One can try to improve upon the CFRAC algorithm in three ways:
(1) Find a way of generating Qs that avoids the large amount of multiprecise arithmetic required by CFRAC.
The polynomials generate Qs that are somewhat larger than those from CFRAC, but because we used a polynomial to generate them we can try to factor them much more quickly.
What makes this method faster is that the numbers can be made much smaller than with QS or CFRAC. The algorithm involves some fairly intricate mathematics and is beyond the scope of this pracnique.