

import inspect
import tempfile
import unittest

from huggingface_hub import hf_hub_download

from transformers import is_torch_available
from transformers.testing_utils import is_flaky, require_torch, slow, torch_device

from ...test_configuration_common import ConfigTester
from ...test_modeling_common import ModelTesterMixin, floats_tensor, ids_tensor
from ...test_pipeline_mixin import PipelineTesterMixin


a_     :    List[Any]     		=   1e-4

if is_torch_available():
      import torch

      from transformers import AutoformerConfig, AutoformerForPrediction, AutoformerModel
      from transformers.models.autoformer.modeling_autoformer import AutoformerDecoder, AutoformerEncoder




@require_torch
class   UpperCamelCase			:

    def __init__(      self			:   str				,     snake_case__			:   List[Any]				,     snake_case__			:   Any=1_6				,     snake_case__			:   Optional[Any]=1_3				,     snake_case__			:   List[Any]=7				,     snake_case__			:   List[Any]=1_4				,     snake_case__			:   Tuple=1_0				,     snake_case__			:   Union[str, Any]=1_9				,     snake_case__			:   int=5				,     snake_case__			:   Tuple=4				,     snake_case__			:   Tuple=True				,     snake_case__			:   Optional[Any]=1_6				,     snake_case__			:   Optional[int]=2				,     snake_case__			:   List[str]=4				,     snake_case__			:   List[str]=4				,     snake_case__			:   str="gelu"				,     snake_case__			:   Dict=0.1				,     snake_case__			:   int=0.1				,     snake_case__			:   str=[1, 2, 3, 4, 5]				,     snake_case__			:   List[str]=2_5				,     snake_case__			:   Any=5				,     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      d_model
     SCREAMING_SNAKE_CASE		    =      parent
     SCREAMING_SNAKE_CASE		    =      batch_size
     SCREAMING_SNAKE_CASE		    =      prediction_length
     SCREAMING_SNAKE_CASE		    =      context_length
     SCREAMING_SNAKE_CASE		    =      cardinality
     SCREAMING_SNAKE_CASE		    =      num_time_features
     SCREAMING_SNAKE_CASE		    =      lags_sequence
     SCREAMING_SNAKE_CASE		    =      embedding_dimension
     SCREAMING_SNAKE_CASE		    =      is_training
     SCREAMING_SNAKE_CASE		    =      hidden_size
     SCREAMING_SNAKE_CASE		    =      num_hidden_layers
     SCREAMING_SNAKE_CASE		    =      num_attention_heads
     SCREAMING_SNAKE_CASE		    =      intermediate_size
     SCREAMING_SNAKE_CASE		    =      hidden_act
     SCREAMING_SNAKE_CASE		    =      hidden_dropout_prob
     SCREAMING_SNAKE_CASE		    =      attention_probs_dropout_prob

     SCREAMING_SNAKE_CASE		    =      context_length
     SCREAMING_SNAKE_CASE		    =      prediction_length + label_length
     SCREAMING_SNAKE_CASE		    =      label_length

     SCREAMING_SNAKE_CASE		    =      moving_average
     SCREAMING_SNAKE_CASE		    =      autocorrelation_factor

    def UpperCamelCase      (      self			:   Dict     ):

     """simple docstring"""






     return AutoformerConfig(
         d_model=self.d_model				,     encoder_layers=self.num_hidden_layers				,     decoder_layers=self.num_hidden_layers				,     encoder_attention_heads=self.num_attention_heads				,     decoder_attention_heads=self.num_attention_heads				,     encoder_ffn_dim=self.intermediate_size				,     decoder_ffn_dim=self.intermediate_size				,     dropout=self.hidden_dropout_prob				,     attention_dropout=self.attention_probs_dropout_prob				,     prediction_length=self.prediction_length				,     context_length=self.context_length				,     label_length=self.label_length				,     lags_sequence=self.lags_sequence				,     num_time_features=self.num_time_features				,     num_static_categorical_features=1				,     cardinality=[self.cardinality]				,     embedding_dimension=[self.embedding_dimension]				,     moving_average=self.moving_average				,     )

    def UpperCamelCase      (      self			:   Optional[int]				,     snake_case__			:   Tuple     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      config.context_length + max(config.lags_sequence     )

     SCREAMING_SNAKE_CASE		    =      ids_tensor([self.batch_size, 1]				,     config.cardinality[0]     )
     SCREAMING_SNAKE_CASE		    =      floats_tensor([self.batch_size, _past_length, config.num_time_features]     )
     SCREAMING_SNAKE_CASE		    =      floats_tensor([self.batch_size, _past_length]     )
     SCREAMING_SNAKE_CASE		    =      floats_tensor([self.batch_size, _past_length]     ) > 0.5

     # decoder inputs
     SCREAMING_SNAKE_CASE		    =      floats_tensor([self.batch_size, config.prediction_length, config.num_time_features]     )
     SCREAMING_SNAKE_CASE		    =      floats_tensor([self.batch_size, config.prediction_length]     )

     SCREAMING_SNAKE_CASE		    =      {
         'past_values': past_values,
         'static_categorical_features': static_categorical_features,
         'past_time_features': past_time_features,
         'past_observed_mask': past_observed_mask,
         'future_time_features': future_time_features,
         'future_values': future_values,
     }
     return inputs_dict

    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      self.get_config()
     SCREAMING_SNAKE_CASE		    =      self.prepare_autoformer_inputs_dict(snake_case__     )
     return config, inputs_dict

    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      self.prepare_config_and_inputs()
     return config, inputs_dict




    def UpperCamelCase      (      self			:   Optional[int]				,     snake_case__			:   str				,     snake_case__			:   Dict     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      AutoformerModel(config=snake_case__     ).to(snake_case__     ).eval()
     SCREAMING_SNAKE_CASE		    =      model(**snake_case__     )

     SCREAMING_SNAKE_CASE		    =      outputs.encoder_last_hidden_state
     SCREAMING_SNAKE_CASE		    =      outputs.last_hidden_state

     with tempfile.TemporaryDirectory() as tmpdirname:
      SCREAMING_SNAKE_CASE		    =      model.get_encoder()
      encoder.save_pretrained(snake_case__     )
      SCREAMING_SNAKE_CASE		    =      AutoformerEncoder.from_pretrained(snake_case__     ).to(snake_case__     )

     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      model.create_network_inputs(**snake_case__     )
     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      model.decomposition_layer(transformer_inputs[:, : config.context_length, ...]     )

     SCREAMING_SNAKE_CASE		    =      torch.cat(
         (transformer_inputs[:, : config.context_length, ...], feature[:, : config.context_length, ...])				,     dim=-1				,     )
     SCREAMING_SNAKE_CASE		    =      encoder(inputs_embeds=snake_case__     )[0]
     self.parent.assertTrue((encoder_last_hidden_state_a - encoder_last_hidden_state).abs().max().item() < 1E-3     )

     SCREAMING_SNAKE_CASE		    =      (
         torch.mean(transformer_inputs[:, : config.context_length, ...]				,     dim=1     )
         .unsqueeze(1     )
         .repeat(1				,     config.prediction_length				,     1     )
     )
     SCREAMING_SNAKE_CASE		    =      torch.zeros(
         [transformer_inputs.shape[0], config.prediction_length, transformer_inputs.shape[2]]				,     device=enc_input.device				,     )

     SCREAMING_SNAKE_CASE		    =      torch.cat(
         (
             torch.cat((seasonal_input[:, -config.label_length :, ...], zeros)				,     dim=1     ),
             feature[:, config.context_length - config.label_length :, ...],
         )				,     dim=-1				,     )
     SCREAMING_SNAKE_CASE		    =      torch.cat(
         (
             torch.cat((trend_input[:, -config.label_length :, ...], mean)				,     dim=1     ),
             feature[:, config.context_length - config.label_length :, ...],
         )				,     dim=-1				,     )

     with tempfile.TemporaryDirectory() as tmpdirname:
      SCREAMING_SNAKE_CASE		    =      model.get_decoder()
      decoder.save_pretrained(snake_case__     )
      SCREAMING_SNAKE_CASE		    =      AutoformerDecoder.from_pretrained(snake_case__     ).to(snake_case__     )

     SCREAMING_SNAKE_CASE		    =      decoder(
         trend=snake_case__				,     inputs_embeds=snake_case__				,     encoder_hidden_states=snake_case__				,     )[0]

     self.parent.assertTrue((last_hidden_state_a - last_hidden_state).abs().max().item() < 1E-3     )




@require_torch
class   UpperCamelCase			(  SCREAMING_SNAKE_CASE						,      SCREAMING_SNAKE_CASE						,      unittest.TestCase ):
    __UpperCamelCase    =(AutoformerModel, AutoformerForPrediction) if is_torch_available() else ()
    __UpperCamelCase    =(AutoformerForPrediction,) if is_torch_available() else ()
    __UpperCamelCase    ={"feature-extraction": AutoformerModel} if is_torch_available() else {}
    __UpperCamelCase    =False
    __UpperCamelCase    =False
    __UpperCamelCase    =False
    __UpperCamelCase    =False
    __UpperCamelCase    =False
    __UpperCamelCase    =False

    def UpperCamelCase      (      self			:   Optional[int]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      AutoformerModelTester(self     )
     SCREAMING_SNAKE_CASE		    =      ConfigTester(self				,     config_class=snake_case__				,     has_text_modality=snake_case__     )

    def UpperCamelCase      (      self			:   Optional[Any]     ):

     """simple docstring"""






     self.config_tester.run_common_tests()

    def UpperCamelCase      (      self			:   Union[str, Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      self.model_tester.prepare_config_and_inputs()
     for model_class in self.all_model_classes:
      SCREAMING_SNAKE_CASE		    =      model_class(snake_case__     )

      with tempfile.TemporaryDirectory() as tmpdirname:
       model.save_pretrained(snake_case__     )
       SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      model_class.from_pretrained(snake_case__				,     output_loading_info=snake_case__     )
      self.assertEqual(info['missing_keys']				,     []     )

    def UpperCamelCase      (      self			:   str     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      self.model_tester.prepare_config_and_inputs_for_common()
     self.model_tester.check_encoder_decoder_model_standalone(*snake_case__     )

    @unittest.skip(reason='Model has no tokens embeddings'     )
    def UpperCamelCase      (      self			:   str     ):

     """simple docstring"""






     pass

    def UpperCamelCase      (      self			:   Dict     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      inspect.signature(getattr(snake_case__				,     'forward'     )     )
     # The main input is the name of the argument after `self`
     SCREAMING_SNAKE_CASE		    =      list(model_signature.parameters.keys()     )[1]
     self.assertEqual(AutoformerModel.main_input_name				,     snake_case__     )

    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      self.model_tester.prepare_config_and_inputs_for_common()

     for model_class in self.all_model_classes:
      SCREAMING_SNAKE_CASE		    =      model_class(snake_case__     )
      SCREAMING_SNAKE_CASE		    =      inspect.signature(model.forward     )
      # signature.parameters is an OrderedDict => so arg_names order is deterministic
      SCREAMING_SNAKE_CASE		    =      [*signature.parameters.keys()]

      SCREAMING_SNAKE_CASE		    =      [
          'past_values',
          'past_time_features',
          'past_observed_mask',
          'static_categorical_features',
          'static_real_features',
          'future_values',
          'future_time_features',
      ]

      if model.__class__.__name__ in ["AutoformerForPrediction"]:
       expected_arg_names.append('future_observed_mask'     )

      expected_arg_names.extend(
          [
              'decoder_attention_mask',
              'head_mask',
              'decoder_head_mask',
              'cross_attn_head_mask',
              'encoder_outputs',
              'past_key_values',
              'output_hidden_states',
              'output_attentions',
              'use_cache',
              'return_dict',
          ]     )

      self.assertListEqual(arg_names[: len(snake_case__     )]				,     snake_case__     )

    def UpperCamelCase      (      self			:   Union[str, Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE		    =      self.model_tester.prepare_config_and_inputs_for_common()
     SCREAMING_SNAKE_CASE		    =      True

     SCREAMING_SNAKE_CASE		    =      getattr(self.model_tester				,     'seq_length'				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      getattr(self.model_tester				,     'decoder_seq_length'				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      getattr(self.model_tester				,     'encoder_seq_length'				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      getattr(self.model_tester				,     'd_model'				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      getattr(self.model_tester				,     'num_attention_heads'				,     snake_case__     )
     SCREAMING_SNAKE_CASE		    =      d_model // num_attention_heads

     for model_class in self.all_model_classes:
      SCREAMING_SNAKE_CASE		    =      True
      SCREAMING_SNAKE_CASE		    =      False
      SCREAMING_SNAKE_CASE		    =      True
      SCREAMING_SNAKE_CASE		    =      model_class(snake_case__     )
      model.to(snake_case__     )
      model.eval()
      with torch.no_grad():
       SCREAMING_SNAKE_CASE		    =      model(**self._prepare_for_class(snake_case__				,     snake_case__     )     )
      SCREAMING_SNAKE_CASE		    =      outputs.encoder_attentions if config.is_encoder_decoder else outputs.attentions
      self.assertEqual(len(snake_case__     )				,     self.model_tester.num_hidden_layers     )

      # check that output_attentions also work using config
      del inputs_dict["output_attentions"]
      SCREAMING_SNAKE_CASE		    =      True
      SCREAMING_SNAKE_CASE		    =      model_class(snake_case__     )
      model.to(snake_case__     )
      model.eval()
      with torch.no_grad():
       SCREAMING_SNAKE_CASE		    =      model(**self._prepare_for_class(snake_case__				,     snake_case__     )     )
      SCREAMING_SNAKE_CASE		    =      outputs.encoder_attentions
      self.assertEqual(len(snake_case__     )				,     self.model_tester.num_hidden_layers     )

      self.assertListEqual(
          list(attentions[0].shape[-3:]     )				,     [self.model_tester.num_attention_heads, encoder_seq_length, dim]				,     )
      SCREAMING_SNAKE_CASE		    =      len(snake_case__     )

      SCREAMING_SNAKE_CASE		    =      7

      if "last_hidden_state" in outputs:
       correct_outlen += 1

      if "trend" in outputs:
       correct_outlen += 1

      if "past_key_values" in outputs:
       correct_outlen += 1  # past_key_values have been returned

      if "loss" in outputs:
       correct_outlen += 1

      if "params" in outputs:
       correct_outlen += 1

      self.assertEqual(snake_case__				,     snake_case__     )

      # decoder attentions
      SCREAMING_SNAKE_CASE		    =      outputs.decoder_attentions
      self.assertIsInstance(snake_case__				,     (list, tuple)     )
      self.assertEqual(len(snake_case__     )				,     self.model_tester.num_hidden_layers     )
      self.assertListEqual(
          list(decoder_attentions[0].shape[-3:]     )				,     [self.model_tester.num_attention_heads, decoder_seq_length, dim]				,     )

      # cross attentions
      SCREAMING_SNAKE_CASE		    =      outputs.cross_attentions
      self.assertIsInstance(snake_case__				,     (list, tuple)     )
      self.assertEqual(len(snake_case__     )				,     self.model_tester.num_hidden_layers     )
      self.assertListEqual(
          list(cross_attentions[0].shape[-3:]     )				,     [self.model_tester.num_attention_heads, decoder_seq_length, dim]				,     )

     # Check attention is always last and order is fine
     SCREAMING_SNAKE_CASE		    =      True
     SCREAMING_SNAKE_CASE		    =      True
     SCREAMING_SNAKE_CASE		    =      model_class(snake_case__     )
     model.to(snake_case__     )
     model.eval()
     with torch.no_grad():
      SCREAMING_SNAKE_CASE		    =      model(**self._prepare_for_class(snake_case__				,     snake_case__     )     )

     self.assertEqual(out_len + 2				,     len(snake_case__     )     )

     SCREAMING_SNAKE_CASE		    =      outputs.encoder_attentions if config.is_encoder_decoder else outputs.attentions

     self.assertEqual(len(snake_case__     )				,     self.model_tester.num_hidden_layers     )
     self.assertListEqual(
         list(self_attentions[0].shape[-3:]     )				,     [self.model_tester.num_attention_heads, encoder_seq_length, dim]				,     )




    @is_flaky()
    def UpperCamelCase      (      self			:   Tuple     ):

     """simple docstring"""






     super().test_retain_grad_hidden_states_attentions()


def 							__lowerCAmelCase     (							_UpperCamelCase		:	Optional[int]="train-batch.pt"						)		->				Tuple:
 '''simple docstring'''


 SCREAMING_SNAKE_CASE		    =      hf_hub_download(repo_id='hf-internal-testing/tourism-monthly-batch'		,				filename=_UpperCamelCase		,				repo_type='dataset'						)
 SCREAMING_SNAKE_CASE		    =      torch.load(_UpperCamelCase		,				map_location=_UpperCamelCase						)
 return batch




@require_torch
@slow
class   UpperCamelCase			(  unittest.TestCase ):

    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      AutoformerModel.from_pretrained('huggingface/autoformer-tourism-monthly'     ).to(snake_case__     )
     SCREAMING_SNAKE_CASE		    =      prepare_batch()

     with torch.no_grad():
      SCREAMING_SNAKE_CASE		    =      model(
          past_values=batch['past_values']				,     past_time_features=batch['past_time_features']				,     past_observed_mask=batch['past_observed_mask']				,     static_categorical_features=batch['static_categorical_features']				,     future_values=batch['future_values']				,     future_time_features=batch['future_time_features']				,     )[0]

     SCREAMING_SNAKE_CASE		    =      torch.Size(
         (6_4, model.config.prediction_length + model.config.label_length, model.config.feature_size)     )
     self.assertEqual(output.shape				,     snake_case__     )

     SCREAMING_SNAKE_CASE		    =      torch.tensor(
         [[0.3_593, -1.3_398, 0.6_330], [0.2_279, 1.5_396, -0.1_792], [0.0_450, 1.3_225, -0.2_335]]				,     device=snake_case__     )
     self.assertTrue(torch.allclose(output[0, :3, :3]				,     snake_case__				,     atol=snake_case__     )     )

    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      AutoformerForPrediction.from_pretrained('huggingface/autoformer-tourism-monthly'     ).to(snake_case__     )
     SCREAMING_SNAKE_CASE		    =      prepare_batch('val-batch.pt'     )
     with torch.no_grad():
      SCREAMING_SNAKE_CASE		    =      model(
          past_values=batch['past_values']				,     past_time_features=batch['past_time_features']				,     past_observed_mask=batch['past_observed_mask']				,     static_categorical_features=batch['static_categorical_features']				,     ).encoder_last_hidden_state
     SCREAMING_SNAKE_CASE		    =      torch.Size((6_4, model.config.context_length, model.config.d_model)     )
     self.assertEqual(output.shape				,     snake_case__     )

     SCREAMING_SNAKE_CASE		    =      torch.tensor(
         [[-0.0_734, -0.9_036, 0.8_358], [4.7_186, 2.4_113, 1.9_581], [1.7_953, 2.3_558, 1.2_970]]				,     device=snake_case__     )
     self.assertTrue(torch.allclose(output[0, :3, :3]				,     snake_case__				,     atol=snake_case__     )     )




    def UpperCamelCase      (      self			:   List[Any]     ):

     """simple docstring"""






     SCREAMING_SNAKE_CASE		    =      AutoformerForPrediction.from_pretrained('huggingface/autoformer-tourism-monthly'     ).to(snake_case__     )
     SCREAMING_SNAKE_CASE		    =      prepare_batch('val-batch.pt'     )
     with torch.no_grad():
      SCREAMING_SNAKE_CASE		    =      model.generate(
          static_categorical_features=batch['static_categorical_features']				,     past_time_features=batch['past_time_features']				,     past_values=batch['past_values']				,     future_time_features=batch['future_time_features']				,     past_observed_mask=batch['past_observed_mask']				,     )
     SCREAMING_SNAKE_CASE		    =      torch.Size((6_4, model.config.num_parallel_samples, model.config.prediction_length)     )
     self.assertEqual(outputs.sequences.shape				,     snake_case__     )

     SCREAMING_SNAKE_CASE		    =      torch.tensor([3_130.6_763, 4_056.5_293, 7_053.0_786]				,     device=snake_case__     )
     SCREAMING_SNAKE_CASE		    =      outputs.sequences.mean(dim=1     )
     self.assertTrue(torch.allclose(mean_prediction[0, -3:]				,     snake_case__				,     rtol=1E-1     )     )




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}""")

