



import numpy as np

import datasets


a__			=		"""
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/]
"""

a__			=		"""\
@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}
}
"""

a__			=		"""
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 						snake_case  (		datasets.Metric     ):



   '''simple docstring'''






   def    UpperCamelCase_     (    self							:	Tuple) -> Union[str, Any]:

    """simple docstring"""







    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    UpperCamelCase_     (    self							:	Union[str, Any] ,						lowerCAmelCase							:	Union[str, Any] ,						lowerCAmelCase							:	List[Any]) -> Union[str, Any]:

    """simple docstring"""







    _snake_case			:     List[Any] 		=    np.array(_SCREAMING_SNAKE_CASE)
    _snake_case			:     Optional[Any] 		=    np.array(_SCREAMING_SNAKE_CASE)

    # 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
    _snake_case			:     str 		=    X - np.mean(_SCREAMING_SNAKE_CASE)
    _snake_case			:     List[str] 		=    np.cov(reference_distribution.T)
    try:
     _snake_case			:     Optional[Any] 		=    np.linalg.inv(_SCREAMING_SNAKE_CASE)
    except np.linalg.LinAlgError:
     _snake_case			:     Optional[int] 		=    np.linalg.pinv(_SCREAMING_SNAKE_CASE)
    _snake_case			:     Optional[Any] 		=    np.dot(_SCREAMING_SNAKE_CASE ,						_SCREAMING_SNAKE_CASE)
    _snake_case			:     Optional[int] 		=    np.dot(_SCREAMING_SNAKE_CASE ,						X_minus_mu.T).diagonal()

    return {"mahalanobis": mahal_dist}




from math import isclose, sqrt


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ):
  A_      :				List[str]		     =					point_y / 4 / point_x
  A_      :				Union[str, Any]		     =					2 * normal_gradient / (1 + normal_gradient * normal_gradient)
  A_      :				int		     =					(1 - normal_gradient * normal_gradient) / (
      1 + normal_gradient * normal_gradient
  )
  A_      :				List[str]		     =					(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

  # to find the next point, solve the simultaeneous equations:
  # y^2 + 4x^2 = 100
  # y - b = m * (x - a)
  # ==> A x^2 + B x + C = 0
  A_      :				List[str]		     =					outgoing_gradient**2 + 4
  A_      :				List[str]		     =					2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
  A_      :				Any		     =					(point_y - outgoing_gradient * point_x) ** 2 - 100

  A_      :				str		     =					(
      -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)
  A_      :				Any		     =					(
      -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)

  # two solutions, one of which is our input point
  A_      :				int		     =					x_minus if isclose(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ) else x_plus
  A_      :				Any		     =					point_y + outgoing_gradient * (next_x - point_x)

  return next_x, next_y, outgoing_gradient


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE = 1.4 ,			SCREAMING_SNAKE_CASE = -9.6       ):
  A_      :				int		     =					0
  A_      :				float		     =					first_x_coord
  A_      :				float		     =					first_y_coord
  A_      :				float		     =					(1_0.1 - point_y) / (0.0 - point_x)

  while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
    A_     ,       A_     ,       A_      :				List[str]		     =					next_point(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )
    num_reflections += 1

  return num_reflections


if __name__ == "__main__":
     print(F'''{solution() = }''')




