





"""simple docstring"""




from __future__ import annotations

from math import gcd





def 						UpperCAmelCase__		(     _UpperCAmelCase ,					_UpperCAmelCase = 2 ,					_UpperCAmelCase = 1 ,					_UpperCAmelCase = 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(_UpperCAmelCase ,					_UpperCAmelCase ,					_UpperCAmelCase					) -> int:
    return (pow(_UpperCAmelCase ,					2					) + step) % modulus

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

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      A_  :       Union[str, Any]										=  rand_fn(_UpperCAmelCase ,					_UpperCAmelCase ,					_UpperCAmelCase					)
      A_  :       Tuple										=  rand_fn(_UpperCAmelCase ,					_UpperCAmelCase ,					_UpperCAmelCase					)
      A_  :       List[str]										=  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``.
      A_  :       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.
    A_  :       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_     :						Optional[int]						=					parser.parse_args()

   lowerCamelCase_     :						List[str]						=					pollard_rho(args.num, attempts=args.attempts)
   if divisor is None:
      print(F"{args.num} is probably prime")
   else:
      lowerCamelCase_     :						List[str]						=					args.num // divisor
      print(F"{args.num} = {divisor} * {quotient}")






"""simple docstring"""




from typing import Optional, Union

import torch
from torch import nn

from ...configuration_utils import ConfigMixin, register_to_config
from ...models.modeling_utils import ModelMixin

class  _UpperCAmelCase  (   UpperCAmelCase__			,   UpperCAmelCase__		):



      '''simple docstring'''

      @register_to_config
      def __init__(     self					,      snake_case_ = 7_6_8					,      ):

        """simple docstring"""

        super().__init__()

        A_  :       Optional[int]										=  nn.Parameter(torch.zeros(1					,      snake_case_  )  )
        A_  :       Optional[int]										=  nn.Parameter(torch.ones(1					,      snake_case_  )  )

      def 					lowerCamelCase_			(     self					,      snake_case_ = None					,      snake_case_ = None					,      ):

        """simple docstring"""

        A_  :       str										=  nn.Parameter(self.mean.to(snake_case_  ).to(snake_case_  )  )
        A_  :       Optional[int]										=  nn.Parameter(self.std.to(snake_case_  ).to(snake_case_  )  )
        return self

      def 					lowerCamelCase_			(     self					,      snake_case_  ):

        """simple docstring"""

        A_  :       Tuple										=  (embeds - self.mean) * 1.0 / self.std
        return embeds

      def 					lowerCamelCase_			(     self					,      snake_case_  ):

        """simple docstring"""

        A_  :       List[str]										=  (embeds * self.std) + self.mean
        return embeds