Alternatively, just use the method in the article (scale the uniform random distribution) and if your point is more than 1u away from the origin then throw it o

Alternatively, just use the method in the article (scale the uniform random distribution) and if your point is more than 1u away from the origin then throw it out and try again. You would throw out 36% of the points, but you might get that back from not using trig functions.