code: 
  '''simple docstring'''
  
  
  
  
  
  from __future__ import annotations
  
  from math import pi, pow
  def   snake_case_       (							_a     :							int | float			):
        if side_length < 0:
              raise ValueError("vol_cube() only accepts non-negative values"			)
        return pow(_a ,					3			)
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or radius < 0:
              raise ValueError("vol_spherical_cap() only accepts non-negative values"			)
        # Volume is 1/3 pi * height squared * (3 * radius - height)
        return 1 / 3 * pi * pow(_a ,					2			) * (3 * radius - height)
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							float			):
        if radius_a < 0 or radius_a < 0 or centers_distance < 0:
              raise ValueError("vol_spheres_intersect() only accepts non-negative values"			)
        if centers_distance == 0:
              return vol_sphere(min(_a ,					_a			)			)
  
        snake_case_						       =							(
            (radius_a - radius_a + centers_distance)
            * (radius_a + radius_a - centers_distance)
            / (2 * centers_distance)
        )
        snake_case_						       =							(
            (radius_a - radius_a + centers_distance)
            * (radius_a + radius_a - centers_distance)
            / (2 * centers_distance)
        )
  
        return vol_spherical_cap(_a ,					_a			) + vol_spherical_cap(_a ,					_a			)
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							float			):
  
        if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
              raise ValueError(
                  "vol_spheres_union() only accepts non-negative values, non-zero radius"			)
  
        if centers_distance == 0:
              return vol_sphere(max(_a ,					_a			)			)
  
        return (
            vol_sphere(_a			)
            + vol_sphere(_a			)
            - vol_spheres_intersect(_a ,					_a ,					_a			)
        )
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							float			):
        if width < 0 or height < 0 or length < 0:
              raise ValueError("vol_cuboid() only accepts non-negative values"			)
        return float(width * height * length			)
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or area_of_base < 0:
              raise ValueError("vol_cone() only accepts non-negative values"			)
        return area_of_base * height / 3.0
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or radius < 0:
              raise ValueError("vol_right_circ_cone() only accepts non-negative values"			)
        return pi * pow(_a ,					2			) * height / 3.0
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or area_of_base < 0:
              raise ValueError("vol_prism() only accepts non-negative values"			)
        return float(area_of_base * height			)
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or area_of_base < 0:
              raise ValueError("vol_pyramid() only accepts non-negative values"			)
        return area_of_base * height / 3.0
  def   snake_case_       (							_a     :							float			):
        if radius < 0:
              raise ValueError("vol_sphere() only accepts non-negative values"			)
        # Volume is 4/3 * pi * radius cubed
        return 4 / 3 * pi * pow(_a ,					3			)
  def   snake_case_       (							_a     :							float			):
        if radius < 0:
              raise ValueError("vol_hemisphere() only accepts non-negative values"			)
        # Volume is radius cubed * pi * 2/3
        return pow(_a ,					3			) * pi * 2 / 3
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if height < 0 or radius < 0:
              raise ValueError("vol_circular_cylinder() only accepts non-negative values"			)
        # Volume is radius squared * height * pi
        return pow(_a ,					2			) * height * pi
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							float			):
        # Volume - (outer_radius squared - inner_radius squared) * pi * height
        if inner_radius < 0 or outer_radius < 0 or height < 0:
              raise ValueError(
                  "vol_hollow_circular_cylinder() only accepts non-negative values"			)
        if outer_radius <= inner_radius:
              raise ValueError("outer_radius must be greater than inner_radius"			)
        return pi * (pow(_a ,					2			) - pow(_a ,					2			)) * height
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							float			):
        # Volume is 1/3 * pi * height *
        #           (radius_1 squared + radius_2 squared + radius_1 * radius_2)
        if radius_a < 0 or radius_a < 0 or height < 0:
              raise ValueError("vol_conical_frustum() only accepts non-negative values"			)
        return (
            1
            / 3
            * pi
            * height
            * (pow(_a ,					2			) + pow(_a ,					2			) + radius_a * radius_a)
        )
  def   snake_case_       (							_a     :							float ,					_a     :							float			):
        if torus_radius < 0 or tube_radius < 0:
              raise ValueError("vol_torus() only accepts non-negative values"			)
        return 2 * pow(_a ,					2			) * torus_radius * pow(_a ,					2			)
  def   snake_case_       (							):
        print("Volumes:"			)
        print(F'''Cube: {vol_cube(2			) = }'''			)  # = 8
        print(F'''Cuboid: {vol_cuboid(2 ,					2 ,					2			) = }'''			)  # = 8
        print(F'''Cone: {vol_cone(2 ,					2			) = }'''			)  # ~= 1.33
        print(F'''Right Circular Cone: {vol_right_circ_cone(2 ,					2			) = }'''			)  # ~= 8.38
        print(F'''Prism: {vol_prism(2 ,					2			) = }'''			)  # = 4
        print(F'''Pyramid: {vol_pyramid(2 ,					2			) = }'''			)  # ~= 1.33
        print(F'''Sphere: {vol_sphere(2			) = }'''			)  # ~= 33.5
        print(F'''Hemisphere: {vol_hemisphere(2			) = }'''			)  # ~= 16.75
        print(F'''Circular Cylinder: {vol_circular_cylinder(2 ,					2			) = }'''			)  # ~= 25.1
        print(F'''Torus: {vol_torus(2 ,					2			) = }'''			)  # ~= 157.9
        print(F'''Conical Frustum: {vol_conical_frustum(2 ,					2 ,					4			) = }'''			)  # ~= 58.6
        print(F'''Spherical cap: {vol_spherical_cap(1 ,					2			) = }'''			)  # ~= 5.24
        print(F'''Spheres intersetion: {vol_spheres_intersect(2 ,					2 ,					1			) = }'''			)  # ~= 21.21
        print(F'''Spheres union: {vol_spheres_union(2 ,					2 ,					1			) = }'''			)  # ~= 45.81
        print(
            F'''Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1 ,					2 ,					3			) = }'''			)  # ~= 28.3
  
  
  if __name__ == "__main__":
       main()
  
  
style_context: 
  '''simple docstring'''
  
  
  
  
  
  from __future__ import annotations
  
  A           =   [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]
  
  
  
  class  lowerCAmelCase__   :
  
  
  
    '''simple docstring'''
  
  
  
  
  
  
    def __init__(self  : Optional[int]	, lowerCAmelCase_  : int    )   ->       None:
          snake_case_						       =							-1
          snake_case_						       =							-1
          snake_case_						       =							[-1] * length
          snake_case_						       =							None
          snake_case_						       =							None
  
  
    def __repr__(self  : List[Any]    )   ->       str:
          return f'''Node(min_value={self.minn} max_value={self.maxx})'''
  
  def   snake_case_       (							_a     :							list[int]			):
        snake_case_						       =							Node(len(_a			)			)
        snake_case_       ,	snake_case_						       =							min(_a			), max(_a			)
        # Leaf node case where the node contains only one unique value
        if root.minn == root.maxx:
              return root
        snake_case_						       =							(root.minn + root.maxx) // 2
  
        snake_case_						       =							[]
        snake_case_						       =							[]
  
        for index, num in enumerate(_a			):
              if num <= pivot:
                    left_arr.append(_a			)
              else:
                    right_arr.append(_a			)
              snake_case_						       =							len(_a			)
        snake_case_						       =							build_tree(_a			)
        snake_case_						       =							build_tree(_a			)
        return root
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int			):
        if index < 0 or node is None:
              return 0
        # Leaf node cases
        if node.minn == node.maxx:
              return index + 1 if node.minn == num else 0
        snake_case_						       =							(node.minn + node.maxx) // 2
        if num <= pivot:
              # go the left subtree and map index to the left subtree
              return rank_till_index(node.left ,					_a ,					node.map_left[index] - 1			)
        else:
              # go to the right subtree and map index to the right subtree
              return rank_till_index(node.right ,					_a ,					index - node.map_left[index]			)
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if start > end:
              return 0
        snake_case_						       =							rank_till_index(_a ,					_a ,					_a			)
        snake_case_						       =							rank_till_index(_a ,					_a ,					start - 1			)
        return rank_till_end - rank_before_start
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if index > (end - start) or start > end or node is None:
              return -1
        # Leaf node case
        if node.minn == node.maxx:
              return node.minn
        # Number of elements in the left subtree in interval [start, end]
        snake_case_						       =							node.map_left[end] - (
            node.map_left[start - 1] if start else 0
        )
        if num_elements_in_left_tree > index:
              return quantile(
                  node.left ,					_a ,					(node.map_left[start - 1] if start else 0) ,					node.map_left[end] - 1 ,					)
        else:
              return quantile(
                  node.right ,					index - num_elements_in_left_tree ,					start - (node.map_left[start - 1] if start else 0) ,					end - node.map_left[end] ,					)
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if (
            start > end
            or node is None
            or start_num > end_num
            or node.minn > end_num
            or node.maxx < start_num
        ):
              return 0
        if start_num <= node.minn and node.maxx <= end_num:
              return end - start + 1
        snake_case_						       =							range_counting(
            node.left ,					(node.map_left[start - 1] if start else 0) ,					node.map_left[end] - 1 ,					_a ,					_a ,					)
        snake_case_						       =							range_counting(
            node.right ,					start - (node.map_left[start - 1] if start else 0) ,					end - node.map_left[end] ,					_a ,					_a ,					)
        return left + right
  
  
  if __name__ == "__main__":
       import doctest
  
       doctest.testmod()
  
  