# Complexity of the Schönhage–Strassen algorithm

In the Wikipedia article, the complexity is listed as $O(n \cdot \log (n) \cdot \log (\log (n)))$, where $n$ is the number of bits.

Would the real bound be given by setting $n=\frac{b}{w}$, where $b$ is the number of bits and $w$ is the word length in the computer? i.e., if I assume that numbers of size $\log (n)$ can be multiplied in $O(1)$ time, then can two numbers of size $r \log (n)$ be multiplied in $O(r \cdot \log (r) \cdot \log( \log (r)))$?