If I am trying to approximate $x = \sqrt D$ such that D is a square free integer I can use Diophantine approximation and the Fundamental Recurrence Formulas to find a rational approximation $\dfrac{a}{b}$ for $\sqrt D$ to a given accuracy $M$ where $a, b \in \mathbb Z$ and $|a|, |b| \le M$. I noticed that Hurwitz theorem gives an upper bound for these approximations to $\dfrac{1}{\sqrt 5b^2}$ and the Lagrange/Markov spectrum demonstrates an improved upper bound of $\dfrac{1}{\sqrt 8b^2}$ if irrational numbers equivalent to $\phi$ the golden ratio are excluded.
So, if I instead use $a, b \in \mathbb Z[\phi]$ and $a$ and $b$ are still bounded by $M$, how much more accurate an approximation can I get given some $M$? How would I go about finding good values of $a$ and $b$?