



from __future__ import annotations

class  snake_case__:

					"""simple docstring"""






					def __init__(				self   :      Optional[Any] ,  SCREAMING_SNAKE_CASE   :      Optional[int]	):
									lowercase__	:       Any						=	order

									# a_{0} ... a_{k}
									lowercase__	:       Dict						=	[1.0] + [0.0] * order
									# b_{0} ... b_{k}
									lowercase__	:       List[str]						=	[1.0] + [0.0] * order

									# x[n-1] ... x[n-k]
									lowercase__	:       str						=	[0.0] * self.order
									# y[n-1] ... y[n-k]
									lowercase__	:       Tuple						=	[0.0] * self.order







					def 					snake_case	(				self   :      List[str] ,  SCREAMING_SNAKE_CASE   :      Dict ,  SCREAMING_SNAKE_CASE   :      Union[str, Any]	):



									if len(__lowerCAmelCase	) < self.order:
													lowercase__	:       Union[str, Any]						=	[1.0, *a_coeffs]

									if len(__lowerCAmelCase	) != self.order + 1:
													lowercase__	:       Optional[Any]						=	(
													    f"""Expected a_coeffs to have {self.order + 1} elements """
													    f"""for {self.order}-order filter, got {len(__lowerCAmelCase	)}"""
													)
													raise ValueError(__lowerCAmelCase	)

									if len(__lowerCAmelCase	) != self.order + 1:
													lowercase__	:       str						=	(
													    f"""Expected b_coeffs to have {self.order + 1} elements """
													    f"""for {self.order}-order filter, got {len(__lowerCAmelCase	)}"""
													)
													raise ValueError(__lowerCAmelCase	)

									lowercase__	:       str						=	a_coeffs
									lowercase__	:       str						=	b_coeffs







					def 					snake_case	(				self   :      Dict ,  SCREAMING_SNAKE_CASE   :      List[str]	):
									lowercase__	:       Union[str, Any]						=	0.0

									# Start at index 1 and do index 0 at the end.
									for i in range(1 ,  self.order + 1	):
													result += (
													    self.b_coeffs[i] * self.input_history[i - 1]
													    - self.a_coeffs[i] * self.output_history[i - 1]
													)

									lowercase__	:       Optional[Any]						=	(result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

									lowercase__	:       Union[str, Any]						=	self.input_history[:-1]
									lowercase__	:       Optional[int]						=	self.output_history[:-1]

									lowercase__	:       List[str]						=	sample
									lowercase__	:       Dict						=	result

									return result





"""simple docstring"""






from math import pow, sqrt


def      UpperCAmelCase__			(							*lowerCAmelCase__	:float						)					->       bool:
      '''simple docstring'''


      lowercase								=      len(lowerCAmelCase__						) > 0 and all(value > 0.0 for value in values						)
      return result


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError("""Input Error: Molar mass values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(effusion_rate * sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(effusion_rate / sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(molar_mass / pow(effusion_rate_a / effusion_rate_a   ,						2						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )






def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(pow(effusion_rate_a / effusion_rate_a   ,						2						) / molar_mass   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )



