Fix conj and abs. Add abs2. Make one(Dual) return a dual number.

[andreasnoack]

I think these are fixes. See e.g. here

[mlubin]

conj is a bit confusing to think about, but I do know that conj{T<:Real}(d::Dual{T}) must be equal to d, simply because conj(x::Real) = x. Otherwise the derivative information isn't correct. We've chosen to define functions so that f(d::Dual) = Dual(f(real(d)),dual(d)*f'(real(d))) always, even if this conflicts with the algebraic definition.

[papamarkou]

@andreasnoackjensen, @mlubin, what choice would you like us to make? I agree with @andreasnoackjensen' changes, as it seems more natural to me. Nevertheless, I see your point @mlubin, we don't want to violate differentiation.

[andreasnoack]

When you want to use Dual for differentiation of real functions you shouldn't use conj in your code, right? Wouldn't it be better to document that instead of having non-standard behaviour?

[papamarkou]

Yes, I agree with your remark and suggestion @andreasnoackjensen. @mlubin, if you don't object, I would agree for these changes to be merged.

[mlubin]

Unfortunately conj is used in many places in Base, for example, in dot and other vector operations. If we make this change, then dot of two Vector{Dual}s will not give the correct derivatives. I think it makes more sense to define a function conjdual that has the algebraic definition. Of course, this behavior should be well documented.

[andreasnoack]

@mlubin I see your point, and differentiation is the main point of the DualNumbers, but it just feels wrong not to have the math definitions right.