qqbar: higher-degree guessing hack

[fredrik-johansson]

qqbar binary operations guess and check for a rational solution when the inputs have high degree. This could be generalized to higher degree guesses (2? 4?) when the inputs have high enough degree, as long as things don't slow down noticeably in the generic case.

[fredrik-johansson]

Simple attempt to guess high-degree results:

if (dx >= 8 && dy >= 8)
{
    slong d, bits, prec;
    acb_t z, z1, z2;
    qqbar_t q;
    int found = 0;

    d = n_gcd(qqbar_degree(x), qqbar_degree(y));
    bits = qqbar_height_bits(x) + qqbar_height_bits(y) + 5;
    prec = d * bits;

    acb_init(z);
    acb_init(z1);
    acb_init(z2);
    qqbar_init(q);

    qqbar_get_acb(z1, x, prec);
    qqbar_get_acb(z2, y, prec);

    if (op == 0)
        acb_add(z, z1, z2, prec);
    else if (op == 1)
        acb_sub(z, z1, z2, prec);
    else if (op == 2)
        acb_mul(z, z1, z2, prec);
    else
        acb_div(z, z1, z2, prec);

    if (qqbar_guess(q, z, d, bits, 0, prec))
    {
        fmpz_poly_t H, Q;

        fmpz_poly_init(H);
        fmpz_poly_init(Q);

        qqbar_fmpz_poly_composed_op(H, QQBAR_POLY(x), QQBAR_POLY(y), op);

        if (fmpz_poly_divides(Q, H, QQBAR_POLY(q)))
        {
            arb_fmpz_poly_evaluate_acb(z, Q, z, prec);
            if (!acb_contains_zero(z))
            {
                qqbar_swap(res, q);
                found = 1;
            }
        }

        fmpz_poly_clear(H);
        fmpz_poly_clear(Q);
    }

    acb_clear(z);
    acb_clear(z1);
    acb_clear(z2);
    qqbar_clear(q);

    printf("found [%ld, %ld : %ld, bits = %ld, prec = %ld] = %i   {%ld, %ld}\n", dx, dy, d, bits, prec, found, found * qqbar_degree(res), found * qqbar_height_bits(res));

    if (found)
        return;
}

This seems to slow things down, so it needs to be done much more selectively and/or cleverly.