Continuous Binary Search over Metric Space

Question

I am studying the binary search problem on continuous intervals using a non-linear distance metric D. In this interval search, I'm restricting myself to binary queries, meaning the only information available is whether the current x is less than x∗. The goal is to find an ε-approximation x' to the actual result x* while making sure that D(x', x∗) ≤ ε. I'm also working on establishing upper and lower bounds on query complexity for this problem.

By distance metric, I mean a function D that measures the distance/difference between any two points of the interval, D(·,·) → R0+ ∪ ∞. This article presents an algorithm closely related to my work, involving continuous search and ε-approximated output.

my_sqrt(x):
    epsilon = 0.001 
    L = 0, R = x
    while (R - L > epsilon):
        mid = L + (R - L)/2
        if (mid*mid <= x):
            L = mid
        else
            R = mid
    return L + (R - L)/2

It utilizes a linear function (absolute value) for the distance metric, but my focus is mainly on incorporating a non-linear metric, such as the square root of log ratio. For any pair of points x and x' within the interval, the distance is computed as D(x, x') = sqrt(log(x) - log(x')) = sqrt(log(x/x')).

I'm thinking of calling this problem metric binary search, but I'm curious if there's any existing research or a commonly used term for this type of algorithm. Additionally, I'm interested in finding out if there's been any study on an unbounded version of this algorithm (like how the binary search extends to unbounded binary search).