'''simple docstring'''




import math as m






def     snake_case__							(      _A: List[Any]				,       _A: int				,       _A: int=0.001			)   ->     List[Any]:

		'''simple docstring'''


		return (f(a + h			) - f(a - h			)) / (2 * h)






def     snake_case__							(      _A: Dict				,       _A: List[str]=0				,       _A: Optional[int]=100				,       _A: List[str]=0.0_001				,       _A: Union[str, Any]=1e-6				,       _A: List[Any]=False			)   ->     Optional[int]:

		'''simple docstring'''


		lowerCAmelCase        =  xa  # set the initial guess
		lowerCAmelCase        =  [a]
		lowerCAmelCase        =  abs(f(_A			)			)
		lowerCAmelCase        =  lambda _A			: calc_derivative(_A				,       _A				,       h=_A			)  # noqa: E731  Derivative of f(x)
		for _ in range(_A			):
				if fa(_A			) == 0:
						raise ValueError("""No converging solution found"""			)
				lowerCAmelCase        =  a - f(_A			) / fa(_A			)  # Calculate the next estimate
				if logsteps:
						steps.append(_A			)
				if error < maxerror:
						break
		else:
				raise ValueError("""Iteration limit reached, no converging solution found"""			)
		if logsteps:
				# If logstep is true, then log intermediate steps
				return a, error, steps
		return a, error


if __name__ == "__main__":
						from matplotlib import pyplot as plt

						__lowercase				   =  lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
						__lowercase   ,					__lowercase   ,					__lowercase				   =  newton_raphson(
						    f, xa=1_0, maxiter=1_0_0_0, step=1e-6, logsteps=True
						)
						plt.plot([abs(f(x)) for x in steps])
						plt.xlabel('''step''')
						plt.ylabel('''error''')
						plt.show()
						print(f'solution = {{{solution:f}}}, error = {{{error:f}}}')





'''simple docstring'''




from __future__ import annotations

__lowercase				   =  [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]






class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase):




					"""simple docstring"""


					lowerCAmelCase        =  -1
					lowerCAmelCase        =  -1
					lowerCAmelCase        =  [-1] * length
					lowerCAmelCase        =  None
					lowerCAmelCase        =  None







			def __repr__(       self):




					"""simple docstring"""


					return f"Node(min_value={self.minn} max_value={self.maxx})"







def     snake_case__							(      _A: list[int]			)   ->     Node | None:

		'''simple docstring'''


		lowerCAmelCase        =  Node(len(_A			)			)
		lowerCAmelCase ,					lowerCAmelCase        =  min(_A			), max(_A			)
		# Leaf node case where the node contains only one unique value
		if root.minn == root.maxx:
				return root
		lowerCAmelCase        =  (root.minn + root.maxx) // 2

		lowerCAmelCase        =  []
		lowerCAmelCase        =  []

		for index, num in enumerate(_A			):
				if num <= pivot:
						left_arr.append(_A			)
				else:
						right_arr.append(_A			)
				lowerCAmelCase        =  len(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		return root






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		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
		lowerCAmelCase        =  (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			)   ->     int:

		'''simple docstring'''


		if start > end:
				return 0
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       _A			)
		lowerCAmelCase        =  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			)   ->     int:

		'''simple docstring'''


		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]
		lowerCAmelCase        =  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			)   ->     int:

		'''simple docstring'''


		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
		lowerCAmelCase        =  range_counting(
		    node.left				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       _A				,       _A				,       )
		lowerCAmelCase        =  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()




