I’m fully aware of how IEEE 754 works, and that there is—to quote my own comment—“a much more complicated generator that takes more care to ensure ideal rounding at the limits of floating-point precision”. If you’re looking for somebody clueless to argue at, you’ll need to go find someone else.
But these details aren’t relevant to this conversation:
• We’re not simulating the weather, we’re drawing dots in a circle—and anyone who cares about the dots being shifted from true uniformity by tiny fractions of the radius of an electron is going to have objections to any conceivable computerized representation of the numbers involved.
• It’s quite reasonable to implicitly treat this discussion as a question within the domain of true mathematical reals, and quite unreasonable to chastise someone as incorrect and careless for doing so.
• It’s still false that “rand(rand()) == rand() * rand() relies on uniform distributions”. This identity between distributions holds with the naïve multiplicative implementation of rand() and any underlying random generator, even a blatantly skewed one, in either the mathematical or IEEE reals, exactly. (And with a more precise IEEE generator, it will hold in as close of an approximate sense as one can expect any mathematical identity to hold when translated to IEEE, which from my perspective is just fine in this context. My only real interest in even acknowledging IEEE here is to support the point that commenters above were correct to consider the identity intuitively obvious, even when you try to throw in that well-actually.)
> We’re not simulating the weather, we’re drawing dots in a circle
Ah, so we don't care about mathematical rigor, yet we're discussing a proof of the pure mathematical behavior?
> It’s quite reasonable to implicitly treat this discussion as a question within the domain of true mathematical reals,
So we do care about mathematical rigor?
> It’s still false that “rand(rand()) == rand() * rand() relies on uniform distributions”.
Consider a distribution A(r) uniform on [0,r) except it never returns 1/4. This is a perfectly valid distribution, extremely close to being the same (in the mathematical sense) as the uniform distribution itself.
A(A(1)) will never be 1/4. A(1) * A(1) may be 1/4. Thus the claim needing uniform (or a proof for whatever distribution you want) requires careful checking. And if you don't like this distribution, you can fiddle with ever larger sets (in cardinality, then lebesgue measure zero) where you tweak things, and you'll soon discover that without uniform, you cannot make the simplification. And I demonstrated that the actual PRNGs are not uniform, so much care is needed to analyze. (They're not even continuous domain or range)
Yes, if you define the rng circularly, then you can make the assumption, but then you cannot make the claim of 1/2 per term, since they are not uniform. If you define the rng more carefully, then they often don't commute (maybe never?).
For example, defining the rng in another reasonable (yet still naive) fashion to keep more bits:
float rnd(float max)
i = int_rand(N) // 0 to N-1, try weird N like N = 12345678
f = i*max
return f/N
this does not satisfy rnd(rnd()) == rnd()*rnd() in general, as you can check easily in code.
Thus, to make the claim you can swap them requires for the mathematical part true uniformity (my claim way up above) and for the algorithm sense it requires knowing details about the underlying algorithm. Simply having a rnd(max) blackbox algorithm is not enough.
> Yes, if you define the rng circularly, then you can make the assumption,
We’re agreed then.
> but then you cannot make the claim of 1/2 per term, since they are not uniform.
As mathematicians, we’re allowed to make multi-step arguments. The first step rand(rand()) = rand() * rand() holds for any distribution that scales multiplicatively with the argument, as we’ve agreed. The second step E[rand() * rand()] = E[rand()] * E[rand()] holds for any two independent distributions with finite expectation. The third step E[rand()] = 1/2 holds for this particular distribution. An ideal mathematical rand() satisfies all three of these properties, so this argument that E[rand(rand())] = 1/4 is valid in the mathematical reals. (And it’s approximately valid in the IEEE reals, which is all one typically asks of the IEEE reals.)
But these details aren’t relevant to this conversation:
• We’re not simulating the weather, we’re drawing dots in a circle—and anyone who cares about the dots being shifted from true uniformity by tiny fractions of the radius of an electron is going to have objections to any conceivable computerized representation of the numbers involved.
• It’s quite reasonable to implicitly treat this discussion as a question within the domain of true mathematical reals, and quite unreasonable to chastise someone as incorrect and careless for doing so.
• It’s still false that “rand(rand()) == rand() * rand() relies on uniform distributions”. This identity between distributions holds with the naïve multiplicative implementation of rand() and any underlying random generator, even a blatantly skewed one, in either the mathematical or IEEE reals, exactly. (And with a more precise IEEE generator, it will hold in as close of an approximate sense as one can expect any mathematical identity to hold when translated to IEEE, which from my perspective is just fine in this context. My only real interest in even acknowledging IEEE here is to support the point that commenters above were correct to consider the identity intuitively obvious, even when you try to throw in that well-actually.)