

from __future__ import annotations

from math import gcd


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

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

  while True:
   # At each iteration, the tortoise moves one step and the hare moves two.
   SCREAMING_SNAKE_CASE		    =      rand_fn(_UpperCamelCase		,				_UpperCamelCase		,				_UpperCamelCase						)
   SCREAMING_SNAKE_CASE		    =      rand_fn(_UpperCamelCase		,				_UpperCamelCase		,				_UpperCamelCase						)
   SCREAMING_SNAKE_CASE		    =      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``.
   SCREAMING_SNAKE_CASE		    =      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.
  SCREAMING_SNAKE_CASE		    =      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_     :    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",
      )
      a_     :    Tuple     		=   parser.parse_args()

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




# Copyright 2021 The HuggingFace Team. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

import argparse

from ...utils.dataclasses import (
    ComputeEnvironment,
    DistributedType,
    DynamoBackend,
    PrecisionType,
    SageMakerDistributedType,
)
from ..menu import BulletMenu


a_     :    str     		=   [
    "EAGER",
    "AOT_EAGER",
    "INDUCTOR",
    "NVFUSER",
    "AOT_NVFUSER",
    "AOT_CUDAGRAPHS",
    "OFI",
    "FX2TRT",
    "ONNXRT",
    "IPEX",
]


def 							__lowerCAmelCase     (							_UpperCamelCase		:	int		,				_UpperCamelCase		:	List[Any]=None		,				_UpperCamelCase		:	Dict=None		,				_UpperCamelCase		:	str=None						)		->				Any:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      True
 while ask_again:
  SCREAMING_SNAKE_CASE		    =      input(_UpperCamelCase						)
  try:
   if default is not None and len(_UpperCamelCase						) == 0:
    return default
   return convert_value(_UpperCamelCase						) if convert_value is not None else result
  except Exception:
   if error_message is not None:
    print(_UpperCamelCase						)


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Optional[Any]		,				_UpperCamelCase		:	Union[str, Any]=[]		,				_UpperCamelCase		:	Optional[int]=None		,				_UpperCamelCase		:	Optional[Any]=0						)		->				List[Any]:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      BulletMenu(_UpperCamelCase		,				_UpperCamelCase						)
 SCREAMING_SNAKE_CASE		    =      menu.run(default_choice=_UpperCamelCase						)
 return convert_value(_UpperCamelCase						) if convert_value is not None else result


def 							__lowerCAmelCase     (							_UpperCamelCase		:	List[str]						)		->				int:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      int(_UpperCamelCase						)
 return ComputeEnvironment(['LOCAL_MACHINE', 'AMAZON_SAGEMAKER'][value]						)


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Tuple						)		->				Tuple:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      int(_UpperCamelCase						)
 return DistributedType(['NO', 'MULTI_CPU', 'MULTI_XPU', 'MULTI_GPU', 'MULTI_NPU', 'TPU'][value]						)


def 							__lowerCAmelCase     (							_UpperCamelCase		:	List[Any]						)		->				List[str]:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      int(_UpperCamelCase						)
 return DynamoBackend(DYNAMO_BACKENDS[value]						).value


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Union[str, Any]						)		->				Dict:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      int(_UpperCamelCase						)
 return PrecisionType(['no', 'fp16', 'bf16', 'fp8'][value]						)


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Tuple						)		->				str:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      int(_UpperCamelCase						)
 return SageMakerDistributedType(['NO', 'DATA_PARALLEL', 'MODEL_PARALLEL'][value]						)


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Dict						)		->				str:
 '''simple docstring'''


 return {"yes": True, "no": False}[value.lower()]




class   UpperCamelCase			(  argparse.RawDescriptionHelpFormatter ):

    def UpperCamelCase      (      self			:   Union[str, Any]				,     snake_case__			:   Tuple				,     snake_case__			:   Optional[Any]				,     snake_case__			:   List[str]				,     snake_case__			:   Dict     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      super()._format_usage(snake_case__				,     snake_case__				,     snake_case__				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      usage.replace('<command> [<args>] '				,     ''     )
     return usage

