Sorting floating-point values, and NaN

[Vladimir Panteleev]

On Tuesday, 12 November 2013 at 19:26:12 UTC, tn wrote:
> But 'a<b' is transitive for floating points. The transitivity 
> condition (`a<b && b<c` implies `a<c`) says nothing about the 
> cases where a and b (or b and c) are incomparable. Transitivity 
> is not the problem

Whoops, you're right. I meant transitivity of the negated 
expression: !(a>b) && !(b>c) implies !(a>c). This fails for [3, 
NaN, 1].

>> Sort et al don't know how to check "a>=b"... and they can't 
>> use "a<b" either, because !(a<b) && !(b<a) implies a==b.
>
> This is the problem, that is, the fact that with '<' you can't 
> tell the difference between equal values and incomparable 
> values (one or both are NaNs). On the other hand, with '<=' you 
> can. See:
>
>  (a<b) && !(b<a)  -->  a<b
> !(a<b) &&  (b<a)  -->  b<a
>  (a<b) &&  (b<a)  -->  (not possible)
> !(a<b) && !(b<a)  -->  a==b OR incomparable
>
> but
>
>  (a<=b) && !(b<=a)  -->  a<b
> !(a<=b) &&  (b<=a)  -->  b<a
>  (a<=b) &&  (b<=a)  -->  a==b
> !(a<=b) && !(b<=a)  -->  incomparable
>
> Thus, the correct solution would be to modify the functions to 
> take "lessOrEqual" as a template parameter instead of "less".

Too late for that now... just like it's too late to change IEEE 
comparison logic.

> If "less" is kept, then we need another template argument to 
> separate equality and incomparability (something like 
> "equals(a,b)", or "incomparable(a,b)").

Is this to allow generic partial ordering (rather than to fix 
sorting NaNs specifically)? I remember seeing a thread about 
partial ordering and opCmp:
http://forum.dlang.org/post/dpfbtu$1ocf$1@digitaldaemon.com