Say we have two unit vectors $\hat{u}, \hat{v} \in \mathbb{R}^n$ where $\hat{u} = (u_1,...,u_n)$ and $\hat{v}$ approximates $\hat{u}$. $~\hat{v} = (u_1+\epsilon, ...,u_n+\epsilon)$ where $\epsilon = \frac{1}{poly(n)}$.
We have an algorithm which takes the product of all elements and outputs the product. For the exact case (with $\hat{u}$), $x = \prod_i^nu_i$. For the approximate case (with $\hat{v}$), $x^* = \prod_i^n(u_i + \epsilon)$.
What is the error of $x^*$ with respect to $x$? How do we bound the error of $x^*$ in terms of epsilon? How is the error analysis typically handled in this situation?