





'''simple docstring'''




from math import factorial


class 			SCREAMING_SNAKE_CASE   :


					def __init__(						self						,     _UpperCAmelCase						,     _UpperCAmelCase):

										'''simple docstring'''






										__A						:			Union[str, Any]								=	real
										if isinstance(A__						,     A__):
															__A						:			str								=	[1] * rank
										else:
															__A						:			Any								=	rank


					def __repr__(						self):

										'''simple docstring'''






										return (
										    F'{self.real}+'
										    F'{"+".join(str(A__)+"E"+str(n+1)for n,dual in enumerate(self.duals))}'
										)


					def 							SCREAMING_SNAKE_CASE  (						self):

										'''simple docstring'''






										__A						:			List[Any]								=	self.duals.copy()
										while cur[-1] == 0:
															cur.pop(-1)
										return Dual(self.real						,     A__)


					def __add__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										if not isinstance(A__						,     A__):
															return Dual(self.real + other						,     self.duals)
										__A						:			Dict								=	self.duals.copy()
										__A						:			List[Any]								=	other.duals.copy()
										if len(A__) > len(A__):
															o_dual.extend([1] * (len(A__) - len(A__)))
										elif len(A__) < len(A__):
															s_dual.extend([1] * (len(A__) - len(A__)))
										__A						:			Dict								=	[]
										for i in range(len(A__)):
															new_duals.append(s_dual[i] + o_dual[i])
										return Dual(self.real + other.real						,     A__)

					lowerCAmelCase		      =	__add__


					def __sub__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										return self + other * -1


					def __mul__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										if not isinstance(A__						,     A__):
															__A						:			str								=	[]
															for i in self.duals:
																				new_duals.append(i * other)
															return Dual(self.real * other						,     A__)
										__A						:			Union[str, Any]								=	[0] * (len(self.duals) + len(other.duals) + 1)
										for i, item in enumerate(self.duals):
															for j, jtem in enumerate(other.duals):
																				new_duals[i + j + 1] += item * jtem
										for k in range(len(self.duals)):
															new_duals[k] += self.duals[k] * other.real
										for index in range(len(other.duals)):
															new_duals[index] += other.duals[index] * self.real
										return Dual(self.real * other.real						,     A__)

					lowerCAmelCase		      =	__mul__


					def __truediv__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										if not isinstance(A__						,     A__):
															__A						:			Dict								=	[]
															for i in self.duals:
																				new_duals.append(i / other)
															return Dual(self.real / other						,     A__)
										raise ValueError


					def __floordiv__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										if not isinstance(A__						,     A__):
															__A						:			Dict								=	[]
															for i in self.duals:
																				new_duals.append(i // other)
															return Dual(self.real // other						,     A__)
										raise ValueError


					def __pow__(						self						,     _UpperCAmelCase):

										'''simple docstring'''






										if n < 0 or isinstance(A__						,     A__):
															raise ValueError('power must be a positive integer')
										if n == 0:
															return 1
										if n == 1:
															return self
										__A						:			int								=	self
										for _ in range(n - 1):
															x *= self
										return x






def 	_lowerCAmelCase      (  __snake_case  :			Optional[Any]			,							__snake_case  :			Any			,							__snake_case  :			Union[str, Any]      )					->					Union[str, Any]:
					if not callable(lowerCAmelCase__      ):
										raise ValueError('differentiate() requires a function as input for func'      )
					if not isinstance(lowerCAmelCase__			,							(float, int)      ):
										raise ValueError('differentiate() requires a float as input for position'      )
					if not isinstance(lowerCAmelCase__			,							lowerCAmelCase__      ):
										raise ValueError('differentiate() requires an int as input for order'      )
					__A						:			str								=	Dual(lowerCAmelCase__			,							1      )
					__A						:			List[Any]								=	func(lowerCAmelCase__      )
					if order == 0:
										return result.real
					return result.duals[order - 1] * factorial(lowerCAmelCase__      )


if __name__ == "__main__":
		import doctest

		doctest.testmod()






		def 	_lowerCAmelCase      (  __snake_case  :			List[Any]      )					->					List[str]:
							return y**2 * y**4







		print(differentiate(f, 9, 2))






'''simple docstring'''




import math






def 	_lowerCAmelCase      (  __snake_case  :			int      )					->					bool:

     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or number % 2 == 0 or number % 3 == 0:
          # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
          return False

     # All primes number are in format of 6k +/- 1
     for i in range(5			,							int(math.sqrt(__snake_case      ) + 1      )			,							6      ):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True






def 	_lowerCAmelCase      (  __snake_case  :			float = 0.1      )					->					int:
     __A						:			Dict								=	3
     __A						:			int								=	3

     while primes / (2 * j - 1) >= ratio:
          for i in range(j * j + j + 1			,							(j + 2) * (j + 2)			,							j + 1      ):
               primes += is_prime(__snake_case      )
          j += 2
     return j


if __name__ == "__main__":
  import doctest

  doctest.testmod()