NaN partials and comparison

[aplavin]

Some operations generate Dual with NaN partials, and I'm not sure comparison operators are defined consistently for them...

julia> xs = Dual(0.0,NaN)

# it's not zero,
julia> iszero(xs)
false
# no, definitely not zero,
julia> xs ≈ 0
false

# it's a regular finite number:
julia> isnan(xs)
false
julia> isfinite(xs)
true

# moreover, it's a positive number
julia> xs > 0
true

# but adding this positive number, or multiplying it by anything still gives the same number back:
julia> xs + xs === xs
true
julia> xs * 1e100 === xs
true
# these are the defining properties of a zero
# shouldn't it be considered a zero really?

Intuitively I would say that the most natural thing is to compare equal to zero, but maybe there are other self-consistent choices as well.

[devmotion]

julia> iszero(xs)
false

julia> xs ≈ 0
false

This is correct.

---

julia> xs > 0
true

This is wrong. Analogously to

julia> NaN > 0
false

julia> NaN < 0
false

we'd want

julia> xs > 0
false

julia> xs < 0
false

The problem seems to be that < on ForwardDiff.Partials falls back to isless (which then uses the generic implementation for AbstractVector). Similar to ForwardDiff.Dual, to handle these floating point comparisons correctly both < and isless should be defined for ForwardDiff.Partials - the first one defining a partial order (due to NaN) and the second one a total order (ordering NaN after regular numbers).

---

julia> xs + xs === xs
true

julia> xs * 1e100 === xs
true

This is correct, it's analogous to

julia> NaN + NaN === NaN
true

julia> NaN * 1e100 === NaN
true