Make atan2(0,0) consistent

[kcrisman]

This is bad:

    sage: atan2(0,0)
    0
    sage: atan2(0,0,hold=True)
    arctan2(0, 0)
    sage: atan2(0,0,hold=True).n()
    ValueError: arctan2(0,0) undefined
    sage: atan2(0,0,hold=True).simplify()
    -----------------------------------------------------
    TypeError: Error executing code in Maxima
    CODE:
    sage1 : atan2(0,0)$
    Maxima ERROR:
    
    atan2: atan2(0,0) is undefined.
     -- an error. To debug this try: debugmode(true);

Probably we should make sure that atan2(0,0) always raises an error. Wolfram Alpha/Mathematica? gives the interval from -pi to pi, which seems as good as any answer, and who knows what others do.

Originally reported at ​ask.sagemath.org.

[vbraun]

I made a patch to pynac yield the following. I'll try to push the patch into the new pynac release at #9880.

sage: atan2(0,0)
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)

/home/vbraun/opt/sage-4.7.1.alpha2/devel/sage-main/<ipython console> in <module>()

/home/vbraun/Sage/sage/local/lib/python2.6/site-packages/sage/symbolic/function.so in sage.symbolic.function.GinacFunction.__call__ (sage/symbolic/function.cpp:6712)()

/home/vbraun/Sage/sage/local/lib/python2.6/site-packages/sage/symbolic/function.so in sage.symbolic.function.Function.__call__ (sage/symbolic/function.cpp:4650)()

RuntimeError: arctan2_eval(): arctan2(0,0) encountered

[kcrisman]

Should that be reported upstream? Just curious.

[burcin]

GiNaC probably got atan2(0,0) -> 0 from Maple:

    |\^/|     Maple 12 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2008
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> arctan(0,0);
                                       0

MMA prints an error like Maxima:

In[5]:= ArcTan[0,0]

ArcTan::indet: Indeterminate expression ArcTan[0, 0] encountered.

Out[5]= Interval[{-Pi, Pi}]

Titus points out that with Volker's patch, arg(0) in Sage also gives an error. All of MMA, Maple and Maxima evaluate that to 0 though.

[burcin]

Pynac 0.2.4 (#12950), includes the patch Volker mentioned in comment:2. Attached to that ticket there is also a patch by Titus fixing failing doctests in symbolics related files and modifying the arg() function so that arg(0) still returns 0.

Apparently this change breaks some tests in 3d plotting. We should open a ticket with that and make this one depend on it. Titus, can you handle that?

[kcrisman]

Apparently this change breaks some tests in 3d plotting. We should open a ticket with that and make this one depend on it. Titus, can you handle that?

And please cc me on that ticket, thanks.

[burcin]

This was fixed with #12950.