Evaluating an fmpz_poly or fmpq_poly at a qqbar argument just uses Horner's rule at the moment, requiring repeated expensive qqbar arithmetic operations. It should be possible to do much better, i.e. finding an annihilating polynomial in one go using linear algebra (at least so I've been told -- I don't know the algorithm myself).
Evaluating an fmpz_poly or fmpq_poly at a qqbar argument just uses Horner's rule at the moment, requiring repeated expensive qqbar arithmetic operations. It should be possible to do much better, i.e. finding an annihilating polynomial in one go using linear algebra (at least so I've been told -- I don't know the algorithm myself).
The same applies to powering.