


'''simple docstring'''



from __future__ import annotations

from math import gcd







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__ = 2 ,      UpperCamelCase__ = 1 ,      UpperCamelCase__ = 3 ,      ):
			# A value less than 2 can cause an infinite loop in the algorithm.
			if num < 2:
						raise ValueError("""The input value cannot be less than 2"""					)

			# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
			# algorithm struggles to find factors that are divisible by two.
			# As a workaround, we specifically check for two and even inputs.
			#   See: https://math.stackexchange.com/a/2856214/165820
			if num > 2 and num % 2 == 0:
						return 2

			# Pollard's Rho algorithm requires a function that returns pseudorandom
			# values between 0 <= X < ``num``.  It doesn't need to be random in the
			# sense that the output value is cryptographically secure or difficult
			# to calculate, it only needs to be random in the sense that all output
			# values should be equally likely to appear.
			# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
			# However, the success of Pollard's algorithm isn't guaranteed and is
			# determined in part by the initial seed and the chosen random function.
			# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
			# where ``C`` is a value that we can modify between each attempt.
			def rand_fn(UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__					) -> int:
						return (pow(UpperCamelCase__ ,      2					) + step) % modulus

			for _ in range(UpperCamelCase__					):
						# These track the position within the cycle detection logic.
						UpperCAmelCase__  :					Any        =     seed
						UpperCAmelCase__  :					Union[str, Any]        =     seed

						while True:
									# At each iteration, the tortoise moves one step and the hare moves two.
									UpperCAmelCase__  :					str        =     rand_fn(UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__					)
									UpperCAmelCase__  :					int        =     rand_fn(UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__					)
									UpperCAmelCase__  :					List[Any]        =     rand_fn(UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__					)

									# At some point both the tortoise and the hare will enter a cycle whose
									# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
									# the tortoise and hare will end up on the same value modulo ``p``.
									# We can detect when this happens because the position difference between
									# the tortoise and the hare will share a common divisor with ``num``.
									UpperCAmelCase__  :					Any        =     gcd(hare - tortoise ,      UpperCamelCase__					)

									if divisor == 1:
												# No common divisor yet, just keep searching.
												continue
									else:
												# We found a common divisor!
												if divisor == num:
															# Unfortunately, the divisor is ``num`` itself and is useless.
															break
												else:
															# The divisor is a nontrivial factor of ``num``!
															return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
						UpperCAmelCase__  :					Optional[Any]        =     hare

						# The new step value for the random function can just be incremented.
						# At first the results will be similar to what the old function would
						# have produced, but the value will quickly diverge after a bit.
						step += 1

			# We haven't found a divisor within the requested number of attempts.
			# We were unlucky or ``num`` itself is actually prime.
			return None


if __name__ == "__main__":
		import argparse

		__A	=argparse.ArgumentParser()
		parser.add_argument(
		    'num',
		    type=int,
		    help='The value to find a divisor of',
		)
		parser.add_argument(
		    '--attempts',
		    type=int,
		    default=3,
		    help='The number of attempts before giving up',
		)
		__A	=parser.parse_args()

		__A	=pollard_rho(args.num, attempts=args.attempts)
		if divisor is None:
				print(f"""{args.num} is probably prime""")
		else:
				__A	=args.num // divisor
				print(f"""{args.num} = {divisor} * {quotient}""")



'''simple docstring'''



from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean







def     _UpperCamelCase      (		UpperCamelCase__					):

			# A local function to see if a dot lands in the circle.
			def is_in_circle(UpperCamelCase__ ,      UpperCamelCase__					) -> bool:
						UpperCAmelCase__  :					List[str]        =     sqrt((x**2) + (y**2)					)
						# Our circle has a radius of 1, so a distance
						# greater than 1 would land outside the circle.
						return distance_from_centre <= 1

			# The proportion of guesses that landed in the circle
			UpperCAmelCase__  :					Optional[Any]        =     mean(
			    int(is_in_circle(uniform(-1.0 ,      1.0					) ,      uniform(-1.0 ,      1.0					)					)					)
			    for _ in range(UpperCamelCase__					)					)
			# The ratio of the area for circle to square is pi/4.
			UpperCAmelCase__  :					int        =     proportion * 4
			print(f'''The estimated value of pi is {pi_estimate}'''					)
			print(f'''The numpy value of pi is {pi}'''					)
			print(f'''The total error is {abs(pi - pi_estimate					)}'''					)







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__ = 0.0 ,      UpperCamelCase__ = 1.0 ,      ):
			return mean(
			    function_to_integrate(uniform(UpperCamelCase__ ,      UpperCamelCase__					)					) for _ in range(UpperCamelCase__					)					) * (max_value - min_value)







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__ = 0.0 ,      UpperCamelCase__ = 1.0					):

			def identity_function(UpperCamelCase__					) -> float:
						return x

			UpperCAmelCase__  :					List[str]        =     area_under_curve_estimator(
			    UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__ ,      UpperCamelCase__					)
			UpperCAmelCase__  :					Optional[int]        =     (max_value * max_value - min_value * min_value) / 2

			print("""******************"""					)
			print(f'''Estimating area under y=x where x varies from {min_value} to {max_value}'''					)
			print(f'''Estimated value is {estimated_value}'''					)
			print(f'''Expected value is {expected_value}'''					)
			print(f'''Total error is {abs(estimated_value - expected_value					)}'''					)
			print("""******************"""					)







def     _UpperCamelCase      (		UpperCamelCase__					):

			def function_to_integrate(UpperCamelCase__					) -> float:
						return sqrt(4.0 - x * x					)

			UpperCAmelCase__  :					Dict        =     area_under_curve_estimator(
			    UpperCamelCase__ ,      UpperCamelCase__ ,      0.0 ,      2.0					)

			print("""******************"""					)
			print("""Estimating pi using area_under_curve_estimator"""					)
			print(f'''Estimated value is {estimated_value}'''					)
			print(f'''Expected value is {pi}'''					)
			print(f'''Total error is {abs(estimated_value - pi					)}'''					)
			print("""******************"""					)


if __name__ == "__main__":
		import doctest

		doctest.testmod()