Python/dynamic_programming/fast_fibonacci.py

#!/usr/bin/env python3

"""
This program calculates the nth Fibonacci number in O(log(n)).
It's possible to calculate F(1_000_000) in less than a second.
"""

from __future__ import annotations

import sys


def fibonacci(n: int) -> int:
    """
    return F(n)
    >>> [fibonacci(i) for i in range(13)]
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
    """
    if n < 0:
        raise ValueError("Negative arguments are not supported")
    return _fib(n)[0]


# returns (F(n), F(n-1))
def _fib(n: int) -> tuple[int, int]:
    if n == 0:  # (F(0), F(1))
        return (0, 1)

    # F(2n) = F(n)[2F(n+1) - F(n)]
    # F(2n+1) = F(n+1)^2+F(n)^2
    a, b = _fib(n // 2)
    c = a * (b * 2 - a)
    d = a * a + b * b
    return (d, c + d) if n % 2 else (c, d)


if __name__ == "__main__":
    n = int(sys.argv[1])
    print(f"fibonacci({n}) is {fibonacci(n)}")