






'''simple docstring'''
from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
a_					= 2_9_9_7_9_2_4_5_8

# Symbols
a_	,				a_	,				a_	,				a_					= symbols('ct x y z')





def  _a(  UpperCamelCase__				:						float							):




      '''simple docstring'''






      if velocity > c:
            raise ValueError('''Speed must not exceed light speed 299,792,458 [m/s]!'''							)
      elif velocity < 1:
            # Usually the speed should be much higher than 1 (c order of magnitude)
            raise ValueError('''Speed must be greater than or equal to 1!'''							)

      return velocity / c





def  _a(  UpperCamelCase__				:						float							):




      '''simple docstring'''






      return 1 / sqrt(1 - beta(UpperCamelCase__							) ** 2							)





def  _a(  UpperCamelCase__				:						float							):




      '''simple docstring'''






      return np.array(
          [
              [gamma(UpperCamelCase__							), -gamma(UpperCamelCase__							) * beta(UpperCamelCase__							), 0, 0],
              [-gamma(UpperCamelCase__							) * beta(UpperCamelCase__							), gamma(UpperCamelCase__							), 0, 0],
              [0, 0, 1, 0],
              [0, 0, 0, 1],
          ]							)





def  _a(  UpperCamelCase__				:						float,	UpperCamelCase__				:						np.ndarray | None = None							):




      '''simple docstring'''






      if event is None:
            SCREAMING_SNAKE_CASE__     :				Optional[Any]							 =np.array([ct, x, y, z]							)  # Symbolic four vector
      else:
            event[0] *= c  # x0 is ct (speed of light * time)

      return transformation_matrix(UpperCamelCase__							) @ event


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  # Example of symbolic vector:
  a_					= transform(2_9_9_7_9_2_4_5)
  print('Example of four vector: ')
  print(F'''ct\' = {four_vector[0]}''')
  print(F'''x\' = {four_vector[1]}''')
  print(F'''y\' = {four_vector[2]}''')
  print(F'''z\' = {four_vector[3]}''')

  # Substitute symbols with numerical values
  a_					= {ct: c, x: 1, y: 1, z: 1}
  a_					= [four_vector[i].subs(sub_dict) for i in range(4)]

  print(F'''\n{numerical_vector}''')







'''simple docstring'''
class      __SCREAMING_SNAKE_CASE      :




 def __init__(      self	:							List[Any]					,      __lowercase	:							int      )	->						None:
       SCREAMING_SNAKE_CASE__     :				Union[str, Any]									=size
       SCREAMING_SNAKE_CASE__     :				List[Any]									=[0] * size
       SCREAMING_SNAKE_CASE__     :				str									=[0] * size




 @staticmethod
 def      __magic_name__	(      __lowercase	:							int      )	->						int:
       return index | (index + 1)




 @staticmethod
 def      __magic_name__	(      __lowercase	:							int      )	->						int:
       return (index & (index + 1)) - 1




 def      __magic_name__	(      self	:							Dict					,      __lowercase	:							int					,      __lowercase	:							int      )	->						None:
       SCREAMING_SNAKE_CASE__     :				List[str]									=value
       while index < self.size:
             SCREAMING_SNAKE_CASE__     :				Any									=self.get_prev(__lowercase      ) + 1
             if current_left_border == index:
                   SCREAMING_SNAKE_CASE__     :				List[str]									=value
             else:
                   SCREAMING_SNAKE_CASE__     :				str									=max(__lowercase					,      __lowercase					,      __lowercase      )
             SCREAMING_SNAKE_CASE__     :				Optional[Any]									=self.get_next(__lowercase      )




 def      __magic_name__	(      self	:							Optional[int]					,      __lowercase	:							int					,      __lowercase	:							int      )	->						int:
       right -= 1  # Because of right is exclusive
       SCREAMING_SNAKE_CASE__     :				str									=0
       while left <= right:
             SCREAMING_SNAKE_CASE__     :				Optional[Any]									=self.get_prev(__lowercase      )
             if left <= current_left:
                   SCREAMING_SNAKE_CASE__     :				List[Any]									=max(__lowercase					,      self.tree[right]      )
                   SCREAMING_SNAKE_CASE__     :				Any									=current_left
             else:
                   SCREAMING_SNAKE_CASE__     :				Optional[Any]									=max(__lowercase					,      self.arr[right]      )
                   right -= 1
       return result


if __name__ == "__main__":
  import doctest

  doctest.testmod()