Uniform over the real numbers.

It’s a little unclear what you mean by that without further explanation. Do you mean that conceptually one selects a real number at random and then rounds that real number to the closest representable float? (And if so, which rounding mode?)

Floats aren't uniformly distributed over the reals (e.g. larger floats are sparser) so the above approach simply tries to account for that when generating the number by using a precisely counteracting non-uniform selection method which will result in even distribution in relation to where in the reals the picks are. It does so directly.