





'''simple docstring'''


from __future__ import annotations

from math import gcd


def 						__a(SCREAMING_SNAKE_CASE_	:       int		,							SCREAMING_SNAKE_CASE_	:       int = 2		,							SCREAMING_SNAKE_CASE_	:       int = 1		,							SCREAMING_SNAKE_CASE_	:       int = 3		,							):



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

    for _ in range(SCREAMING_SNAKE_CASE_     ):
        # These track the position within the cycle detection logic.
        _lowerCAmelCase						      =   seed
        _lowerCAmelCase						      =   seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            _lowerCAmelCase						      =   rand_fn(SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_     )
            _lowerCAmelCase						      =   rand_fn(SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_     )
            _lowerCAmelCase						      =   rand_fn(SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_		,							SCREAMING_SNAKE_CASE_     )

            # 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``.
            _lowerCAmelCase						      =   gcd(hare - tortoise		,							SCREAMING_SNAKE_CASE_     )

            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.
        _lowerCAmelCase						      =   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

       _SCREAMING_SNAKE_CASE									=				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",
       )
       _SCREAMING_SNAKE_CASE									=				parser.parse_args()

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









'''simple docstring'''


from typing import TYPE_CHECKING

from ....utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


_SCREAMING_SNAKE_CASE									=				{
    "configuration_mctct": ["MCTCT_PRETRAINED_CONFIG_ARCHIVE_MAP", "MCTCTConfig"],
    "feature_extraction_mctct": ["MCTCTFeatureExtractor"],
    "processing_mctct": ["MCTCTProcessor"],
}


try:
       if not is_torch_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       _SCREAMING_SNAKE_CASE									=				[
           "MCTCT_PRETRAINED_MODEL_ARCHIVE_LIST",
           "MCTCTForCTC",
           "MCTCTModel",
           "MCTCTPreTrainedModel",
       ]


if TYPE_CHECKING:
       from .configuration_mctct import MCTCT_PRETRAINED_CONFIG_ARCHIVE_MAP, MCTCTConfig
       from .feature_extraction_mctct import MCTCTFeatureExtractor
       from .processing_mctct import MCTCTProcessor

       try:
              if not is_torch_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_mctct import MCTCT_PRETRAINED_MODEL_ARCHIVE_LIST, MCTCTForCTC, MCTCTModel, MCTCTPreTrainedModel

else:
       import sys

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


