Did you try polynomial preprocessing methods, like Knuth's and Estrin's methods? https://en.wikipedia.org/wiki/Polynomial_evaluation#Evaluati... they let you compute polynomials with half the multiplications of Horner's method, and I used them in the past to improve the speed of the exponential function in Boost.
yes, Estrin's method is the update
Sorry, I said that wrong. Estrin's doesn't reduce the number of multiplications.
If your goal is reducing the number of multiplications, I imagine it would make sense to factor that polynomial into degree-1 and degree-2 factors.