



"""simple docstring"""







from math import isclose, sqrt

def 				UpperCamelCase_    (    lowerCAmelCase__      :				float    ,   lowerCAmelCase__      :				float    ,   lowerCAmelCase__      :				float						)						->   tuple[float, float, float]:

  """simple docstring"""




  lowerCAmelCase_   :				Dict								= point_y / 4 / point_x
  lowerCAmelCase_   :				Dict								= 2 * normal_gradient / (1 + normal_gradient * normal_gradient)
  lowerCAmelCase_   :				Dict								= (1 - normal_gradient * normal_gradient) / (
      1 + normal_gradient * normal_gradient
  )
  lowerCAmelCase_   :				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_   :				str								= outgoing_gradient**2 + 4
  lowerCAmelCase_   :				Tuple								= 2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
  lowerCAmelCase_   :				Optional[int]								= (point_y - outgoing_gradient * point_x) ** 2 - 100

  lowerCAmelCase_   :				List[Any]								= (
      -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
  ) / (2 * quadratic_term)
  lowerCAmelCase_   :				Dict								= (
      -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
  ) / (2 * quadratic_term)

  # two solutions, one of which is our input point
  lowerCAmelCase_   :				Tuple								= x_minus if isclose(lowerCAmelCase__    ,   lowerCAmelCase__						) else x_plus
  lowerCAmelCase_   :				List[Any]								= point_y + outgoing_gradient * (next_x - point_x)

  return next_x, next_y, outgoing_gradient

def 				UpperCamelCase_    (    lowerCAmelCase__      :				float = 1.4    ,   lowerCAmelCase__      :				float = -9.6						)						->   int:

  """simple docstring"""




  lowerCAmelCase_   :				int								= 0
  lowerCAmelCase_   :				float								= first_x_coord
  lowerCAmelCase_   :				float								= first_y_coord
  lowerCAmelCase_   :				float								= (10.1 - point_y) / (0.0 - point_x)

  while not (-0.01 <= point_x <= 0.01 and point_y > 0):
    lowerCAmelCase_						,lowerCAmelCase_						,lowerCAmelCase_   :				str								= next_point(lowerCAmelCase__    ,   lowerCAmelCase__    ,   lowerCAmelCase__						)
    num_reflections += 1

  return num_reflections


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










"""simple docstring"""







import random

def 				UpperCamelCase_    (    lowerCAmelCase__      :				int    ,   lowerCAmelCase__      :				float    ,   lowerCAmelCase__      :				bool = False						)						->   dict:

  """simple docstring"""




  lowerCAmelCase_   :				dict								= {i: [] for i in range(lowerCAmelCase__						)}

  # if probability is greater or equal than 1, then generate a complete graph
  if probability >= 1:
    return complete_graph(lowerCAmelCase__						)
  # if probability is lower or equal than 0, then return a graph without edges
  if probability <= 0:
    return graph

  # for each couple of nodes, add an edge from u to v
  # if the number randomly generated is greater than probability probability
  for i in range(lowerCAmelCase__						):
    for j in range(i + 1    ,   lowerCAmelCase__						):
      if random.random() < probability:
        graph[i].append(lowerCAmelCase__						)
        if not directed:
          # if the graph is undirected, add an edge in from j to i, either
          graph[j].append(lowerCAmelCase__						)
  return graph

def 				UpperCamelCase_    (    lowerCAmelCase__      :				int						)						->   dict:

  """simple docstring"""




  return {
      i: [j for j in range(lowerCAmelCase__						) if i != j] for i in range(lowerCAmelCase__						)
  }


if __name__ == "__main__":
     import doctest

     doctest.testmod()





