[Dylan16807]

Signed infinity represents aggressive rounding, but you still know roughly what number it is. It works out well to let it still participate in equality and ordering. NaN can be created in many different ways and there is no way to say basically anything about what numbers it could be related to.

You could arbitrarily make NaN sort as if it was a certain value, and that would be useful when you want to sort a big array, but it would have unpleasant side effects when you're doing math. IEEE decided "always false" was less likely to cause problems, but to be clear you get problems no matter what you choose.

[Retric]

My point was you can have multiple types of “Not a Number” used at the same time.

Defining just error, undefined, positive infinity, and negative infinity is far from an exhaustive list but they obviously should be treated differently in many contexts.

[Dylan16807]

Well let's look at wikipedia's list of things that cause IEEE NaN, since they already split out Infinity.

0/0, ∞/∞, ∞%n, n%0, ∞-∞, results with imaginary components

The first five have no meaningful approximation or way to interact with anything. And there's no good way to pretend a single float is a complex number.

So those results get the "this doesn't exist" treatment. Coder's choice if NaN triggers errors or not.

Would you try to define any more behavior for any of those NaNs?

[Retric]

Only if you wanted to treat them as more severe types of NaN, Aka ∞-∞ - ∞ can evaluate to ∞-∞, but ∞-∞ / 0 = 0/0.

Though really that choice is just about what kind of errors you’re showing users.