exponential additive functional equation

Let $S$ be a semigroup with no identity element and $m:S\to \Bbb C$ be given function($m\not\equiv 0$) satisfying the exponential functional equation
$$
m(x+y)=m(x)m(y)
$$
for all $x, y\in S$. Find all solutions $f:S\to \Bbb C$ satisfying

the equation
\begin{equation}
f(x+y)=f(x)m(y)+f(y)m(x) \tag 1
\end{equation}
for all $x, y\in S$.

Remark. If $S$ is a group, then using the fact that $m(x)\ne 0$ for all $x\in S$ and dividing $(1)$ by $m(x+y)$, we have
$$
f(x)=m(x)A(x)
$$

for all $x\in S$, where $A$ is an additive function.