I like the idea of an random number generator that exploits the full precision of the floats. However, the issue with the naive method is loss of accuracy, not

I like the idea of an random number generator that exploits the full precision of the floats. However, the issue with the naive method is loss of accuracy, not bias. When the article states If k is 64, a natural choice for the source of bits, then because the set {0, 1, ..., 2^53 - 1} is represented exactly, whereas many subsets of {2^53, 2^53 + 1, ..., 2^64 - 1} are rounded to common floating-point numbers, outputs in [0, 2^-11) are underrepresented. this is very misleading, because the points that are "underrepresented" are closer together. The discrete probability density created by the naive method is a perfectly good approximation to the uniform distribution. Furthermore, although as the article shows, we can do better, this can, and must, come at a cost. Since the small numbers occur with much lower probability, the only way to generate them (without importance sampling) is by generating much more randomness (1088 bits in the article). Generating random bits is expensive, and this uses 17 times more randomness than the naive method. EDIT: it would be possible to avoid generating all the extra bits in most cases, and so an efficient average time algorithm should be possible, but the worst case still requires 1088 bits to be generated.