:- initialization(main).

main :-
    exponentiation((e,0),(0,pi),(A,B)),
    add((A,B),(1,0),(C,D)),
    C < 2.220446049250313e-16,
    D < 2.220446049250313e-16,
    write('[] = "PASS".'),
    nl.

exponentiation((A,B),(C,D),(E,F)) :-
    polar((A,B),(R,T)),
    E is R^C*exp(-D*T)*cos(D*log(R)+C*T),
    F is R^C*exp(-D*T)*sin(D*log(R)+C*T).

polar((X,Y),(R,Tp)) :-
    R is sqrt(X**2+Y**2),
    T is acos(abs(X)/R),
    angular(X,Y,T,Tp).

angular(X,Y,T,Tp) :-
    X >= 0,
    Y >= 0,
    Tp = T.
angular(X,Y,T,Tp) :-
    X < 0,
    Y >= 0,
    Tp is pi-T.
angular(X,Y,T,Tp) :-
    X < 0,
    Y < 0,
    Tp is T+pi.
angular(X,Y,T,Tp) :-
    X >= 0,
    Y < 0,
    Tp is 2*pi-T.

minus((X1,X2),(Y1,Y2)) :-
    Y1 is -X1,
    Y2 is -X2.

minus((X1,X2),(Y1,Y2),(Z1,Z2)) :-
    Z1 is X1-Y1,
    Z2 is X2-Y2.

add((X1,X2),(Y1,Y2),(Z1,Z2)) :-
    Z1 is X1+Y1,
    Z2 is X2+Y2.

times((X1,X2),(Y1,Y2),(Z1,Z2)) :-
    Z1 is X1*Y1-X2*Y2,
    Z2 is X1*Y2+X2*Y1.

inverse((X1,X2),(Y1,Y2)) :-
    Y1 is X1/(X1**2+X2**2),
    Y2 is -X2/(X1**2+X2**2).

divide(X,Y,Z) :-
    inverse(Y,Yp),
    times(X,Yp,Z).