What is the rationale for all comparisons returning false for IEEE754 NaN values?

To throw in yet another analogy. If I hand you two boxes, and tell you that neither of them contains an apple, would you tell me that the boxes contain the same thing?

NaN contains no information about what something is, just what it isn't. Therefore these elements can never definitely be said to be equal.

NaN can be thought of as an undefined state/number. similar to the concept of 0/0 being undefined or sqrt(-3) (in the real number system where the floating point lives).

NaN is used as a sort of placeholder for this undefined state. Mathematically speaking, undefined is not equal to undefined. Neither can you say an undefined value is greater or less than another undefined value. Therefore all comparisons return false.

This behaviour is also advantageous in the cases where you compare sqrt(-3) to sqrt(-2). They would both return NaN but they are not equivalent even though they return the same value. Therefore having equality always returning false when dealing with NaN is the desired behaviour.

I was a member of the IEEE-754 committee, I'll try to help clarify things a bit.

First off, floating-point numbers are not real numbers, and floating-point arithmetic does not satisfy the axioms of real arithmetic. Trichotomy is not the only property of real arithmetic that does not hold for floats, nor even the most important. For example:

• Addition is not associative.
• The distributive law does not hold.
• There are floating-point numbers without inverses.

I could go on. It is not possible to specify a fixed-size arithmetic type that satisfies all of the properties of real arithmetic that we know and love. The 754 committee has to decide to bend or break some of them. This is guided by some pretty simple principles:

1. When we can, we match the behavior of real arithmetic.
2. When we can't, we try to make the violations as predictable and as easy to diagnose as possible.

Regarding your comment "that doesn't mean that the correct answer is false", this is wrong. The predicate (y < x) asks whether y is less than x. If y is NaN, then it is not less than any floating-point value x, so the answer is necessarily false.

I mentioned that trichotomy does not hold for floating-point values. However, there is a similar property that does hold. Clause 5.11, paragraph 2 of the 754-2008 standard:

Four mutually exclusive relations are possible: less than, equal, greater than, and unordered. The last case arises when at least one operand is NaN. Every NaN shall compare unordered with everything, including itself.

As far as writing extra code to handle NaNs goes, it is usually possible (though not always easy) to structure your code in such a way that NaNs fall through properly, but this is not always the case. When it isn't, some extra code may be necessary, but that's a small price to pay for the convenience that algebraic closure brought to floating-point arithmetic.

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Addendum: Many commenters have argued that it would be more useful to preserve reflexivity of equality and trichotomy on the grounds that adopting NaN != NaN doesn’t seem to preserve any familiar axiom. I confess to having some sympathy for this viewpoint, so I thought I would revisit this answer and provide a bit more context.

My understanding from talking to Kahan is that NaN != NaN originated out of two pragmatic considerations:

• That x == y should be equivalent to x - y == 0 whenever possible (beyond being a theorem of real arithmetic, this makes hardware implementation of comparison more space-efficient, which was of utmost importance at the time the standard was developed — note, however, that this is violated for x = y = infinity, so it’s not a great reason on its own; it could have reasonably been bent to (x - y == 0) or (x and y are both NaN)).

• More importantly, there was no isnan( ) predicate at the time that NaN was formalized in the 8087 arithmetic; it was necessary to provide programmers with a convenient and efficient means of detecting NaN values that didn’t depend on programming languages providing something like isnan( ) which could take many years. I’ll quote Kahan’s own writing on the subject:

Were there no way to get rid of NaNs, they would be as useless as Indefinites on CRAYs; as soon as one were encountered, computation would be best stopped rather than continued for an indefinite time to an Indefinite conclusion. That is why some operations upon NaNs must deliver non-NaN results. Which operations? … The exceptions are C predicates “ x == x ” and “ x != x ”, which are respectively 1 and 0 for every infinite or finite number x but reverse if x is Not a Number ( NaN ); these provide the only simple unexceptional distinction between NaNs and numbers in languages that lack a word for NaN and a predicate IsNaN(x).

Note that this is also the logic that rules out returning something like a “Not-A-Boolean”. Maybe this pragmatism was misplaced, and the standard should have required isnan( ), but that would have made NaN nearly impossible to use efficiently and conveniently for several years while the world waited for programming language adoption. I’m not convinced that would have been a reasonable tradeoff.

To be blunt: the result of NaN == NaN isn’t going to change now. Better to learn to live with it than to complain on the internet. If you want to argue that an order relation suitable for containers should also exist, I would recommend advocating that your favorite programming language implement the totalOrder predicate standardized in IEEE-754 (2008). The fact that it hasn’t already speaks to the validity of Kahan’s concern that motivated the current state of affairs.

From the wikipedia article on NaN, the following practices may cause NaNs:

• All mathematical operations> with a NaN as at least one operand
• The divisions 0/0, ∞/∞, ∞/-∞, -∞/∞, and -∞/-∞
• The multiplications 0×∞ and 0×-∞
• The additions ∞ + (-∞), (-∞) + ∞ and equivalent subtractions.
• Applying a function to arguments outside its domain, including taking the square root of a negative number, taking the logarithm of a negative number, taking the tangent of an odd multiple of 90 degrees (or π/2 radians), or taking the inverse sine or cosine of a number which is less than -1 or greater than +1.

Since there is no way to know which of these operations created the NaN, there is no way to compare them that makes sense.