
from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean






def   lowerCAmelCase_						(					_snake_case      :  Dict   )				->		int:







	'''simple docstring'''


	def is_in_circle(_snake_case      :  Any      ,   _snake_case      :  List[str]   ) -> bool:
		__magic_name__		:		str	   =     sqrt((x**2) + (y**2)   )
		# Our circle has a radius of 1, so a distance
		# greater than 1 would land outside the circle.
		return distance_from_centre <= 1

	# The proportion of guesses that landed in the circle
	__magic_name__		:		int	   =     mean(
	    int(is_in_circle(uniform(-1.0      ,   1.0   )      ,   uniform(-1.0      ,   1.0   )   )   )
	    for _ in range(UpperCamelCase__   )   )
	# The ratio of the area for circle to square is pi/4.
	__magic_name__		:		int	   =     proportion * 4
	print(F'''The estimated value of pi is {pi_estimate}'''   )
	print(F'''The numpy value of pi is {pi}'''   )
	print(F'''The total error is {abs(pi - pi_estimate   )}'''   )






def   lowerCAmelCase_						(					_snake_case      :  Optional[int]      ,   _snake_case      :  Optional[Any]      ,   _snake_case      :  Optional[Any] = 0.0      ,   _snake_case      :  int = 1.0      ,   )				->		Optional[int]:







	'''simple docstring'''


	return mean(
	    function_to_integrate(uniform(UpperCamelCase__      ,   UpperCamelCase__   )   ) for _ in range(UpperCamelCase__   )   ) * (max_value - min_value)






def   lowerCAmelCase_						(					_snake_case      :  List[str]      ,   _snake_case      :  Union[str, Any] = 0.0      ,   _snake_case      :  Union[str, Any] = 1.0   )				->		Dict:







	'''simple docstring'''


	def identity_function(_snake_case      :  List[str]   ) -> float:
		return x

	__magic_name__		:		Optional[int]	   =     area_under_curve_estimator(
	    UpperCamelCase__      ,   UpperCamelCase__      ,   UpperCamelCase__      ,   UpperCamelCase__   )
	__magic_name__		:		str	   =     (max_value * max_value - min_value * min_value) / 2

	print("******************"   )
	print(F'''Estimating area under y=x where x varies from {min_value} to {max_value}'''   )
	print(F'''Estimated value is {estimated_value}'''   )
	print(F'''Expected value is {expected_value}'''   )
	print(F'''Total error is {abs(estimated_value - expected_value   )}'''   )
	print("******************"   )






def   lowerCAmelCase_						(					_snake_case      :  Union[str, Any]   )				->		Union[str, Any]:







	'''simple docstring'''


	def function_to_integrate(_snake_case      :  Tuple   ) -> float:
		return sqrt(4.0 - x * x   )

	__magic_name__		:		Any	   =     area_under_curve_estimator(
	    UpperCamelCase__      ,   UpperCamelCase__      ,   0.0      ,   2.0   )

	print("******************"   )
	print("Estimating pi using area_under_curve_estimator"   )
	print(F'''Estimated value is {estimated_value}'''   )
	print(F'''Expected value is {pi}'''   )
	print(F'''Total error is {abs(estimated_value - pi   )}'''   )
	print("******************"   )


if __name__ == "__main__":
				import doctest

				doctest.testmod()



def 					__lowerCamelCase	( ):







							'''simple docstring'''






							return [list(range(1000 - i   ,							-1000 - i   ,							-1      )      ) for i in range(1000      )]


_UpperCAmelCase		:   Union[str, Any]										=       generate_large_matrix()
_UpperCAmelCase		:   Tuple										=       (
    [[4, 3, 2, -1], [3, 2, 1, -1], [1, 1, -1, -2], [-1, -1, -2, -3]],
    [[3, 2], [1, 0]],
    [[7, 7, 6]],
    [[7, 7, 6], [-1, -2, -3]],
    grid,
)





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							assert all(row == sorted(UpperCamelCase__   ,							reverse=UpperCamelCase__      ) for row in grid      )
							assert all(list(UpperCamelCase__      ) == sorted(UpperCamelCase__   ,							reverse=UpperCamelCase__      ) for col in zip(*UpperCamelCase__      )      )





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    0
							snake_case_							=    len(UpperCamelCase__      ) - 1

							# Edge cases such as no values or all numbers are negative.
							if not array or array[0] < 0:
														return 0

							while right + 1 > left:
														snake_case_							=    (left + right) // 2
														snake_case_							=    array[mid]

														# Num must be negative and the index must be greater than or equal to 0.
														if num < 0 and array[mid - 1] >= 0:
																					return mid

														if num >= 0:
																					snake_case_							=    mid + 1
														else:
																					snake_case_							=    mid - 1
    # No negative numbers so return the last index of the array + 1 which is the length.
							return len(UpperCamelCase__      )





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    0
							snake_case_							=    len(grid[0]      )

							for i in range(len(UpperCamelCase__      )      ):
														snake_case_							=    find_negative_index(grid[i][:bound]      )
														total += bound
							return (len(UpperCamelCase__      ) * len(grid[0]      )) - total





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							return len([number for row in grid for number in row if number < 0]      )





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    0
							for row in grid:
														for i, number in enumerate(UpperCamelCase__      ):
																					if number < 0:
																												total += len(UpperCamelCase__      ) - i
																												break
							return total





def 					__lowerCamelCase	( ):







							'''simple docstring'''






							from timeit import timeit

							print('Running benchmarks'      )
							snake_case_							=    (
							    'from __main__ import count_negatives_binary_search, '
							    'count_negatives_brute_force, count_negatives_brute_force_with_break, grid'
							)
							for func in (
							    "count_negatives_binary_search",  # took 0.7727 seconds
							    "count_negatives_brute_force_with_break",  # took 4.6505 seconds
							    "count_negatives_brute_force",  # took 12.8160 seconds
							):
														snake_case_							=    timeit(F'''{func}(grid=grid)'''   ,							setup=UpperCamelCase__   ,							number=500      )
														print(F'''{func}() took {time:0.4f} seconds'''      )


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					benchmark()




