How to bi-unify a term's operator in Prolog?

Question

I just asked the question in How to unify a term's operator in Prolog? and got good answers.

However, it doesn't completely solve my problem, i.e. =.. only works for single-side.

For example,

test(Expr) :-
  1 + 2 = Expr.
test(Expr) :-
  1 * 2 = Expr.

?- E =.. [Op, X, Y], test(E). % E can not be a variable!
ERROR: Arguments are not sufficiently instantiated

I hope the query returns

Op = (+),
X = 1,
Y = 2;
Op = (*),
X = 1,
Y = 2.

Very thanks.

Answer 1

expr_op_x_y(Expr, Op, X, Y) :-
   when((nonvar(Expr) ; nonvar(Op)), Expr =.. [Op, X, Y]).

?- expr_op_x_y(Expr, Op, X, Y).
   when((nonvar(Expr);nonvar(Op)),Expr=..[Op,X,Y]).
?- expr_op_x_y(Expr, Op, X, Y), Op = (+).
   Expr = X+Y, Op = (+).
?- expr_op_x_y(Expr, Op, X, Y), Expr = 1+2.
   Expr = 1+2, Op = (+), X = 1, Y = 2.

This could also be called a boomy univ for entirely historical reasons.