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.