


"""simple docstring"""


from math import isclose, sqrt




def   snake_case		(     UpperCamelCase__       :						float			,				UpperCamelCase__       :						float			,				UpperCamelCase__       :						float    )   ->      tuple[float, float, float]:
      lowerCamelCase					:					List[Any]        =	point_y / 4 / point_x
      lowerCamelCase					:					Dict        =	2 * normal_gradient / (1 + normal_gradient * normal_gradient)
      lowerCamelCase					:					Tuple        =	(1 - normal_gradient * normal_gradient) / (
          1 + normal_gradient * normal_gradient
      )
      lowerCamelCase					:					Optional[int]        =	(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
      lowerCamelCase					:					Tuple        =	outgoing_gradient**2 + 4
      lowerCamelCase					:					List[Any]        =	2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
      lowerCamelCase					:					Optional[int]        =	(point_y - outgoing_gradient * point_x) ** 2 - 100

      lowerCamelCase					:					Dict        =	(
          -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term    )
      ) / (2 * quadratic_term)
      lowerCamelCase					:					Union[str, Any]        =	(
          -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term    )
      ) / (2 * quadratic_term)

      # two solutions, one of which is our input point
      lowerCamelCase					:					Union[str, Any]        =	x_minus if isclose(UpperCamelCase__			,				UpperCamelCase__    ) else x_plus
      lowerCamelCase					:					str        =	point_y + outgoing_gradient * (next_x - point_x)

      return next_x, next_y, outgoing_gradient




def   snake_case		(     UpperCamelCase__       :						float = 1.4			,				UpperCamelCase__       :						float = -9.6    )   ->      int:
      lowerCamelCase					:					int        =	0
      lowerCamelCase					:					float        =	first_x_coord
      lowerCamelCase					:					float        =	first_y_coord
      lowerCamelCase					:					float        =	(1_0.1 - point_y) / (0.0 - point_x)

      while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
            lowerCamelCase     ,			lowerCamelCase     ,			lowerCamelCase					:					Any        =	next_point(UpperCamelCase__			,				UpperCamelCase__			,				UpperCamelCase__    )
            num_reflections += 1

      return num_reflections


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








"""simple docstring"""


from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_torch_available,
)


__lowerCamelCase      :Optional[Any]	    =		{
    'configuration_encodec': [
        'ENCODEC_PRETRAINED_CONFIG_ARCHIVE_MAP',
        'EncodecConfig',
    ],
    'feature_extraction_encodec': ['EncodecFeatureExtractor'],
}

try:
    if not is_torch_available():
        raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
    pass
else:
    __lowerCamelCase      :Union[str, Any]	    =		[
        'ENCODEC_PRETRAINED_MODEL_ARCHIVE_LIST',
        'EncodecModel',
        'EncodecPreTrainedModel',
    ]

if TYPE_CHECKING:
    from .configuration_encodec import (
        ENCODEC_PRETRAINED_CONFIG_ARCHIVE_MAP,
        EncodecConfig,
    )
    from .feature_extraction_encodec import EncodecFeatureExtractor

    try:
        if not is_torch_available():
            raise OptionalDependencyNotAvailable()
    except OptionalDependencyNotAvailable:
        pass
    else:
        from .modeling_encodec import (
            ENCODEC_PRETRAINED_MODEL_ARCHIVE_LIST,
            EncodecModel,
            EncodecPreTrainedModel,
        )

else:
    import sys

    __lowerCamelCase      :Dict	    =		_LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)




