Let $v_1,v_2 \in \{0,1\}^n$. Denote $v_1v_2=((v_1)_1 (v_2)_1, \ldots, (v_1)_n (v_2)_n)$ and $|v|=\sum v_{i}$.
\begin{align}
{\scriptsize
f(v_1, v_2) = \sum_{x_1=0}^{|v_1|} \sum_{x_2=0}^{|v_2|} \sum_{d=0}^{|v_1 v_2|} \frac{1}{2^{|v_1|+|v_2|-|v_1 v_2|}} \biggl| {|v_1| - |v_1 v_2| \choose x_1 - d} {|v_2| - |v_1 v_2| \choose x_2 - d} - {|v_1| - |v_1 v_2| \choose x_1 + 1 - d} {|v_2| - |v_1 v_2| \choose x_2 + 1 - d} \biggr|.}
\end{align}
I want to estimate $f(v_1, v_2)$ when $|v_1|, |v_2| \to \infty$. 

As a first step, I obtain
\begin{align}
{ \scriptsize
f(v_1, v_2) = \sum_{x_1=0}^{|v_1|} \sum_{x_2=0}^{|v_2|} \sum_{d=0}^{|v_1v_2|}  \frac{1}{2^{|v_1|+|v_2|-|v_1v_2|}} \biggl| \left( 1- \frac{(v_1-|v_1v_2|-x_1+d)(v_2-|v_1||v_2|-x_2+d)}{(x_1+1-d)(x_2+1-d)} \right) {|v_1| - |v_1v_2| \choose x_1 - d} \biggr|, }
\end{align}

How to estimate $f(v_1,v_2)$? Thank you very much.