#include <stdio.h>

int main() {
    int n, t1, t2, t, a, b, msb, temp, i;
    printf("Enter n: ");
    scanf("%d", &n);

    if (n == 0) {
        printf("Fibonacci = 0\n");
        return 0;
    }

    a = 0; // F(0)
    b = 1; // F(1)

    // Find the highest set bit position
    msb = 0;
    temp = n;
    while (temp > 1) {
        temp >>= 1;
        msb++;
    }

    // Bottom-up doubling method
    for (i = msb; i >= 0; i--) {
        t1 = a * ((b << 1) - a);   // F(2k)
        t2 = a * a + b * b;        // F(2k+1)
        a = t1;
        b = t2;

        if ((n >> i) & 1) {  // If bit is set
            int t = a + b;  // move to next This moves from  (F(2k),F(2k+1)) (F(2k),F(2k+1)) → (F(2k+1),F(2k+2))(F(2k+1),F(2k+2)).
            a = b;	
            b = t;
        }
    }

    printf("Fibonacci = %d\n", a);
    return 0;
}

/*
 * Binary of 5 = 101

Highest bit index = 2

Loop steps (showing only a = F(k)):

Start: (a,b) = (0,1)

Bit 2: doubling → (0,1) → (0,1). Bit=1 → shift → (1,1)

Bit 1: doubling → (1,1) → (1,2). Bit=0 → no shift

Bit 0: doubling → (1,2) → (3,5). Bit=1 → shift → (5,8)

Done: a = 5 = F(5). 
 * */