mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-24 13:31:07 +00:00
62 lines
2.0 KiB
Python
62 lines
2.0 KiB
Python
|
from decimal import Decimal, getcontext
|
|||
|
|
from math import ceil, factorial
|
|||
|
|
|
|||
|
|
|
|||
|
|
def pi(precision: int) -> str:
|
|||
|
|
"""
|
|||
|
|
The Chudnovsky algorithm is a fast method for calculating the digits of PI,
|
|||
|
|
based on Ramanujan’s PI formulae.
|
|||
|
|
|
|||
|
|
https://en.wikipedia.org/wiki/Chudnovsky_algorithm
|
|||
|
|
|
|||
|
|
PI = constant_term / ((multinomial_term * linear_term) / exponential_term)
|
|||
|
|
where constant_term = 426880 * sqrt(10005)
|
|||
|
|
|
|||
|
|
The linear_term and the exponential_term can be defined iteratively as follows:
|
|||
|
|
L_k+1 = L_k + 545140134 where L_0 = 13591409
|
|||
|
|
X_k+1 = X_k * -262537412640768000 where X_0 = 1
|
|||
|
|
|
|||
|
|
The multinomial_term is defined as follows:
|
|||
|
|
6k! / ((3k)! * (k!) ^ 3)
|
|||
|
|
where k is the k_th iteration.
|
|||
|
|
|
|||
|
|
This algorithm correctly calculates around 14 digits of PI per iteration
|
|||
|
|
|
|||
|
|
>>> pi(10)
|
|||
|
|
'3.14159265'
|
|||
|
|
>>> pi(100)
|
|||
|
|
'3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706'
|
|||
|
|
>>> pi('hello')
|
|||
|
|
Traceback (most recent call last):
|
|||
|
|
...
|
|||
|
|
TypeError: Undefined for non-integers
|
|||
|
|
>>> pi(-1)
|
|||
|
|
Traceback (most recent call last):
|
|||
|
|
...
|
|||
|
|
ValueError: Undefined for non-natural numbers
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
if not isinstance(precision, int):
|
|||
|
|
raise TypeError("Undefined for non-integers")
|
|||
|
|
elif precision < 1:
|
|||
|
|
raise ValueError("Undefined for non-natural numbers")
|
|||
|
|
|
|||
|
|
getcontext().prec = precision
|
|||
|
|
num_iterations = ceil(precision / 14)
|
|||
|
|
constant_term = 426880 * Decimal(10005).sqrt()
|
|||
|
|
multinomial_term = 1
|
|||
|
|
exponential_term = 1
|
|||
|
|
linear_term = 13591409
|
|||
|
|
partial_sum = Decimal(linear_term)
|
|||
|
|
for k in range(1, num_iterations):
|
|||
|
|
multinomial_term = factorial(6 * k) // (factorial(3 * k) * factorial(k) ** 3)
|
|||
|
|
linear_term += 545140134
|
|||
|
|
exponential_term *= -262537412640768000
|
|||
|
|
partial_sum += Decimal(multinomial_term * linear_term) / exponential_term
|
|||
|
|
return str(constant_term / partial_sum)[:-1]
|
|||
|
|
|
|||
|
|
|
|||
|
|
if __name__ == "__main__":
|
|||
|
|
n = 50
|
|||
|
|
print(f"The first {n} digits of pi is: {pi(n)}")
|