

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					int	)				->    Optional[int]:
				_UpperCAmelCase    :						Union[str, Any]							=      0
				_UpperCAmelCase    :						Dict							=      len(lowerCAmelCase	)

				for i in range(n - 1	):
								for j in range(i + 1					, lowerCAmelCase	):
												if arr[i] > arr[j]:
																num_inversions += 1

				return num_inversions

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					List[Any]	)				->    List[Any]:
				if len(lowerCAmelCase	) <= 1:
								return arr, 0
				_UpperCAmelCase    :						int							=      len(lowerCAmelCase	) // 2
				_UpperCAmelCase    :						Optional[Any]							=      arr[0:mid]
				_UpperCAmelCase    :						List[Any]							=      arr[mid:]

				_UpperCAmelCase						,							_UpperCAmelCase    :						Any							=      count_inversions_recursive(lowerCAmelCase	)
				_UpperCAmelCase						,							_UpperCAmelCase    :						List[Any]							=      count_inversions_recursive(lowerCAmelCase	)
				_UpperCAmelCase						,							_UpperCAmelCase    :						str							=      _count_cross_inversions(lowerCAmelCase					, lowerCAmelCase	)

				_UpperCAmelCase    :						Dict							=      inversion_p + inversions_q + cross_inversions
				return c, num_inversions

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Union[str, Any]					, lowerCAmelCase:					List[Any]	)				->    Tuple:
				_UpperCAmelCase    :						int							=      []
				_UpperCAmelCase    :						str							=      0
				while i < len(lowerCAmelCase	) and j < len(lowerCAmelCase	):
								if p[i] > q[j]:
												# if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
												# These are all inversions. The claim emerges from the
												# property that P is sorted.
												num_inversion += len(lowerCAmelCase	) - i
												r.append(q[j]	)
												j += 1
								else:
												r.append(p[i]	)
												i += 1

				if i < len(lowerCAmelCase	):
								r.extend(p[i:]	)
				else:
								r.extend(q[j:]	)

				return r, num_inversion

def 							__SCREAMING_SNAKE_CASE							(			)				->    Union[str, Any]:
				_UpperCAmelCase    :						Optional[int]							=      [10, 2, 1, 5, 5, 2, 11]

				# this arr has 8 inversions:
				# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

				_UpperCAmelCase    :						List[Any]							=      count_inversions_bf(lowerCAmelCase	)
				_UpperCAmelCase						,							_UpperCAmelCase    :						Optional[Any]							=      count_inversions_recursive(lowerCAmelCase	)

				assert num_inversions_bf == num_inversions_recursive == 8

				print("number of inversions = "					, lowerCAmelCase	)

				# testing an array with zero inversion (a sorted arr_1)

				arr_a.sort()
				_UpperCAmelCase    :						Optional[Any]							=      count_inversions_bf(lowerCAmelCase	)
				_UpperCAmelCase						,							_UpperCAmelCase    :						List[Any]							=      count_inversions_recursive(lowerCAmelCase	)

				assert num_inversions_bf == num_inversions_recursive == 0
				print("number of inversions = "					, lowerCAmelCase	)

				# an empty list should also have zero inversions
				_UpperCAmelCase    :						Union[str, Any]							=      []
				_UpperCAmelCase    :						Dict							=      count_inversions_bf(lowerCAmelCase	)
				_UpperCAmelCase						,							_UpperCAmelCase    :						int							=      count_inversions_recursive(lowerCAmelCase	)

				assert num_inversions_bf == num_inversions_recursive == 0
				print("number of inversions = "					, lowerCAmelCase	)


if __name__ == "__main__":
					main()





from __future__ import annotations

from decimal import Decimal
from math import *  # noqa: F403

from sympy import diff

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					str					, lowerCAmelCase:					float | Decimal					, lowerCAmelCase:					float = 10**-10	)				->    float:
				_UpperCAmelCase    :						Optional[int]							=      a
				while True:
								_UpperCAmelCase    :						Tuple							=      Decimal(lowerCAmelCase	) - (
								    Decimal(eval(lowerCAmelCase	)	) / Decimal(eval(str(diff(lowerCAmelCase	)	)	)	)  # noqa: S307
								)
								# This number dictates the accuracy of the answer
								if abs(eval(lowerCAmelCase	)	) < precision:  # noqa: S307
												return float(lowerCAmelCase	)


# Let's Execute
if __name__ == "__main__":
					# Find root of trigonometric function
					# Find value of pi
					print(F'''The root of sin(x) = 0 is {newton_raphson("sin(x)", 2)}''')
					# Find root of polynomial
					print(F'''The root of x**2 - 5*x + 2 = 0 is {newton_raphson("x**2 - 5*x + 2", 0.4)}''')
					# Find Square Root of 5
					print(F'''The root of log(x) - 1 = 0 is {newton_raphson("log(x) - 1", 2)}''')
					# Exponential Roots
					print(F'''The root of exp(x) - 1 = 0 is {newton_raphson("exp(x) - 1", 0)}''')


