For an example with equality, consider the order-$16$ central product of $D_4$ and $C_4$, i.e., the group (of order $16$) $G=NK$ with $N\cong D_4$, $K\cong C_4$, $|N\cap K|=2$ and $C_G(N)=K$. This can be written as a semidirect product $D_4\rtimes C_2$ (the product of two order-$4$ elements of $N$ and of $K$ is an involution, providing a complement to $N$), but not as a direct product (or $Z(G)$ would contain more than one involution). Still, from the definition, it's easy to see that there are twice as many conjugacy classes as in $D_4$ (the class of $g\in N$ is in natural bijection with the one of $xg$ where $x$ is a generator of $K$), i.e., $r(G)=r(D_4)\cdot r(C_2)$.