




'''simple docstring'''






from itertools import count
from math import asin, pi, sqrt

def       __lowerCAmelCase (   snake_case__						):
			return (
			    (1 - 2 * point) * sqrt(point - point**2						) + 2 * point + asin(sqrt(1 - point						)						)
			) / 4

def       __lowerCAmelCase (   snake_case__						):
			__UpperCamelCase				:      Optional[Any]									=       (circles_number + 1 - sqrt(2 * circles_number						)) / (
			    2 * (circles_number**2 + 1)
			)
			__UpperCamelCase				:      Tuple									=       circles_number * intersection_y

			__UpperCamelCase				:      Any									=       intersection_x * intersection_y / 2
			__UpperCamelCase				:      Dict									=       circle_bottom_arc_integral(
			    1 / 2						) - circle_bottom_arc_integral(snake_case__						)

			return triangle_area + concave_region_area

def       __lowerCAmelCase (   snake_case__ = 1 / 1_000						):
			__UpperCamelCase				:      str									=       (1 - pi / 4) / 4

			for n in count(1						):
						if concave_triangle_area(snake_case__						) / l_section_area < fraction:
									return n

			return -1


if __name__ == "__main__":
				print(f'{solution() = }')









'''simple docstring'''






from typing import Optional, Tuple

from ...tokenization_utils_fast import PreTrainedTokenizerFast
from ...utils import logging
from .tokenization_openai import OpenAIGPTTokenizer


_lowerCAmelCase				     = logging.get_logger(__name__)

_lowerCAmelCase				     = {'''vocab_file''': '''vocab.json''', '''merges_file''': '''merges.txt''', '''tokenizer_file''': '''tokenizer.json'''}

_lowerCAmelCase				     = {
    '''vocab_file''': {'''openai-gpt''': '''https://huggingface.co/openai-gpt/resolve/main/vocab.json'''},
    '''merges_file''': {'''openai-gpt''': '''https://huggingface.co/openai-gpt/resolve/main/merges.txt'''},
    '''tokenizer_file''': {'''openai-gpt''': '''https://huggingface.co/openai-gpt/resolve/main/tokenizer.json'''},
}

_lowerCAmelCase				     = {
    '''openai-gpt''': 512,
}



class  A		(			SCREAMING_SNAKE_CASE__		):




				'''simple docstring'''





				A				     =    VOCAB_FILES_NAMES
				A				     =    PRETRAINED_VOCAB_FILES_MAP
				A				     =    PRETRAINED_POSITIONAL_EMBEDDINGS_SIZES
				A				     =    ["input_ids", "attention_mask"]
				A				     =    OpenAIGPTTokenizer
				def __init__(self			,  _UpperCAmelCase=None			,  _UpperCAmelCase=None			,  _UpperCAmelCase=None			,  _UpperCAmelCase="<unk>"			,  **_UpperCAmelCase      )     ->	str:
							super().__init__(_UpperCAmelCase			,  _UpperCAmelCase			,  tokenizer_file=_UpperCAmelCase			,  unk_token=_UpperCAmelCase			,  **_UpperCAmelCase      )
				@property
				def 	a_	(self      )     ->	Dict:
							return True





				def 	a_	(self			,  _UpperCAmelCase			,  _UpperCAmelCase = None      )     ->	Tuple[str]:
							__UpperCamelCase				:      Optional[Any]									=       self._tokenizer.model.save(_UpperCAmelCase			,  name=_UpperCAmelCase      )
							return tuple(_UpperCAmelCase      )



