






"""simple docstring"""



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


def   SCREAMING_SNAKE_CASE__					(     snake_case					:   float							)->		float:






     '''simple docstring'''



     return (
         (1 - 2 * point) * sqrt(point - point**2							) + 2 * point + asin(sqrt(1 - point							)							)
     ) / 4


def   SCREAMING_SNAKE_CASE__					(     snake_case					:   int							)->		float:






     '''simple docstring'''



     UpperCAmelCase__	:						Optional[int]       =       (circles_number + 1 - sqrt(2 * circles_number							)) / (
         2 * (circles_number**2 + 1)
     )
     UpperCAmelCase__	:						Any       =       circles_number * intersection_y

     UpperCAmelCase__	:						Any       =       intersection_x * intersection_y / 2
     UpperCAmelCase__	:						Optional[Any]       =       circle_bottom_arc_integral(
         1 / 2							) - circle_bottom_arc_integral(snake_case							)

     return triangle_area + concave_region_area


def   SCREAMING_SNAKE_CASE__					(     snake_case					:   float = 1 / 1000							)->		int:






     '''simple docstring'''



     UpperCAmelCase__	:						Optional[int]       =       (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"""



#
# Sort large text files in a minimum amount of memory
#
import argparse
import os




class  lowerCAmelCase__ :
   SCREAMING_SNAKE_CASE_         ='''block_{0}.dat'''
   def __init__(		self			:  Tuple       , snake_case__			:  List[str]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						List[Any]       =       filename
        UpperCAmelCase__	:						List[Any]       =       []
   def 			__a       (		self			:  Optional[Any]       , snake_case__			:  Union[str, Any]       , snake_case__			:  Union[str, Any]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Dict       =       self.BLOCK_FILENAME_FORMAT.format(snake_case__       )
        with open(snake_case__       , "w"       ) as file:
             file.write(snake_case__       )
        self.block_filenames.append(snake_case__       )
   def 			__a       (		self			:  List[str]       ):


        '''simple docstring'''
        return self.block_filenames
   def 			__a       (		self			:  Optional[Any]       , snake_case__			:  List[str]       , snake_case__			:  str=None       ):


        '''simple docstring'''
        UpperCAmelCase__	:						int       =       0
        with open(self.filename       ) as file:
             while True:
                  UpperCAmelCase__	:						Union[str, Any]       =       file.readlines(snake_case__       )

                  if lines == []:
                       break

                  if sort_key is None:
                       lines.sort()
                  else:
                       lines.sort(key=snake_case__       )

                  self.write_block("".join(snake_case__       )       , snake_case__       )
                  i += 1




   def 			__a       (		self			:  Optional[Any]       ):


        '''simple docstring'''
        map(os.remove       , self.block_filenames       )




class  lowerCAmelCase__ :
   def 			__a       (		self			:  Union[str, Any]       , snake_case__			:  Optional[Any]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Any       =       -1
        UpperCAmelCase__	:						Tuple       =       None

        for i in range(len(snake_case__       )       ):
             if min_str is None or choices[i] < min_str:
                  UpperCAmelCase__	:						List[str]       =       i

        return min_index




class  lowerCAmelCase__ :
   def __init__(		self			:  Dict       , snake_case__			:  Optional[Any]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Optional[Any]       =       files
        UpperCAmelCase__	:						str       =       set()
        UpperCAmelCase__	:						Any       =       len(snake_case__       )
        UpperCAmelCase__	:						Tuple       =       {i: None for i in range(self.num_buffers       )}
   def 			__a       (		self			:  List[str]       ):


        '''simple docstring'''
        return {
            i: self.buffers[i] for i in range(self.num_buffers       ) if i not in self.empty
        }
   def 			__a       (		self			:  str       ):


        '''simple docstring'''
        for i in range(self.num_buffers       ):
             if self.buffers[i] is None and i not in self.empty:
                  UpperCAmelCase__	:						Tuple       =       self.files[i].readline()

                  if self.buffers[i] == "":
                       self.empty.add(snake_case__       )
                       self.files[i].close()

        if len(self.empty       ) == self.num_buffers:
             return False

        return True




   def 			__a       (		self			:  List[str]       , snake_case__			:  Tuple       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Any       =       self.buffers[index]
        UpperCAmelCase__	:						Union[str, Any]       =       None

        return value




class  lowerCAmelCase__ :
   def __init__(		self			:  Optional[Any]       , snake_case__			:  int       ):


        '''simple docstring'''
        UpperCAmelCase__	:						str       =       merge_strategy
   def 			__a       (		self			:  Dict       , snake_case__			:  Dict       , snake_case__			:  Optional[int]       , snake_case__			:  Optional[int]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Dict       =       FilesArray(self.get_file_handles(snake_case__       , snake_case__       )       )
        with open(snake_case__       , "w"       , snake_case__       ) as outfile:
             while buffers.refresh():
                  UpperCAmelCase__	:						Any       =       self.merge_strategy.select(buffers.get_dict()       )
                  outfile.write(buffers.unshift(snake_case__       )       )




   def 			__a       (		self			:  Tuple       , snake_case__			:  Dict       , snake_case__			:  List[str]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						str       =       {}

        for i in range(len(snake_case__       )       ):
             UpperCAmelCase__	:						List[Any]       =       open(filenames[i]       , "r"       , snake_case__       )  # noqa: UP015

        return files




class  lowerCAmelCase__ :
   def __init__(		self			:  str       , snake_case__			:  Optional[Any]       ):


        '''simple docstring'''
        UpperCAmelCase__	:						List[str]       =       block_size
   def 			__a       (		self			:  Optional[Any]       , snake_case__			:  Optional[int]       , snake_case__			:  Any=None       ):


        '''simple docstring'''
        UpperCAmelCase__	:						Union[str, Any]       =       self.get_number_blocks(snake_case__       , self.block_size       )
        UpperCAmelCase__	:						Union[str, Any]       =       FileSplitter(snake_case__       )
        splitter.split(self.block_size       , snake_case__       )

        UpperCAmelCase__	:						int       =       FileMerger(NWayMerge()       )
        UpperCAmelCase__	:						int       =       self.block_size / (num_blocks + 1)
        merger.merge(splitter.get_block_filenames()       , filename + ".out"       , snake_case__       )

        splitter.cleanup()




   def 			__a       (		self			:  Union[str, Any]       , snake_case__			:  Optional[int]       , snake_case__			:  Optional[Any]       ):


        '''simple docstring'''
        return (os.stat(snake_case__       ).st_size / block_size) + 1




def   SCREAMING_SNAKE_CASE__					(     snake_case					:   Tuple							)->		Union[str, Any]:






     '''simple docstring'''



     if string[-1].lower() == "k":
          return int(string[:-1]							) * 1024
     elif string[-1].lower() == "m":
          return int(string[:-1]							) * 1024 * 1024
     elif string[-1].lower() == "g":
          return int(string[:-1]							) * 1024 * 1024 * 1024
     else:
          return int(snake_case							)


def   SCREAMING_SNAKE_CASE__					(     )->		Any:






     '''simple docstring'''



     UpperCAmelCase__	:						Any       =       argparse.ArgumentParser()
     parser.add_argument(
         "-m" ,				"--mem" ,				help="amount of memory to use for sorting" ,				default="100M"							)
     parser.add_argument(
         "filename" ,				metavar="<filename>" ,				nargs=1 ,				help="name of file to sort"							)
     UpperCAmelCase__	:						int       =       parser.parse_args()

     UpperCAmelCase__	:						Optional[int]       =       ExternalSort(parse_memory(args.mem							)							)
     sorter.sort(args.filename[0]							)


if __name__ == "__main__":
 main()


