


"""simple docstring"""


import os
import sys
import tempfile

import torch

from .state import AcceleratorState
from .utils import PrecisionType, PrepareForLaunch, is_mps_available, patch_environment



def   snake_case						(				_a:     List[str]		,   _a:     Dict=()		,   _a:     List[Any]=None		,   _a:     List[Any]="no"		,   _a:     Optional[Any]="29500"							)->		Optional[int]:
      '''simple docstring'''


      lowerCamelCase__           =		False
      lowerCamelCase__           =		False
      if any(key.startswith('KAGGLE'							) for key in os.environ.keys()							):
            lowerCamelCase__           =		True
      elif "IPython" in sys.modules:
            lowerCamelCase__           =		'google.colab' in str(sys.modules['IPython'].get_ipython()							)

      try:
            lowerCamelCase__           =		PrecisionType(mixed_precision.lower()							)
      except ValueError:
            raise ValueError(
                F'Unknown mixed_precision mode: {args.mixed_precision.lower()}. Choose between {PrecisionType.list()}.'							)

      if (in_colab or in_kaggle) and (os.environ.get('TPU_NAME'		,   _a							) is not None):
            # TPU launch
            import torch_xla.distributed.xla_multiprocessing as xmp

            if len(AcceleratorState._shared_state							) > 0:
                  raise ValueError(
                      'To train on TPU in Colab or Kaggle Kernel, the `Accelerator` should only be initialized inside '
                      'your training function. Restart your notebook and make sure no cells initializes an '
                      '`Accelerator`.'							)
            if num_processes is None:
                  lowerCamelCase__           =		8

            lowerCamelCase__           =		PrepareForLaunch(_a		,   distributed_type='TPU'							)
            print(F'Launching a training on {num_processes} TPU cores.'							)
            xmp.spawn(_a		,   args=_a		,   nprocs=_a		,   start_method='fork'							)
      elif in_colab:
            # No need for a distributed launch otherwise as it's either CPU or one GPU.
            if torch.cuda.is_available():
                  print('Launching training on one GPU.'							)
            else:
                  print('Launching training on one CPU.'							)
            function(*_a							)
      else:
            if num_processes is None:
                  raise ValueError(
                      'You have to specify the number of GPUs you would like to use, add `num_processes=...` to your call.'							)

            if num_processes > 1:
                  # Multi-GPU launch
                  from torch.multiprocessing import start_processes
                  from torch.multiprocessing.spawn import ProcessRaisedException

                  if len(AcceleratorState._shared_state							) > 0:
                        raise ValueError(
                            'To launch a multi-GPU training from your notebook, the `Accelerator` should only be initialized '
                            'inside your training function. Restart your notebook and make sure no cells initializes an '
                            '`Accelerator`.'							)

                  if torch.cuda.is_initialized():
                        raise ValueError(
                            'To launch a multi-GPU training from your notebook, you need to avoid running any instruction '
                            'using `torch.cuda` in any cell. Restart your notebook and make sure no cells use any CUDA '
                            'function.'							)

                  # torch.distributed will expect a few environment variable to be here. We set the ones common to each
                  # process here (the other ones will be set be the launcher).
                  with patch_environment(
                      world_size=_a		,   master_addr='127.0.01'		,   master_port=_a		,   mixed_precision=_a							):
                        lowerCamelCase__           =		PrepareForLaunch(_a		,   distributed_type='MULTI_GPU'							)
                        print(F'Launching training on {num_processes} GPUs.'							)
                        try:
                              start_processes(_a		,   args=_a		,   nprocs=_a		,   start_method='fork'							)
                        except ProcessRaisedException as e:
                              if "Cannot re-initialize CUDA in forked subprocess" in e.args[0]:
                                    raise RuntimeError(
                                        'CUDA has been initialized before the `notebook_launcher` could create a forked subprocess. '
                                        'This likely stems from an outside import causing issues once the `notebook_launcher()` is called. '
                                        'Please review your imports and test them when running the `notebook_launcher()` to identify '
                                        'which one is problematic.'							) from e

            else:
                  # No need for a distributed launch otherwise as it's either CPU, GPU or MPS.
                  if is_mps_available():
                        lowerCamelCase__           =		'1'
                        print('Launching training on MPS.'							)
                  elif torch.cuda.is_available():
                        print('Launching training on one GPU.'							)
                  else:
                        print('Launching training on CPU.'							)
                  function(*_a							)






def   snake_case						(				_a:     Tuple		,   _a:     Union[str, Any]=()		,   _a:     List[Any]=2							)->		Union[str, Any]:
      '''simple docstring'''


      from torch.multiprocessing import start_processes

      with tempfile.NamedTemporaryFile() as tmp_file:
            # torch.distributed will expect a few environment variable to be here. We set the ones common to each
            # process here (the other ones will be set be the launcher).
            with patch_environment(
                world_size=_a		,   master_addr='127.0.01'		,   master_port='29500'		,   accelerate_mixed_precision='no'		,   accelerate_debug_rdv_file=tmp_file.name		,   accelerate_use_cpu='yes'		,   ):
                  lowerCamelCase__           =		PrepareForLaunch(_a		,   debug=_a							)
                  start_processes(_a		,   args=_a		,   nprocs=_a		,   start_method='fork'							)







"""simple docstring"""


from __future__ import annotations

from math import gcd



def   snake_case						(				_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.
            lowerCamelCase__           =		seed
            lowerCamelCase__           =		seed

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

   _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",
   )
   _snake_case          =							parser.parse_args()

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



