5.8.3 Converting to a polynomial or rational expression

Some of the methods described so far only work on polynomials or rational functions. GiNaC provides a way to extend the domain of these functions to general expressions by using the temporary replacement algorithm described above. You do this by calling

ex ex::to_polynomial(exmap & m);
ex ex::to_polynomial(lst & l);

or

ex ex::to_rational(exmap & m);
ex ex::to_rational(lst & l);

on the expression to be converted. The supplied exmap or lst will be filled with the generated temporary symbols and their replacement expressions in a format that can be used directly for the subs() method. It can also already contain a list of replacements from an earlier application of .to_polynomial() or .to_rational(), so it's possible to use it on multiple expressions and get consistent results.

The difference between .to_polynomial() and .to_rational() is probably best illustrated with an example:

{
    symbol x("x"), y("y");
    ex a = 2*x/sin(x) - y/(3*sin(x));
    cout << a << endl;

    lst lp;
    ex p = a.to_polynomial(lp);
    cout << " = " << p << "\n   with " << lp << endl;
     // = symbol3*symbol2*y+2*symbol2*x
     //   with {symbol2==sin(x)^(-1),symbol3==-1/3}

    lst lr;
    ex r = a.to_rational(lr);
    cout << " = " << r << "\n   with " << lr << endl;
     // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
     //   with {symbol4==sin(x)}
}

The following more useful example will print ‘sin(x)-cos(x)’:

{
    symbol x("x");
    ex a = pow(sin(x), 2) - pow(cos(x), 2);
    ex b = sin(x) + cos(x);
    ex q;
    exmap m;
    divide(a.to_polynomial(m), b.to_polynomial(m), q);
    cout << q.subs(m) << endl;
}