'''simple docstring'''




import inspect

import jax
import jax.lax as lax
import jax.numpy as jnp

from ..utils import add_start_docstrings
from ..utils.logging import get_logger


_A								=   get_logger(__name__)


_A								=   R'\n    Args:\n        input_ids (`jnp.ndarray` of shape `(batch_size, sequence_length)`):\n            Indices of input sequence tokens in the vocabulary.\n\n            Indices can be obtained using [`PreTrainedTokenizer`]. See [`PreTrainedTokenizer.encode`] and\n            [`PreTrainedTokenizer.__call__`] for details.\n\n            [What are input IDs?](../glossary#input-ids)\n        scores (`jnp.ndarray` of shape `(batch_size, config.vocab_size)`):\n            Prediction scores of a language modeling head. These can be logits for each vocabulary when not using beam\n            search or log softmax for each vocabulary token when using beam search\n        kwargs (`Dict[str, Any]`, *optional*):\n            Additional logits processor specific kwargs.\n\n    Return:\n        `jnp.ndarray` of shape `(batch_size, config.vocab_size)`: The processed prediction scores.\n\n'


class       UpperCAmelCase__  :






   """simple docstring"""
   @add_start_docstrings(_a				)
   def __call__(self       ,				_a       ,				_a				)    ->     jnp.ndarray:
        raise NotImplementedError(
            f'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''				)



class       UpperCAmelCase__  :






   """simple docstring"""
   @add_start_docstrings(_a				)
   def __call__(self       ,				_a       ,				_a				)    ->     jnp.ndarray:
        raise NotImplementedError(
            f'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''				)



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   @add_start_docstrings(_a				)
   def __call__(self       ,				_a       ,				_a       ,				_a       ,				**_a				)    ->     jnp.ndarray:
        for processor in self:
             lowercase_					:       Tuple     				=					inspect.signature(processor.__call__				).parameters
             if len(_a				) > 3:
                  if not all(arg in kwargs for arg in list(function_args.keys()				)[2:]				):
                       raise ValueError(
                           f'''Make sure that all the required parameters: {list(function_args.keys()				)} for '''
                           f'''{processor.__class__} are passed to the logits processor.'''				)
                  lowercase_					:       Tuple     				=					processor(_a       ,				_a       ,				_a       ,				**_a				)
             else:
                  lowercase_					:       List[Any]     				=					processor(_a       ,				_a       ,				_a				)
        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a				)    ->     List[Any]:
        if not isinstance(_a       ,				_a				) or not (temperature > 0):
             raise ValueError(f'''`temperature` has to be a strictly positive float, but is {temperature}'''				)

        lowercase_					:       Optional[Any]     				=					temperature





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_					:       str     				=					scores / self.temperature
        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a = -float('Inf'				)       ,				_a = 1				)    ->     Tuple:
        if not isinstance(_a       ,				_a				) or (top_p < 0 or top_p > 1.0):
             raise ValueError(f'''`top_p` has to be a float > 0 and < 1, but is {top_p}'''				)
        if not isinstance(_a       ,				_a				) or (min_tokens_to_keep < 1):
             raise ValueError(f'''`min_tokens_to_keep` has to be a positive integer, but is {min_tokens_to_keep}'''				)

        lowercase_					:       Dict     				=					top_p
        lowercase_					:       Dict     				=					filter_value
        lowercase_					:       Tuple     				=					min_tokens_to_keep





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_				,lowercase_					:       List[Any]     				=					lax.top_k(_a       ,				scores.shape[-1]				)

        lowercase_					:       Tuple     				=					jnp.full_like(_a       ,				self.filter_value				)
        lowercase_					:       Union[str, Any]     				=					jax.nn.softmax(_a       ,				axis=-1				).cumsum(axis=-1				)
        lowercase_					:       List[str]     				=					cumulative_probs < self.top_p

        # include the token that is higher than top_p as well
        lowercase_					:       Union[str, Any]     				=					jnp.roll(_a       ,				1				)
        score_mask |= score_mask.at[:, 0].set(_a				)

        # min tokens to keep
        lowercase_					:       str     				=					score_mask.at[:, : self.min_tokens_to_keep].set(_a				)

        lowercase_					:       List[Any]     				=					jnp.where(_a       ,				_a       ,				_a				)
        lowercase_					:       Union[str, Any]     				=					jax.lax.sort_key_val(_a       ,				_a				)[-1]

        return next_scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a = -float('Inf'				)       ,				_a = 1				)    ->     Dict:
        if not isinstance(_a       ,				_a				) or top_k <= 0:
             raise ValueError(f'''`top_k` has to be a strictly positive integer, but is {top_k}'''				)

        lowercase_					:       Optional[Any]     				=					max(_a       ,				_a				)
        lowercase_					:       int     				=					filter_value





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_				,lowercase_					:       Union[str, Any]     				=					scores.shape
        lowercase_					:       Union[str, Any]     				=					jnp.full(batch_size * vocab_size       ,				self.filter_value				)

        lowercase_					:       str     				=					min(self.top_k       ,				scores.shape[-1]				)  # Safety check
        lowercase_				,lowercase_					:       Optional[int]     				=					lax.top_k(_a       ,				_a				)
        lowercase_					:       List[Any]     				=					jnp.broadcast_to((jnp.arange(_a				) * vocab_size)[:, None]       ,				(batch_size, topk)				).flatten()
        lowercase_					:       List[Any]     				=					topk_scores.flatten()
        lowercase_					:       List[Any]     				=					topk_indices.flatten() + shift

        lowercase_					:       Optional[Any]     				=					next_scores_flat.at[topk_indices_flat].set(_a				)
        lowercase_					:       Optional[Any]     				=					next_scores_flat.reshape(_a       ,				_a				)
        return next_scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a				)    ->     Any:
        lowercase_					:       Optional[int]     				=					bos_token_id





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_					:       Tuple     				=					jnp.full(scores.shape       ,				-float('inf'				)				)

        lowercase_					:       str     				=					1 - jnp.bool_(cur_len - 1				)

        lowercase_					:       Tuple     				=					jnp.where(_a       ,				new_scores.at[:, self.bos_token_id].set(0				)       ,				_a				)

        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a				)    ->     Dict:
        lowercase_					:       Optional[int]     				=					max_length
        lowercase_					:       Tuple     				=					eos_token_id





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_					:       Optional[Any]     				=					jnp.full(scores.shape       ,				-float('inf'				)				)

        lowercase_					:       List[Any]     				=					1 - jnp.bool_(cur_len - self.max_length + 1				)

        lowercase_					:       List[str]     				=					jnp.where(_a       ,				new_scores.at[:, self.eos_token_id].set(0				)       ,				_a				)

        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a				)    ->     List[str]:
        if not isinstance(_a       ,				_a				) or min_length < 0:
             raise ValueError(f'''`min_length` has to be a positive integer, but is {min_length}'''				)

        if not isinstance(_a       ,				_a				) or eos_token_id < 0:
             raise ValueError(f'''`eos_token_id` has to be a positive integer, but is {eos_token_id}'''				)

        lowercase_					:       int     				=					min_length
        lowercase_					:       int     				=					eos_token_id





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        # create boolean flag to decide if min length penalty should be applied
        lowercase_					:       Tuple     				=					1 - jnp.clip(cur_len - self.min_length       ,				0       ,				1				)

        lowercase_					:       List[Any]     				=					jnp.where(_a       ,				scores.at[:, self.eos_token_id].set(-float('inf'				)				)       ,				_a				)

        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a				)    ->     Optional[Any]:
        lowercase_					:       Optional[Any]     				=					list(_a				)
        lowercase_					:       List[Any]     				=					begin_index





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     Optional[Any]:
        lowercase_					:       Any     				=					1 - jnp.bool_(cur_len - self.begin_index				)

        lowercase_					:       List[str]     				=					jnp.where(_a       ,				scores.at[:, self.begin_suppress_tokens].set(-float('inf'				)				)       ,				_a				)

        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a				)    ->     Any:
        lowercase_					:       str     				=					list(_a				)





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        lowercase_					:       Optional[int]     				=					scores.at[..., self.suppress_tokens].set(-float('inf'				)				)

        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a				)    ->     Tuple:
        lowercase_					:       List[Any]     				=					dict(_a				)
        # Converts the dictionary of format {index: token} containing the tokens to be forced to an array, where the
        # index of the array corresponds to the index of the token to be forced, for XLA compatibility.
        # Indexes without forced tokens will have a negative value.
        lowercase_					:       Dict     				=					jnp.ones((max(force_token_map.keys()				) + 1)       ,				dtype=jnp.intaa				) * -1
        for index, token in force_token_map.items():
             if token is not None:
                  lowercase_					:       int     				=					force_token_array.at[index].set(_a				)
        lowercase_					:       Any     				=					jnp.intaa(_a				)





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     jnp.ndarray:
        def _force_token(_a				):
             lowercase_					:       Optional[Any]     				=					scores.shape[0]
             lowercase_					:       Optional[Any]     				=					self.force_token_array[generation_idx]

             lowercase_					:       Optional[Any]     				=					jnp.ones_like(_a       ,				dtype=scores.dtype				) * -float('inf'				)
             lowercase_					:       List[Any]     				=					jnp.zeros((batch_size, 1)       ,				dtype=scores.dtype				)
             lowercase_					:       Optional[int]     				=					lax.dynamic_update_slice(_a       ,				_a       ,				(0, current_token)				)
             return new_scores

        lowercase_					:       Optional[Any]     				=					lax.cond(
            cur_len >= self.force_token_array.shape[0]       ,				lambda: scores       ,				lambda: lax.cond(
                self.force_token_array[cur_len] >= 0       ,				lambda: _force_token(_a				)       ,				lambda: scores       ,				)       ,				)
        return scores



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a       ,				_a				)    ->     Dict:
        lowercase_					:       Union[str, Any]     				=					generate_config.eos_token_id
        lowercase_					:       Dict     				=					generate_config.no_timestamps_token_id
        lowercase_					:       Tuple     				=					generate_config.no_timestamps_token_id + 1

        lowercase_					:       Dict     				=					decoder_input_length + 1

        if generate_config.is_multilingual:
             # room for language token and task token
             self.begin_index += 2
        if hasattr(_a       ,				'max_initial_timestamp_index'				):
             lowercase_					:       Optional[int]     				=					generate_config.max_initial_timestamp_index
        else:
             lowercase_					:       Dict     				=					model_config.vocab_size
        if self.max_initial_timestamp_index is None:
             lowercase_					:       Optional[int]     				=					model_config.vocab_size





   def __call__(self       ,				_a       ,				_a       ,				_a				)    ->     Optional[int]:
        # suppress <|notimestamps|> which is handled by without_timestamps
        lowercase_					:       Dict     				=					scores.at[:, self.no_timestamps_token_id].set(-float('inf'				)				)

        def handle_pairs(_a       ,				_a				):
             lowercase_					:       Optional[int]     				=					jnp.where((cur_len - self.begin_index) >= 1       ,				_a       ,				_a				)
             lowercase_					:       List[Any]     				=					jnp.where(
                 input_ids_k[cur_len - 1] >= self.timestamp_begin       ,				True and last_was_timestamp       ,				_a       ,				)

             lowercase_					:       List[Any]     				=					jnp.where((cur_len - self.begin_index) < 2       ,				_a       ,				_a				)
             lowercase_					:       Optional[Any]     				=					jnp.where(
                 input_ids_k[cur_len - 2] >= self.timestamp_begin       ,				_a       ,				_a       ,				)

             return jnp.where(
                 _a       ,				jnp.where(
                     penultimate_was_timestamp > 0       ,				scores_k.at[self.timestamp_begin :].set(-float('inf'				)				)       ,				scores_k.at[: self.eos_token_id].set(-float('inf'				)				)       ,				)       ,				_a       ,				)

        lowercase_					:       List[str]     				=					jax.vmap(_a				)(_a       ,				_a				)

        lowercase_					:       str     				=					jnp.where(cur_len == self.begin_index       ,				_a       ,				_a				)
        lowercase_					:       Union[str, Any]     				=					jnp.where(
            self.max_initial_timestamp_index is not None       ,				True and apply_max_initial_timestamp       ,				_a       ,				)

        lowercase_					:       Optional[Any]     				=					self.timestamp_begin + self.max_initial_timestamp_index

        lowercase_					:       Optional[int]     				=					jnp.where(
            _a       ,				scores.at[:, last_allowed + 1 :].set(-float('inf'				)				)       ,				_a       ,				)

        # if sum of probability over timestamps is above any other token, sample timestamp
        lowercase_					:       Union[str, Any]     				=					jax.nn.log_softmax(_a       ,				axis=-1				)

        def handle_cumulative_probs(_a       ,				_a				):
             lowercase_					:       Dict     				=					jax.nn.logsumexp(logprobs_k[self.timestamp_begin :]       ,				axis=-1				)
             lowercase_					:       List[Any]     				=					jnp.max(logprobs_k[: self.timestamp_begin]				)
             return jnp.where(
                 timestamp_logprob > max_text_token_logprob       ,				scores_k.at[: self.timestamp_begin].set(-float('inf'				)				)       ,				_a       ,				)

        lowercase_					:       Optional[int]     				=					jax.vmap(_a				)(_a       ,				_a				)

        return scores





'''simple docstring'''




from decimal import Decimal, getcontext
from math import ceil, factorial

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_							):

     if not isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							):
          raise TypeError('Undefined for non-integers'							)
     elif precision < 1:
          raise ValueError('Undefined for non-natural numbers'							)

     lowercase_					:       str     				=					precision
     lowercase_					:       List[str]     				=					ceil(precision / 14							)
     lowercase_					:       Union[str, Any]     				=					426_880 * Decimal(10_005							).sqrt()
     lowercase_					:       List[Any]     				=					1
     lowercase_					:       Optional[int]     				=					13_591_409
     lowercase_					:       Dict     				=					Decimal(SCREAMING_SNAKE_CASE_							)
     for k in range(1  ,       SCREAMING_SNAKE_CASE_							):
          lowercase_					:       List[str]     				=					factorial(6 * k							) // (factorial(3 * k							) * factorial(SCREAMING_SNAKE_CASE_							) ** 3)
          linear_term += 545_140_134
          exponential_term *= -262_537_412_640_768_000
          partial_sum += Decimal(multinomial_term * linear_term							) / exponential_term
     return str(constant_term / partial_sum							)[:-1]


if __name__ == "__main__":
 _A								=   5_0
 print(F"""The first {n} digits of pi is: {pi(n)}""")




