




'''simple docstring'''


from typing import TYPE_CHECKING

from ..utils import _LazyModule


_lowerCAmelCase  		=      {
    "config": [
        "EXTERNAL_DATA_FORMAT_SIZE_LIMIT",
        "OnnxConfig",
        "OnnxConfigWithPast",
        "OnnxSeq2SeqConfigWithPast",
        "PatchingSpec",
    ],
    "convert": ["export", "validate_model_outputs"],
    "features": ["FeaturesManager"],
    "utils": ["ParameterFormat", "compute_serialized_parameters_size"],
}


if TYPE_CHECKING:
  from .config import (
      EXTERNAL_DATA_FORMAT_SIZE_LIMIT,
      OnnxConfig,
      OnnxConfigWithPast,
      OnnxSeqaSeqConfigWithPast,
      PatchingSpec,
  )
  from .convert import export, validate_model_outputs
  from .features import FeaturesManager
  from .utils import ParameterFormat, compute_serialized_parameters_size

else:
  import sys

  _lowerCAmelCase  		=      _LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)









'''simple docstring'''

from __future__ import annotations

from math import gcd



def 	__lowerCamelCase	( lowerCAmelCase_		,     lowerCAmelCase_ = 2		,     lowerCAmelCase_ = 1		,     lowerCAmelCase_ = 3		,     )     -> int | None:
		# 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(lowerCAmelCase_		,     lowerCAmelCase_		,     lowerCAmelCase_    ) -> int:
				return (pow(lowerCAmelCase_		,     2    ) + step) % modulus

		for _ in range(lowerCAmelCase_    ):
				# These track the position within the cycle detection logic.
				_a	: Optional[int]								=				seed
				_a	: str								=				seed

				while True:
						# At each iteration, the tortoise moves one step and the hare moves two.
						_a	: str								=				rand_fn(lowerCAmelCase_		,     lowerCAmelCase_		,     lowerCAmelCase_    )
						_a	: Dict								=				rand_fn(lowerCAmelCase_		,     lowerCAmelCase_		,     lowerCAmelCase_    )
						_a	: Tuple								=				rand_fn(lowerCAmelCase_		,     lowerCAmelCase_		,     lowerCAmelCase_    )

						# 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``.
						_a	: Optional[Any]								=				gcd(hare - tortoise		,     lowerCAmelCase_    )

						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.
				_a	: Union[str, 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		     =  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		     =  parser.parse_args()

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




