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

Question

Why do comparisons of NaN values behave differently from all other values? That is, all comparisons with the operators ==, <=, >=, <, > where one or both values is NaN

Answer 1

It only looks peculiar because most programming environments that allow NaNs do not also allow 3-valued logic. If you throw 3-valued logic into the mix, it becomes consistent:

• (2.7 == 2.7) = true
• (2.7 == 2.6) = false
• (2.7 == NaN) = unknown
• (NaN == NaN) = unknown

Even .NET does not provide a bool? operator==(double v1, double v2) operator, so you are still stuck with the silly (NaN == NaN) = false result.

Answer 2

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.

Answer 3

Because mathematics is the field where numbers "just exist". In computing you must initialize those numbers and keep their state according to your needs. At those old days memory initialization worked in the ways you could never rely on. You never could allow yourself to think about this "oh, that would be initialized with 0xCD all the time, my algo will not broke".

So you need proper non-mixing solvent which is sticky enough to not not letting your algorithm getting sucked into and broken. Good algorithms involving numbers are mostly going to work with relations, and those if() relations will be omitted.

This is just grease which you can put into new variable at creation, instead of programming random hell from computer memory. And your algorithm whatever it is, will not break.

Next, when you still suddenly finding out that your algorithm is producing NaNs, it is possible to clean it out, looking into every branch one at a time. Again, "always false" rule is helping a lot in this.

Answer 4

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.

Answer 5

While I agree that comparisons of NaN with any real number should be unordered, I think there is just cause for comparing NaN with itself. How, for example does one discover the difference between signaling NaNs and quiet NaNs? If we think of the signals as a set of Boolean values (i.e. a bit-vector) one might well ask whether the bit-vectors are the same or different and order the sets accordingly. For example, on decoding a maximum biased exponent, if the significand were left shifted so as to align the most significant bit of the significand on the most significant bit of the binary format, a negative value would be a quiet NaN and any positive value would be a signaling NaN. Zero of course is reserved for infinity and the comparison would be unordered. MSB alignment would allow for the direct comparison of signals even from different binary formats. Two NaNs with the same set of signals would therefore be equivalent and give meaning to equality.