Generate uniformly random float which can return all possible values

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An easy way to generate a random float64 in [0,1) is by generating a uniformly random int in [0,2⁵³) and dividing it by 2⁵³. This is essentially what rand.Float64() is doing. However, not all possible float64 values between 0 and 1 can be generated this way: if the value is lower than 2⁻⁴ for example, the 4 last bits of the significand are always going to be 0. Or, put more simply, the naive method always returns multiples of 2⁻⁵³, and not all floating point numbers between 0 and 1 are multiples of 2⁻⁵³.

How do you generate a uniformly random float64 such as every possible value has a chance of being returned?

For context, I need this because I'm implementing this paper and the assumption "all possible values between 0 and 1 are represented" is essential for the result to hold.

 


Because the binary64 floating point numbers are not uniformly spaced, you cannot generate a uniform distribution which can return all possible values less that 1.

If you omit the requirement uniform you have to generate all representable multiples of the smallest positive denormal number 2^(-1074)and zero.

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