In this paper, our approach allows to refine the results announced by Ebadian et al. [Results Math., 36 (2013), 409--423]. Namely, we reduce the distance between approximate and exact double derivations on Banach algebras and Lie $C^*$-algebras up to $\frac{1}{2^{n-1}}$ and $\frac{1}{2^{n-2}}$ for $n\geq 2$. Indeed, we give a correct utilization of fixed point theory in the sense of Diaz and Margolis [Bull. Amer. Math. Soc., 74 (1968), 305--309] concerning the stability of double derivations.