#5219 rectform(atan2(y, 0)) is wrong for negative y and carries a spurious imaginary part

[David Scherfgen]

Bug report written by Claude.

rectform of atan2(y, 0) for a symbolic real y returns an expression that is off by 2*%pi for y < 0, and that has a nonzero-looking imaginary part -%i*log(abs(y)/sqrt(y^2)), which is zero but is not simplified: sqrt(y^2) was produced with $domain bound to complex inside risplit and is not resimplified to abs(y) once the binding is gone. The real part is not reduced into (-%pi, %pi] either. Any expression that has such an atan2 in an exponent inherits the error, as the last input shows.

(%i1) display2d : false$
(%i2) rectform(atan2(y, 0));
(%o2) (2*%pi*ceiling((2*atan2(0,y)-%pi)/(2*%pi))+%pi)/2-%i*log(abs(y)/sqrt(y^2))
(%i3) subst(y = 2, %);
(%o3) %pi/2
(%i4) subst(y = -2, rectform(atan2(y, 0)));
(%o4) (3*%pi)/2
(%i5) atan2(-2, 0);
(%o5) -%pi/2
(%i6) imagpart(atan2(y, 0));
(%o6) -log(abs(y)/sqrt(y^2))
(%i7) rectform(%e^(%i*atan2(y, 0)/3));
(%o7) %e^(log(abs(y)/sqrt(y^2))/3)*%i*sin((2*%pi*ceiling((2*atan2(0,y)-%pi)/(2*%pi))+%pi)/6)+%e^(log(abs(y)/sqrt(y^2))/3)*cos((2*%pi*ceiling((2*atan2(0,y)-%pi)/(2*%pi))+%pi)/6)

Expected: atan2(y, 0) is real for every real y, so imagpart(atan2(y, 0)) should be 0 and rectform(atan2(y, 0)) an expression equal to %pi/2 for y > 0 and -%pi/2 for y < 0, for instance atan2(y, 0) itself, or %pi/2 - atan2(0, y), which rectform(atan2(y, 1)) and rectform(atan2(1, 0)) get right.

Where: risplit in src/rpart.lisp, the clause for %atan2, which goes through the logarithmic form of the arc tangent; the reduction by the ceiling term is applied to 2*atan2(0, y) and does not bring the result back into the principal range.