Suppose wehave two sequences:
$(a_0, a_1, a_2, \dots, a_{2^n-1})$ $(b_0, b_1, b_2, \dots, b_{2^n-1})$
We also have the following sum:
$\sum_{k=0}^{2^n-1}{a_k \cdot b_k}$
I'd like to know the fastest way to get this sum:
$\sum_{k=0}^{2^n-1}{(a_k+c) \cdot (b_k+d)}$
MY THOUGHTS AND WORK
I thought that some statiscal information could be crucial. We can suppose that other statistics are handy - I will try to find a way to get them. I'm mainly interested in the fastest way to calculate this problem.
I figured that if we could get some kind of geometric mean and perhaps standard deviation for both sequences, we might somehow be able to add $c$ and $d$ to these values and simply multiply them together. I'm not very handy with statistics though, so I really don't know if this is possible.
WHAT I'D LIKE TO DO
I'm hoping that this could become a community wiki so that we could list all methods that might have a shot at being fast, and use feedback to find the best answer.