Is There a Faster Way to Check if an Integer's Square Root is an Integer?

Fastest way to determine if an integer's square root is an integer

Overview

The provided C/C code offers a highly optimized method to determine if an integer's square root is itself an integer. The code leverages various optimizations to significantly improve performance compared to the basic approach of using the built-in Math.sqrt() function.

Implementation Details

1. Quick Failure Checks: The code first performs a series of quick checks to identify cases where the square root cannot be an integer, based on the integer's sign and bit patterns. If any of these checks fail, the function immediately returns false.
2. Modulo 255 Check: To further filter out non-squares, the code calculates the integer modulo 255 (a product of three prime numbers) and checks if the result is known to not be a square. A precomputed table, bad255, is used for this purpose.
3. Dividing Out Powers of 4: The code uses binary searches to divide out powers of 2 and 4 from the input integer, effectively reducing the problem size.
4. Final Check (Hensel's Lemma-Inspired): The final step involves using an iterative process inspired by Hensel's lemma to compute the square root of the reduced integer. The process starts with an initial estimate and iteratively refines it using bitwise operations and modulo tricks.

Improvements over Math.sqrt()

The provided code offers significant speed advantages over the basic approach of using Math.sqrt(). By applying various optimizations and exploiting mathematical properties, the code can determine integer square roots much faster, particularly for large integers.

Complexity Analysis

The time complexity of the code is influenced by the number of iterations needed in the final step. In most cases, a small number of iterations (often less than 10) suffice. Therefore, the overall complexity is approximately O(1).

Usage

The code provides the square() function, which takes an integer as its parameter and returns true if it is a perfect square and false otherwise. It can be easily integrated into any C/C program to quickly and efficiently check for integer square roots.

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