
"""simple docstring"""






from __future__ import annotations

from math import gcd





def __A						(   a_ :							int		,		a_ :							int = 2		,		a_ :							int = 1		,		a_ :							int = 3		,		)->							int | None:




		'''simple docstring'''






		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(a_ :							int		,		a_ :							int		,		a_ :							int     ) -> int:
				return (pow(a_		,		2     ) + step) % modulus

		for _ in range(a_     ):
				# These track the position within the cycle detection logic.
				SCREAMING_SNAKE_CASE       :     str				= seed
				SCREAMING_SNAKE_CASE       :     List[Any]				= seed

				while True:
						# At each iteration, the tortoise moves one step and the hare moves two.
						SCREAMING_SNAKE_CASE       :     Any				= rand_fn(a_		,		a_		,		a_     )
						SCREAMING_SNAKE_CASE       :     Dict				= rand_fn(a_		,		a_		,		a_     )
						SCREAMING_SNAKE_CASE       :     Optional[int]				= rand_fn(a_		,		a_		,		a_     )

						# 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``.
						SCREAMING_SNAKE_CASE       :     str				= gcd(hare - tortoise		,		a_     )

						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.
				SCREAMING_SNAKE_CASE       :     List[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

					lowerCamelCase__ :       int						     =			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",
					)
					lowerCamelCase__ :       Any						     =			parser.parse_args()

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







"""simple docstring"""






from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray





def __A						(   a_ :							float		,		a_ :							float		,		a_ :							bool = False     )->							list[float]:




		'''simple docstring'''






		if radian_mode:
				return [magnitude * cos(a_     ), magnitude * sin(a_     )]
		return [magnitude * cos(radians(a_     )     ), magnitude * sin(radians(a_     )     )]





def __A						(   a_ :							NDArray[floataa]		,		a_ :							NDArray[floataa]		,		a_ :							float = 10**-1     )->							bool:




		'''simple docstring'''






		SCREAMING_SNAKE_CASE       :     NDArray[floataa]				= cross(a_		,		a_     )
		SCREAMING_SNAKE_CASE       :     float				= sum(a_     )
		return abs(a_     ) < eps


if __name__ == "__main__":
					# Test to check if it works
					lowerCamelCase__ :       Optional[Any]						     =			array(
					    [
					        polar_force(7_1_8.4, 180 - 30),
					        polar_force(8_7_9.5_4, 45),
					        polar_force(100, -90),
					    ]
					)

					lowerCamelCase__ :       NDArray[floataa]						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem 1 in image_data/2D_problems.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array(
					    [
					        polar_force(30 * 9.8_1, 15),
					        polar_force(215, 180 - 45),
					        polar_force(264, 90 - 30),
					    ]
					)

					lowerCamelCase__ :       Any						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem in image_data/2D_problems_1.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array([[0, -2000], [0, -1200], [0, 15600], [0, -12400]])

					lowerCamelCase__ :       Optional[int]						     =			array([[0, 0], [6, 0], [10, 0], [12, 0]])

					assert in_static_equilibrium(forces, location)

					import doctest

					doctest.testmod()





