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The x*x vs. x^2 issue that the author presents seems contrived. Interval arithmetic is just a specialized case of arithmetic on random variables. If two values are uncorrelated, why would you use the same variable for them? The ambiguity could be trivially resolved by representing the problem as x*x vs. x*y, where x = (3, 3) and y = (3, 3).

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It took me a while to understand that example.

Basically if you multiply two intervals (-3,3) * (-3,3), it's relevant whether it's actually the same quantity multiplied by itself (in which case, you want the answer 0-9, because you'll always be multiplying something like -1 * -1, because you're sampling once from the interval and then squaring it), or it's two different quantities, both of which are in the same interval, in which case the answer you want is (-9,9), because you could be multiplying something like -3 * 3, because you're sampling twice from the same interval and multiplying the two results together.

That makes sense, at that point you need some concept of identity though, like a sequential ID, so you’d be working with something like (int, float, float) instead. Or allocate on the heap and use the address for identity.

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