Uniform distribution of integers using floating point source

The standard way to get a random integer in the range [0, n) in JavaScript - or any other language that only offers a random() function that returns a float in the range [0,1) - is to use Math.floor(Math.random() * n).

Now the math behind this is trivial assuming we're operating on the set of rational numbers. The question is: With all the complications of IEEE-754 floating point numbers is the resulting distribution actually really uniform?

Considering that the gap between one floating point number and the next higher one increases as they grow larger I would think that this should introduce some kind of bias towards smaller numbers.