Separability of the nearest power of two function I have a function $f$ given by

$f(x)=nearest\ upper\ power\ of\ two\ of\ x$

For example, $f(5)=8$ and $f(2)=2$.

Are $f(x \wedge y)$ and $f(x+y\mod{N})$ separable over some operator. For example, could we do:

$f(x\wedge y)=g(x)g(y)$ 

or 

$f(x+y\mod{N})=g(x)\ op\ g(y)$

where $op$ is some arbitrary operator.

In other words, what can be done algebraically with the nearest power of two function?

In Warren's Hacker's delight, we have that the nearest power of two function is given by:

$
f(x)=1&lt;&lt;(32-nlz(x-1)-1)
$

on a 32 bit register and with $nlz$ being the number of leading zeros. Maybe this is of use?
No, $f(x+y)$ is not separable.

Consider: $f(3+5)=8$.  Also $f(3)=4$ and $f(5)=8$.

At the same time, $f(4+5)=16$.  Also $f(4)=4$ and $f(5)=8$.

It follows that $f(x+y)$ is not determined by $f(x)$ and $f(y)$ (consider $y=5$, $x_1=3$, $x_2=4$), so there is no function $h$ such that $f(x+y)=h(f(x),f(y))$.