Let $x\in \mathbb{C}$. I want to analyze the stability of the numerical algorithm computing $2x$ by $f(x) = fl(x) \oplus fl(x)$. So here is what I did:
$|fl(x)+fl(x)|(1+\epsilon_1)=|fl(a+ib)+fl(a+ib)|(1+\epsilon_1)=|a(1+\epsilon_2)+ib(1+\epsilon_3)+a(1+\epsilon_2+ib(1+\epsilon_3)|(1+\epsilon_1)\le 2|a+ib|(1+2\epsilon_{mach})+O(\epsilon_{mach}^2)=|x+x|(1+2\epsilon_{mach})+O(\epsilon_{mach}^2).$
As far as I understand, this algorithm is at least stable by this inequality, but not backward stable. However, I read in a textbook that this algorithm is actually backward stable.
What am I doing wrong?