#### Fibonacci Number Golden Ration

P
```package com.thealgorithms.maths;

/**
* This class provides methods for calculating Fibonacci numbers using Binet's formula.
* Binet's formula is based on the golden ratio and allows computing Fibonacci numbers efficiently.
*
* @see <a href="https://en.wikipedia.org/wiki/Fibonacci_sequence#Binet's_formula">Binet's formula on Wikipedia</a>
*/
public final class FibonacciNumberGoldenRation {
private FibonacciNumberGoldenRation() {
// Private constructor to prevent instantiation of this utility class.
}

/**
* Compute the limit for 'n' that fits in a long data type.
* Reducing the limit to 70 due to potential floating-point arithmetic errors
* that may result in incorrect results for larger inputs.
*/
public static final int MAX_ARG = 70;

/**
* Calculates the nth Fibonacci number using Binet's formula.
*
* @param n The index of the Fibonacci number to calculate.
* @return The nth Fibonacci number as a long.
* @throws IllegalArgumentException if the input 'n' is negative or exceeds the range of a long data type.
*/
public static long compute(int n) {
if (n < 0) {
throw new IllegalArgumentException("Input 'n' must be a non-negative integer.");
}

if (n > MAX_ARG) {
throw new IllegalArgumentException("Input 'n' is too big to give accurate result.");
}

if (n <= 1) {
return n;
}

// Calculate the nth Fibonacci number using the golden ratio formula
final double sqrt5 = Math.sqrt(5);
final double phi = (1 + sqrt5) / 2;
final double psi = (1 - sqrt5) / 2;
final double result = (Math.pow(phi, n) - Math.pow(psi, n)) / sqrt5;

// Round to the nearest integer and return as a long
return Math.round(result);
}
}
```  