I am writing an algorithm that evaluates the square root of a positive real number $y$. To do this I am using the Newton-Raphton method to approximate the roots to $f(x)=x^2-y$. The $n^{th}$ iteration gives $$x_n=\frac{x_{n-1}^2+y}{2x_{n-1}}$$ as an approximation to $\sqrt{y}$. I found that starting with an initial guess $x_0=1$ works pretty well generally, so an answer to the question below that assumes $x_0=1$ is fine.
My question: is there an exact expression for the minimum $N$ of iterations needed to attain a given precision $p$ in the approximate solution $x_N$? In other words I'm looking for the smallest integer $N$ such that $$\left|\frac{x_N-\sqrt y}{\sqrt y}\right|<p.$$
I've thought about this for a while and played around with the expression for the errors $\epsilon_n = x_n - \sqrt y$ which can be shown to satisfy $\epsilon_{n+1}=\epsilon_n^2\,/\,2x_n$, but I can't find an answer. I've looked around on Google but I couldn't find an answer either.
Any pointers to a solution online or help would be much appreciated. A follow-up would of course be: can $x_0$ be optimised (while being a simple enough expression in terms of $y$) in order to minimise $N$?