Mastering Interview Math: Implement a Custom Square Root Function Without Libraries

Interview Perspective

In technical interviews, the interview is a two‑way selection where both the candidate and the interviewer aim to assess fit and problem‑solving ability.

Problem Statement

Implement a function double sqrt(int v, double t) that returns a value within a given error tolerance t without using library functions such as Math.sqrt.

Basic Linear Approach

A simple method iterates from 0 upward until the square exceeds v , then refines the result by adding t until the square surpasses v . This works but is inefficient for large v or small t.

Binary Search Solution

By treating the problem as searching a sorted range, binary search reduces the number of checks. The search interval can start as [0, v] or tighter [0, v/2]. At each step the midpoint is squared and compared to v , narrowing the interval until the difference is within t.

The termination condition can be expressed as |r² - v| ≤ t·v, where r is the current approximation.

Further Optimizations

For very high precision, the linear method becomes too slow. Possible optimizations include increasing the step size adaptively or switching to more advanced techniques such as Newton’s method, gradient descent, or Taylor series expansions.

Testing Considerations

Robust test cases should cover normal values, edge cases like v = 0 or negative inputs, and verify that the returned interval respects the specified tolerance.

FAQ

Why is v an int? The original LeetCode problem used integers, but the type can be changed.

Does solving this problem guarantee interview success? No, it only demonstrates problem‑solving skills.

Is binary search the only solution? No, brute‑force, Newton’s method, and other mathematical approaches are also valid.