Performance of Apart with complex numbers

Question

Consider the following code

randExpr[] := 
  Together@Expand[(a - b - c - I RandomInteger[{10^3, 10^4}] d + 
       3/(a - b - c - d))^6];
test = randExpr[];
AbsoluteTiming[res1 = Apart[test, a];]

On my system (Fedora 22, Mathematica 10.3, i5-3320M) I get

{12.3737, Null}

Now with the following trivial workaround

myApart[expr_, x_] := 
  Block[{i}, 
    If[  FreeQ[expr, Complex], 
         Apart[expr, x], 
         Apart[expr /. Complex[0, a_] :> i a, x] /. i -> I
    ]
  ];
AbsoluteTiming[res2 = myApart[test, a];]

the output becomes

{0.387141, Null}

and of course

res1 === res2

returns True. I would expect that Apart should be intelligent enough to figure out this particular case automatically, instead of wasting computer time.

Do you think that the current is behavior is worth reporting to the support as a bug, or is it yet something one should expect and tolerate?

I would also be interested in other tricks to make Apart work faster when complex numbers are involved.

P.S. The issue was originally discovered by Rolf Mertig, who provided the given example.

Answer 1

You can always inject some code into the definitions of expressions (such as Apart) which handles the special cases you would like it to treat differently. For example

Unprotect[Apart];

Apart[expr_, x_] := Module[{i},
   Apart[expr /. Complex[re_, im_] :> (re + i im), x] /. i -> I
] /; Not[FreeQ[expr, Complex[_, _]]]

Protect[Apart];

will modify the in-built Apart function, such that it does exactly what you want. Specifically, after evaluating the above, you find that

AbsoluteTiming[res3 = Apart[test, a];]
(* {0.789089, Null} *)

is now just as fast as your myApart function.