Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits).

I want to approximate the fraction:

$$\frac{M}{N} \sim \frac{k}{L+r}$$

where $r$ is at most $L$. In other words, I want to find good approximations to $M/N$ with denominator larger than $L$ but less than $(1+\delta)L$ where $\delta < 1$.

The standard Stern-Brocot binary search tree can give me good approximations, but it gives no guarantee that these approximations have denominator between $L$ and $(1+\delta)L$.

https://www.johndcook.com/blog/2010/10/20/best-rational-approximation/

https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations

Any modifications we can make to the search to give us the desired approximations?