What do you mean when you say you don't like that some numbers are returned more often than others? This seems completely inevitable for a uniform distribution because floating point numbers don't have a uniform density across the real line.

Only if you choose a range like [1,2] with a uniform density of floating point numbers you will see all floating point numbers in the interval - except both bounds of course - with equal probability.

You would have to not return some floating point numbers ever to see all returned floating point numbers in the interval with equal probability but that of course hardly qualifies as a uniform distribution if some perfectly valid floating point numbers will never occur.