atan2 may evaluate incorrectly for symbolic expressions

[cqc-alec]

There is a (7-year-old) TODO comment in the implementation of atan2 which acknowledges that "ideally the answer should depend on the signs of num and dem". Currently, for example, if a is symbolic then atan2(a,a) evaluates to pi/4, whereas the correct value is dependent on the sign of a. So atan2(-1,-1) gives a different result from that obtained by substituting -1 for a in the expression.

The comment mentions is_positive and is_negative but perhaps sign is enough to fix this issue?

[isuruf]

Yes, sign should be enough to fix this issue.

[rikardn]

I think we need to do what sympy does, which is to simply return atan2(a, a). If a is real and non-zero we can refine it using the refine-function, but for general complex a including a=0 I don't think it is desired or even possible to auto-simplify.

[cqc-alec]

I think returning atan2(a, a) would be fine.

But, is atan2(y,x) even defined for non-real arguments? I suppose it could be defined as the principal value of the argument of x + i y for complex x and y, though most implementations including std::atan2 restrict to the reals.

One simplification that can be made is to cancel common factors that are positive constants, like sqrt(2), but I don't know how easy this is to implement.

[isuruf]

Here's a draft PR: #1808