Why ArcTan[1, 0. I ] yields -1.5708+0. I?

Question

Bug introduced in 5.2 or earlier and persisting through 12.0.0

---

Why ArcTan[1, 0. I ] yields -1.5708+0. I ? The result should equal to 0.

Is this a bug?

Answer 1

It looks like a bug.

Perhaps

ArcTan[1, p q + $MinMachineNumber * I]

will be acceptable. The little noise is unlikely to be a numerical problem. The main pitfall is when p q == -$MinMachineNumber * I, which drops you back into the buggy case.

ArcTan[1. + 0. I, $MinMachineNumber*I]
(*  0. + 0. I  *)

Update: Runtime environments for existing code

You can overload ArcTan with code like this:

Internal`InheritedBlock[{ArcTan},
 Unprotect[ArcTan];
 ArcTan[x_, y_] := func[x, y];
 Protect[ArcTan];
 ArcTan[1., 0. I]
 ]

There are several issues with doing this, and the simplest thing to do is to understand your particular use-case and choose an appropriate compromise.

For instance, here is one possibility:

ClearAll[runWithNewArcTan];
SetAttributes[runWithNewArcTan, HoldAll];
runWithNewArcTan[code_] :=
  Internal`InheritedBlock[{ArcTan},
   Unprotect[ArcTan];

   (* keeps packed arrays from being unpacked *)
   ClearAttributes[ArcTan, Listable];

   (* makes all ArcTan[x,y] results Complex;
    * vectorized formula means it still works on lists
    * however, the formula can have rounding errors when y == 0
    * of around $MachineEpsilon in magnitude *)
   ArcTan[x_, y_] /; ! FreeQ[{x, y}, _Complex] :=
     -I Log[(x + I y)/Sqrt[x^2 + y^2]];

   Protect[ArcTan];

   (* run code *)
   code
   ];

Here is another:

ClearAll[runWithNewArcTan];
SetAttributes[runWithNewArcTan, HoldAll];
runWithNewArcTan[code_] :=
  Internal`InheritedBlock[{ArcTan},
   Unprotect[ArcTan];
   (* keeping Listable attribute means packed arrays will be unpacked *)

   (* fix just the buggy values; the patterns tests are not
    * vectorized, so the Listable attribute will unpack packed arrays
    * even when neither definition below is used. *)
   ArcTan[x_, y_] /; 
     Precision[{x, y}] === MachinePrecision && 
       ! FreeQ[{x, y}, _Complex] && Positive[x] && y == 0 := 0. + 0. I;
   ArcTan[x_, y_] /; 
     Precision[{x, y}] === MachinePrecision && 
       ! FreeQ[{x, y}, _Complex] && Negative[x] && y == 0 := Pi + 0. I;
   Protect[ArcTan];

   (* run code *)
   code
   ];

The last one would work well in code that does not use packed arrays. I haven't thought of a simple way that would work exactly like ArcTan[x, y] but fix the bug.