Numerical evaluation of the power function

I would like to suggest to implement the numerical evaluation of the
power function z^a for complex values z or a in float or bigfloat
precision too. We already evaluate numerically for real float or
bigfloat values. Furthermore, we have special code to evaluate the exp
function and the sqrt function.

This could be the implementation for float precision. The bigfloat
implementation is similar.

((and (complex-float-numerical-eval-p gr pot)
      (or (not (ratnump gr)) (not (ratnump pot))))
 (let ((xr ($float ($realpart gr)))
       (xi ($float ($imagpart gr)))
       (yr (if (integerp pot) pot ($float ($realpart pot))))
       (yi ($float ($imagpart pot))))
  (return ($float ($rectform (list '(mexpt)
                                   (add xr (mul '$%i xi))
                                   (if (integerp yr)
                                   yr
                                   (add yr (mul '$%i yi)))))))))

The testsuite and the share_testsuite will have no problems.

The sqrt and exp functions evalutate already numerically:

(%i7) sqrt(1.0+%i);
(%o7) .4550898605622273*%i+1.09868411346781

(%i8) exp(1.0+%i);
(%o8) 2.287355287178842*%i+1.468693939915885

But not the power function in general:

(%i9) (1.0+%i)^2;
(%o9) (%i+1.0)^2

(%i10) (1.0+%i)^(1/3);
(%o10) (%i+1.0)^(1/3)

(%i11) (1.0+%i)^2.5;
(%o11) (%i+1.0)^2.5

That is what we will get:

(%i3) (1.0+%i)^2;
(%o3) 2.0 %i

(%i4) (1.0+%i)^(1/3);
(%o4) .2905145555072514 %i + 1.084215081491351

(%i5) (1.0+%i)^2.5;
(%o5) 2.19736822693562 %i - .9101797211244543

Furthermore, numer and float will work as expected:

(%i5) (1.0+%i)^2,numer;
(%o5) 2.0*%i

(%i6) 1/(1.0+%i)^2,numer;
(%o6) -0.5*%i

(%i8) float((1+%i)^(1/3));
(%o8) .2905145555072514*%i+1.084215081491351


I think a lot of things would become much simpler if the power function
evaluates numerically for complex values too. Furthermore, I think it is
the expected behavior.

Dieter Kaiser