


'''simple docstring'''

from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


lowerCamelCase							:   List[Any]  						=	{
    'configuration_pegasus_x': ['PEGASUS_X_PRETRAINED_CONFIG_ARCHIVE_MAP', 'PegasusXConfig'],
}

try:
 if not is_torch_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 lowerCamelCase							:   Union[str, Any]  						=	[
     'PEGASUS_X_PRETRAINED_MODEL_ARCHIVE_LIST',
     'PegasusXForConditionalGeneration',
     'PegasusXModel',
     'PegasusXPreTrainedModel',
 ]


if TYPE_CHECKING:
 from .configuration_pegasus_x import PEGASUS_X_PRETRAINED_CONFIG_ARCHIVE_MAP, PegasusXConfig

 try:
  if not is_torch_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .modeling_pegasus_x import (
      PEGASUS_X_PRETRAINED_MODEL_ARCHIVE_LIST,
      PegasusXForConditionalGeneration,
      PegasusXModel,
      PegasusXPreTrainedModel,
  )


else:
 import sys

 lowerCamelCase							:   List[Any]  						=	_LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)









'''simple docstring'''






from math import isclose, sqrt


def  snake_case_     (					_lowerCAmelCase  :						float   ,      _lowerCAmelCase  :						float   ,      _lowerCAmelCase  :						float  )      ->   tuple[float, float, float]:
							UpperCAmelCase :       Optional[int] 							=		point_y / 4 / point_x
							UpperCAmelCase :       str 							=		2 * normal_gradient / (1 + normal_gradient * normal_gradient)
							UpperCAmelCase :       Any 							=		(1 - normal_gradient * normal_gradient) / (
							    1 + normal_gradient * normal_gradient
							)
							UpperCAmelCase :       Union[str, Any] 							=		(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
							UpperCAmelCase :       Union[str, Any] 							=		outgoing_gradient**2 + 4
							UpperCAmelCase :       Dict 							=		2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
							UpperCAmelCase :       List[Any] 							=		(point_y - outgoing_gradient * point_x) ** 2 - 100

							UpperCAmelCase :       List[str] 							=		(
							    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term  )
							) / (2 * quadratic_term)
							UpperCAmelCase :       Optional[int] 							=		(
							    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term  )
							) / (2 * quadratic_term)

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

							return next_x, next_y, outgoing_gradient


def  snake_case_     (					_lowerCAmelCase  :						float = 1.4   ,      _lowerCAmelCase  :						float = -9.6  )      ->   int:
							UpperCAmelCase :       int 							=		0
							UpperCAmelCase :       float 							=		first_x_coord
							UpperCAmelCase :       float 							=		first_y_coord
							UpperCAmelCase :       float 							=		(1_0.1 - point_y) / (0.0 - point_x)

							while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
														UpperCAmelCase							,		UpperCAmelCase							,		UpperCAmelCase :       str 							=		next_point(_lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )
														num_reflections += 1

							return num_reflections


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





