Why is complex being deprecated again?

[Lars T. Kyllingstad]

On 15/04/12 23:56, Stewart Gordon wrote:
> On 15/04/2012 22:09, Lars T. Kyllingstad wrote:
>> On 15/04/12 14:29, Stewart Gordon wrote:
> <snip>
>>> My impression was that the plan is to deprecate it once the stuff in
>>> std.complex is complete. std.complex has clearly grown since that
>>> discussion, but it still needs a pure imaginary type (and I don't know
>>> what else at the moment).
>>
>> I absolutely do not think it does. There is nothing you can do with a
>> pure imaginary type
>> that you cannot do with a complex type.
> <snip>
>
> What proof have you of this - and in particular, that the rationale for
> pure imaginary types on the comparison page is wrong?

For any standard type (built-in or library) to be useful, it has to 
actually be used for something.  And in all my years of using D, I have 
never seen a *single* real-world use of the pure imaginary types.

The reason the imaginaries are so seldomly used is precisely because 
there are so few things you can do with them.  Basically, if you do 
anything beyond addition and subtraction, and multiplication with a real 
number, you are back in the complex plane.  And if those operations are 
all you need, the real line is just as good as the imaginary line, and 
you might as well fake it with a real floating-point type.

Yes, complex and imaginary numbers have some quirks and subtleties that 
need to be taken into consideration, but I see this as an implementation 
issue with the complex type, and not a justification for the existence a 
pure imaginary type.

All that said, however, if anyone can demonstrate that the pure 
imaginary types are in fact used in a substantial body of real-world 
code, I will be happy to change my stance on the above.

Now, to address the rationale for pure imaginary types on 
http://dlang.org/cppcomplex.html.

Firstly, in light of what I've said above (and given that I am not wrong 
<g>), the efficiency issue would appear moot.  That leaves the semantic 
issues.  The quote by prof. Kahan mentions that the identities 
sqrt(conj(z))==conj(sqrt(z)) and log(conj(z))==conj(log(z)) should hold 
even when z is a negative real number.  IIRC, f(conj(z))==conj(f(z)) 
holds when f is analytic, but both sqrt and log (conventionally) have 
branch cuts along the negative real axis.

Case in point:  Neither of these identities hold in Mathematica, which 
is considered the state of the art in mathematical software:

     Conjugate[Log[-1]]   evaluates to   -I Pi
     Log[Conjugate[-1]]   evaluates to    I Pi
     Conjugate[Sqrt[-1]]  evaluates to   -I
     Sqrt[Conjugate[-1]]  evaluates to    I

The page also mentions some identity involving infinities which is 
supposed to hold.  This is not obviously true.  An IEEE infinity has 
very little to do with mathematical infinity, it is just a special value 
which means either "this number is too large to be represented by the 
given number of bits", or "this is a result of a divide by positive 
zero".  (The signedness of zero is another quirk of IEEE floats. 
Basically, -0.0 means "a negative number which is arbitrarily close to 
zero", and dividing by it yields an arbitrarily large negative number, 
i.e. -double.infinity.  Mathematically, something/0 makes no sense at 
all.  Personally, I think IEEE made a mistake in defining FP numbers in 
this way.)

It is true that the real line can be extended with elements called plus 
and minus infinity (affinely extended real line, see 
https://en.wikipedia.org/wiki/Extended_real_number), and the IEEE floats 
can be said to approximate this system, but this does not generalise 
directly to complex numbers.  The extended complex plane (see 
https://en.wikipedia.org/wiki/Riemann_sphere) only has one "infinity".

How should the IEEE system be extended to the complex plane, anyway?  If 
we look at the problem in terms of the cartesian representation, we may 
want four infinities, namely:

     infinity + i * infinity
    -infinity + i * infinity
     infinity - i * infinity
    -infinity - i * infinity

If, on the other hand, we take a polar view of things, there are MANY 
possibilies:

     infinity * exp(i * r)  // where r is any real number

Now, we may invent our own rules for operations with complex numbers 
involving infinities, if nothing else for predictability in 
calculations.  (And maybe such rules are well established already?)  But 
it does not justify the existence of a pure imaginary type.

-Lars