
'''simple docstring'''







def 			_a		(	_SCREAMING_SNAKE_CASE		:   list[int]	,			_SCREAMING_SNAKE_CASE		:   list[int]  ):

						# Check if the input is valid
						if not len(_SCREAMING_SNAKE_CASE  ) == len(_SCREAMING_SNAKE_CASE  ) == 3:
												raise ValueError("Please enter a valid equation."  )
						if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
												raise ValueError("Both a & b of two equations can't be zero."  )

						# Extract the coefficients
						_SCREAMING_SNAKE_CASE,							_SCREAMING_SNAKE_CASE,							_SCREAMING_SNAKE_CASE				= equationa
						_SCREAMING_SNAKE_CASE,							_SCREAMING_SNAKE_CASE,							_SCREAMING_SNAKE_CASE				= equationa

						# Calculate the determinants of the matrices
						_SCREAMING_SNAKE_CASE				= aa * ba - aa * ba
						_SCREAMING_SNAKE_CASE				= ca * ba - ca * ba
						_SCREAMING_SNAKE_CASE				= aa * ca - aa * ca

						# Check if the system of linear equations has a solution (using Cramer's rule)
						if determinant == 0:
												if determinant_x == determinant_y == 0:
																		raise ValueError("Infinite solutions. (Consistent system)"  )
												else:
																		raise ValueError("No solution. (Inconsistent system)"  )
						else:
												if determinant_x == determinant_y == 0:
																		# Trivial solution (Inconsistent system)
																		return (0.0, 0.0)
												else:
																		_SCREAMING_SNAKE_CASE				= determinant_x / determinant
																		_SCREAMING_SNAKE_CASE				= determinant_y / determinant
																		# Non-Trivial Solution (Consistent system)
																		return (x, y)

'''simple docstring'''







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



def 			_a		(	_SCREAMING_SNAKE_CASE		:   int  ):

						# A local function to see if a dot lands in the circle.
						def is_in_circle(_SCREAMING_SNAKE_CASE		:   float	,			_SCREAMING_SNAKE_CASE		:   float  ) -> bool:
												_SCREAMING_SNAKE_CASE				= 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
						_SCREAMING_SNAKE_CASE				= mean(
						    int(is_in_circle(uniform(-1.0	,			1.0  )	,			uniform(-1.0	,			1.0  )  )  )
						    for _ in range(_SCREAMING_SNAKE_CASE  )  )
						# The ratio of the area for circle to square is pi/4.
						_SCREAMING_SNAKE_CASE				= 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 			_a		(	_SCREAMING_SNAKE_CASE		:   int	,			_SCREAMING_SNAKE_CASE		:   Callable[[float], float]	,			_SCREAMING_SNAKE_CASE		:   float = 0.0	,			_SCREAMING_SNAKE_CASE		:   float = 1.0	,			):
						return mean(
						    function_to_integrate(uniform(_SCREAMING_SNAKE_CASE	,			_SCREAMING_SNAKE_CASE  )  ) for _ in range(_SCREAMING_SNAKE_CASE  )  ) * (max_value - min_value)



def 			_a		(	_SCREAMING_SNAKE_CASE		:   int	,			_SCREAMING_SNAKE_CASE		:   float = 0.0	,			_SCREAMING_SNAKE_CASE		:   float = 1.0  ):

						def identity_function(_SCREAMING_SNAKE_CASE		:   float  ) -> float:
												return x

						_SCREAMING_SNAKE_CASE				= area_under_curve_estimator(
						    _SCREAMING_SNAKE_CASE	,			_SCREAMING_SNAKE_CASE	,			_SCREAMING_SNAKE_CASE	,			_SCREAMING_SNAKE_CASE  )
						_SCREAMING_SNAKE_CASE				= (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 			_a		(	_SCREAMING_SNAKE_CASE		:   int  ):

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

						_SCREAMING_SNAKE_CASE				= area_under_curve_estimator(
						    _SCREAMING_SNAKE_CASE	,			_SCREAMING_SNAKE_CASE	,			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()