Python/maths/fibonacci.py

# fibonacci.py
"""
Calculates the Fibonacci sequence using iteration, recursion, and a simplified
form of Binet's formula

NOTE 1: the iterative and recursive functions are more accurate than the Binet's
formula function because the iterative function doesn't use floats

NOTE 2: the Binet's formula function is much more limited in the size of inputs
that it can handle due to the size limitations of Python floats
"""

from math import sqrt
from time import time


def time_func(func, *args, **kwargs):
    """
    Times the execution of a function with parameters
    """
    start = time()
    output = func(*args, **kwargs)
    end = time()
    if int(end - start) > 0:
        print(f"{func.__name__} runtime: {(end - start):0.4f} s")
    else:
        print(f"{func.__name__} runtime: {(end - start) * 1000:0.4f} ms")
    return output


def fib_iterative(n: int) -> list[int]:
    """
    Calculates the first n (0-indexed) Fibonacci numbers using iteration
    >>> fib_iterative(0)
    [0]
    >>> fib_iterative(1)
    [0, 1]
    >>> fib_iterative(5)
    [0, 1, 1, 2, 3, 5]
    >>> fib_iterative(10)
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
    >>> fib_iterative(-1)
    Traceback (most recent call last):
    ...
    Exception: n is negative
    """
    if n < 0:
        raise Exception("n is negative")
    if n == 0:
        return [0]
    fib = [0, 1]
    for _ in range(n - 1):
        fib.append(fib[-1] + fib[-2])
    return fib


def fib_recursive(n: int) -> list[int]:
    """
    Calculates the first n (0-indexed) Fibonacci numbers using recursion
    >>> fib_iterative(0)
    [0]
    >>> fib_iterative(1)
    [0, 1]
    >>> fib_iterative(5)
    [0, 1, 1, 2, 3, 5]
    >>> fib_iterative(10)
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
    >>> fib_iterative(-1)
    Traceback (most recent call last):
    ...
    Exception: n is negative
    """

    def fib_recursive_term(i: int) -> int:
        """
        Calculates the i-th (0-indexed) Fibonacci number using recursion
        """
        if i < 0:
            raise Exception("n is negative")
        if i < 2:
            return i
        return fib_recursive_term(i - 1) + fib_recursive_term(i - 2)

    if n < 0:
        raise Exception("n is negative")
    return [fib_recursive_term(i) for i in range(n + 1)]


def fib_binet(n: int) -> list[int]:
    """
    Calculates the first n (0-indexed) Fibonacci numbers using a simplified form
    of Binet's formula:
    https://en.m.wikipedia.org/wiki/Fibonacci_number#Computation_by_rounding

    NOTE 1: this function diverges from fib_iterative at around n = 71, likely
    due to compounding floating-point arithmetic errors

    NOTE 2: this function doesn't accept n >= 1475 because it overflows
    thereafter due to the size limitations of Python floats
    >>> fib_binet(0)
    [0]
    >>> fib_binet(1)
    [0, 1]
    >>> fib_binet(5)
    [0, 1, 1, 2, 3, 5]
    >>> fib_binet(10)
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
    >>> fib_binet(-1)
    Traceback (most recent call last):
    ...
    Exception: n is negative
    >>> fib_binet(1475)
    Traceback (most recent call last):
    ...
    Exception: n is too large
    """
    if n < 0:
        raise Exception("n is negative")
    if n >= 1475:
        raise Exception("n is too large")
    sqrt_5 = sqrt(5)
    phi = (1 + sqrt_5) / 2
    return [round(phi ** i / sqrt_5) for i in range(n + 1)]


if __name__ == "__main__":
    num = 20
    time_func(fib_iterative, num)
    time_func(fib_recursive, num)
    time_func(fib_binet, num)