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# fibonacci.py

"""

Calculates the Fibonacci sequence using iteration, recursion, memoization,

and a simplified form of Binet's formula

NOTE 1: the iterative, recursive, memoization functions are more accurate than

the Binet's formula function because the Binet formula function uses 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

RESULTS: (n = 20)

fib_iterative runtime: 0.0055 ms

fib_recursive runtime: 6.5627 ms

fib_memoization runtime: 0.0107 ms

fib_binet runtime: 0.0174 ms

"""

import functools

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_recursive_cached(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

"""

@functools.cache

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_memoization(n: int) -> list[int]:

"""

Calculates the first n (0-indexed) Fibonacci numbers using memoization

>>> fib_memoization(0)

[0]

>>> fib_memoization(1)

[0, 1]

>>> fib_memoization(5)

[0, 1, 1, 2, 3, 5]

>>> fib_memoization(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")

# Cache must be outside recursuive function

# other it will reset every time it calls itself.

cache: dict[int, int] = {0: 0, 1: 1, 2: 1} # Prefilled cache

def rec_fn_memoized(num: int) -> int:

if num in cache:

return cache[num]

value = rec_fn_memoized(num - 1) + rec_fn_memoized(num - 2)

cache[num] = value

return value

return [rec_fn_memoized(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 = 30

time_func(fib_iterative, num)

time_func(fib_recursive, num) # Around 3s runtime

time_func(fib_recursive_cached, num) # Around 0ms runtime

time_func(fib_memoization, num)

time_func(fib_binet, num)

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