


'''simple docstring'''







from __future__ import annotations

from math import gcd






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

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

              while True:
                     # At each iteration, the tortoise moves one step and the hare moves two.
                     _lowerCAmelCase     :				str       =							rand_fn(_A ,    _A ,    _A       )
                     _lowerCAmelCase     :				Optional[int]       =							rand_fn(_A ,    _A ,    _A       )
                     _lowerCAmelCase     :				Union[str, Any]       =							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     :				Optional[int]       =							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     :				Tuple       =							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     :  Tuple											=    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     :  List[str]											=    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     :  Union[str, Any]											=    args.num // divisor
              print(F'''{args.num} = {divisor} * {quotient}''')





'''simple docstring'''







import argparse
from collections import defaultdict

import yaml


lowerCAmelCase     :  Dict											=    """docs/source/en/_toctree.yml"""






def       lowercase					(_A       ):







       """simple docstring"""







       _lowerCAmelCase     :				Optional[int]       =							defaultdict(_A       )
       _lowerCAmelCase     :				Tuple       =							[]
       _lowerCAmelCase     :				Dict       =							[]
       for doc in doc_list:
              if "local" in doc:
                     counts[doc["local"]] += 1

              if doc["title"].lower() == "overview":
                     overview_doc.append({'local': doc['local'], 'title': doc['title']}       )
              else:
                     new_doc_list.append(_A       )

       _lowerCAmelCase     :				str       =							new_doc_list
       _lowerCAmelCase     :				List[Any]       =							[key for key, value in counts.items() if value > 1]

       _lowerCAmelCase     :				List[str]       =							[]
       for duplicate_key in duplicates:
              _lowerCAmelCase     :				Any       =							list({doc['title'] for doc in doc_list if doc['local'] == duplicate_key}       )
              if len(_A       ) > 1:
                     raise ValueError(
                         f'{duplicate_key} is present several times in the documentation table of content at '
                         '`docs/source/en/_toctree.yml` with different *Title* values. Choose one of those and remove the '
                         'others.'       )
              # Only add this once
              new_doc.append({'local': duplicate_key, 'title': titles[0]}       )

       # Add none duplicate-keys
       new_doc.extend([doc for doc in doc_list if 'local' not in counts or counts[doc['local']] == 1]       )
       _lowerCAmelCase     :				Any       =							sorted(_A ,    key=lambda _A       : s["title"].lower()       )

       # "overview" gets special treatment and is always first
       if len(_A       ) > 1:
              raise ValueError('{doc_list} has two \'overview\' docs which is not allowed.'       )

       overview_doc.extend(_A       )

       # Sort
       return overview_doc






def       lowercase					(_A=False       ):







       """simple docstring"""







       with open(_A ,    encoding='utf-8'       ) as f:
              _lowerCAmelCase     :				Union[str, Any]       =							yaml.safe_load(f.read()       )

       # Get to the API doc
       _lowerCAmelCase     :				Tuple       =							0
       while content[api_idx]["title"] != "API":
              api_idx += 1
       _lowerCAmelCase     :				str       =							content[api_idx]['sections']

       # Then to the model doc
       _lowerCAmelCase     :				Dict       =							0
       while api_doc[scheduler_idx]["title"] != "Schedulers":
              scheduler_idx += 1

       _lowerCAmelCase     :				Optional[int]       =							api_doc[scheduler_idx]['sections']
       _lowerCAmelCase     :				Any       =							clean_doc_toc(_A       )

       _lowerCAmelCase     :				int       =							False
       if new_scheduler_doc != scheduler_doc:
              _lowerCAmelCase     :				str       =							True
              if overwrite:
                     _lowerCAmelCase     :				int       =							new_scheduler_doc

       if diff:
              if overwrite:
                     _lowerCAmelCase     :				Dict       =							api_doc
                     with open(_A ,    'w' ,    encoding='utf-8'       ) as f:
                            f.write(yaml.dump(_A ,    allow_unicode=_A       )       )
              else:
                     raise ValueError(
                         'The model doc part of the table of content is not properly sorted, run `make style` to fix this.'       )






def       lowercase					(_A=False       ):







       """simple docstring"""







       with open(_A ,    encoding='utf-8'       ) as f:
              _lowerCAmelCase     :				Optional[Any]       =							yaml.safe_load(f.read()       )

       # Get to the API doc
       _lowerCAmelCase     :				Optional[Any]       =							0
       while content[api_idx]["title"] != "API":
              api_idx += 1
       _lowerCAmelCase     :				Union[str, Any]       =							content[api_idx]['sections']

       # Then to the model doc
       _lowerCAmelCase     :				Optional[int]       =							0
       while api_doc[pipeline_idx]["title"] != "Pipelines":
              pipeline_idx += 1

       _lowerCAmelCase     :				List[Any]       =							False
       _lowerCAmelCase     :				List[Any]       =							api_doc[pipeline_idx]['sections']
       _lowerCAmelCase     :				Union[str, Any]       =							[]

       # sort sub pipeline docs
       for pipeline_doc in pipeline_docs:
              if "section" in pipeline_doc:
                     _lowerCAmelCase     :				Tuple       =							pipeline_doc['section']
                     _lowerCAmelCase     :				Optional[int]       =							clean_doc_toc(_A       )
                     if overwrite:
                            _lowerCAmelCase     :				str       =							new_sub_pipeline_doc
              new_pipeline_docs.append(_A       )

       # sort overall pipeline doc
       _lowerCAmelCase     :				str       =							clean_doc_toc(_A       )

       if new_pipeline_docs != pipeline_docs:
              _lowerCAmelCase     :				Any       =							True
              if overwrite:
                     _lowerCAmelCase     :				Optional[int]       =							new_pipeline_docs

       if diff:
              if overwrite:
                     _lowerCAmelCase     :				str       =							api_doc
                     with open(_A ,    'w' ,    encoding='utf-8'       ) as f:
                            f.write(yaml.dump(_A ,    allow_unicode=_A       )       )
              else:
                     raise ValueError(
                         'The model doc part of the table of content is not properly sorted, run `make style` to fix this.'       )


if __name__ == "__main__":
       lowerCAmelCase     :  Tuple											=    argparse.ArgumentParser()
       parser.add_argument("""--fix_and_overwrite""", action="""store_true""", help="""Whether to fix inconsistencies.""")
       lowerCAmelCase     :  List[Any]											=    parser.parse_args()

       check_scheduler_doc(args.fix_and_overwrite)
       check_pipeline_doc(args.fix_and_overwrite)

