


'''simple docstring'''






from math import factorial




class 		a__    :





							'''simple docstring'''






							def __init__(		self ,   lowerCamelCase_ ,   lowerCamelCase_       )	->							Tuple:
														lowerCAmelCase__	       =     real
														if isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					lowerCAmelCase__	       =     [1] * rank
														else:
																					lowerCAmelCase__	       =     rank







							def __repr__(		self       )	->							List[str]:
														return (
														    F"""{self.real}+"""
														    F"""{"+".join(str(lowerCamelCase_       )+"E"+str(n+1       )for n,dual in enumerate(self.duals       )       )}"""
														)







							def   __SCREAMING_SNAKE_CASE							(		self       )	->							Dict:
														lowerCAmelCase__	       =     self.duals.copy()
														while cur[-1] == 0:
																					cur.pop(-1       )
														return Dual(self.real ,   lowerCamelCase_       )







							def __add__(		self ,   lowerCamelCase_       )	->							Tuple:
														if not isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					return Dual(self.real + other ,   self.duals       )
														lowerCAmelCase__	       =     self.duals.copy()
														lowerCAmelCase__	       =     other.duals.copy()
														if len(lowerCamelCase_       ) > len(lowerCamelCase_       ):
																					o_dual.extend([1] * (len(lowerCamelCase_       ) - len(lowerCamelCase_       ))       )
														elif len(lowerCamelCase_       ) < len(lowerCamelCase_       ):
																					s_dual.extend([1] * (len(lowerCamelCase_       ) - len(lowerCamelCase_       ))       )
														lowerCAmelCase__	       =     []
														for i in range(len(lowerCamelCase_       )       ):
																					new_duals.append(s_dual[i] + o_dual[i]       )
														return Dual(self.real + other.real ,   lowerCamelCase_       )

							lowercase__      :       Tuple     						=				__add__







							def __sub__(		self ,   lowerCamelCase_       )	->							List[str]:
														return self + other * -1







							def __mul__(		self ,   lowerCamelCase_       )	->							Dict:
														if not isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					lowerCAmelCase__	       =     []
																					for i in self.duals:
																												new_duals.append(i * other       )
																					return Dual(self.real * other ,   lowerCamelCase_       )
														lowerCAmelCase__	       =     [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 ,   lowerCamelCase_       )

							lowercase__      :       Union[str, Any]     						=				__mul__







							def __truediv__(		self ,   lowerCamelCase_       )	->							Tuple:
														if not isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					lowerCAmelCase__	       =     []
																					for i in self.duals:
																												new_duals.append(i / other       )
																					return Dual(self.real / other ,   lowerCamelCase_       )
														raise ValueError







							def __floordiv__(		self ,   lowerCamelCase_       )	->							Union[str, Any]:
														if not isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					lowerCAmelCase__	       =     []
																					for i in self.duals:
																												new_duals.append(i // other       )
																					return Dual(self.real // other ,   lowerCamelCase_       )
														raise ValueError







							def __pow__(		self ,   lowerCamelCase_       )	->							Optional[int]:
														if n < 0 or isinstance(lowerCamelCase_ ,   lowerCamelCase_       ):
																					raise ValueError('''power must be a positive integer'''       )
														if n == 0:
																					return 1
														if n == 1:
																					return self
														lowerCAmelCase__	       =     self
														for _ in range(n - 1       ):
																					x *= self
														return x


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


if __name__ == "__main__":
		import doctest

		doctest.testmod()


		def  _snake_case       (					A	)						->   str:
									return y**2 * y**4






		print(differentiate(f, 9, 2))



'''simple docstring'''






import numpy as np

import datasets


__UpperCAmelCase       =							'''
Compute the Mahalanobis Distance

Mahalonobis distance is the distance between a point and a distribution.
And not between two distinct points. It is effectively a multivariate equivalent of the Euclidean distance.
It was introduced by Prof. P. C. Mahalanobis in 1936
and has been used in various statistical applications ever since
[source: https://www.machinelearningplus.com/statistics/mahalanobis-distance/]
'''

__UpperCAmelCase       =							'''\
@article{de2000mahalanobis,
  title={The mahalanobis distance},
  author={De Maesschalck, Roy and Jouan-Rimbaud, Delphine and Massart, D{\'e}sir{\'e} L},
  journal={Chemometrics and intelligent laboratory systems},
  volume={50},
  number={1},
  pages={1--18},
  year={2000},
  publisher={Elsevier}
}
'''

__UpperCAmelCase       =							'''
Args:
    X: List of datapoints to be compared with the `reference_distribution`.
    reference_distribution: List of datapoints from the reference distribution we want to compare to.
Returns:
    mahalanobis: The Mahalonobis distance for each datapoint in `X`.
Examples:

    >>> mahalanobis_metric = datasets.load_metric("mahalanobis")
    >>> results = mahalanobis_metric.compute(reference_distribution=[[0, 1], [1, 0]], X=[[0, 1]])
    >>> print(results)
    {\'mahalanobis\': array([0.5])}
'''




@datasets.utils.file_utils.add_start_docstrings(_DESCRIPTION    ,    _KWARGS_DESCRIPTION  )
class 		a__    (							datasets.Metric  ):





							'''simple docstring'''






							def   __SCREAMING_SNAKE_CASE							(		self       )	->							Any:
														return datasets.MetricInfo(
														    description=_DESCRIPTION ,   citation=_CITATION ,   inputs_description=_KWARGS_DESCRIPTION ,   features=datasets.Features(
														        {
														            '''X''': datasets.Sequence(datasets.Value('''float''' ,   id='''sequence'''       ) ,   id='''X'''       ),
														        }       ) ,   )







							def   __SCREAMING_SNAKE_CASE							(		self ,   lowerCamelCase_ ,   lowerCamelCase_       )	->							str:
														# convert to numpy arrays
														lowerCAmelCase__	       =     np.array(lowerCamelCase_       )
														lowerCAmelCase__	       =     np.array(lowerCamelCase_       )

														# Assert that arrays are 2D
														if len(X.shape       ) != 2:
																					raise ValueError('''Expected `X` to be a 2D vector'''       )
														if len(reference_distribution.shape       ) != 2:
																					raise ValueError('''Expected `reference_distribution` to be a 2D vector'''       )
														if reference_distribution.shape[0] < 2:
																					raise ValueError(
																					    '''Expected `reference_distribution` to be a 2D vector with more than one element in the first dimension'''       )

														# Get mahalanobis distance for each prediction
														lowerCAmelCase__	       =     X - np.mean(lowerCamelCase_       )
														lowerCAmelCase__	       =     np.cov(reference_distribution.T       )
														try:
																					lowerCAmelCase__	       =     np.linalg.inv(lowerCamelCase_       )
														except np.linalg.LinAlgError:
																					lowerCAmelCase__	       =     np.linalg.pinv(lowerCamelCase_       )
														lowerCAmelCase__	       =     np.dot(lowerCamelCase_ ,   lowerCamelCase_       )
														lowerCAmelCase__	       =     np.dot(lowerCamelCase_ ,   X_minus_mu.T       ).diagonal()

														return {"mahalanobis": mahal_dist}