very close to zero (scale 1/2^63), somewhat zoomed out from zero (scale 1/2^53), centered on 0.5 (scale 1/2^53), and close to one (scale 1/2^53).
Computed with rsc.io/tmp/randdist.