





'''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    ):



    '''simple docstring'''





    def is_in_circle(SCREAMING_SNAKE_CASE_  :					float		,       SCREAMING_SNAKE_CASE_  :					float    ) -> bool:
        _lowerCAmelCase				=      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
    _lowerCAmelCase				=      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.
    _lowerCAmelCase				=      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		,       ):



    '''simple docstring'''





    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    ):



    '''simple docstring'''





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

    _lowerCAmelCase				=      area_under_curve_estimator(
        SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    )
    _lowerCAmelCase				=      (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    ):



    '''simple docstring'''





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

    _lowerCAmelCase				=      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()









'''simple docstring'''


def 	__a(SCREAMING_SNAKE_CASE_  :					str		,       SCREAMING_SNAKE_CASE_  :					bool = False    ):



    '''simple docstring'''





    if not isinstance(SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    ):
        _lowerCAmelCase				=      F'''Expected string as input, found {type(SCREAMING_SNAKE_CASE_    )}'''
        raise ValueError(SCREAMING_SNAKE_CASE_    )
    if not isinstance(SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    ):
        _lowerCAmelCase				=      F'''Expected boolean as use_pascal parameter, found {type(SCREAMING_SNAKE_CASE_    )}'''
        raise ValueError(SCREAMING_SNAKE_CASE_    )

    _lowerCAmelCase				=      input_str.split("_"    )

    _lowerCAmelCase				=      0 if use_pascal else 1

    _lowerCAmelCase				=      words[start_index:]

    _lowerCAmelCase				=      [word[0].upper() + word[1:] for word in words_to_capitalize]

    _lowerCAmelCase				=      "" if use_pascal else words[0]

    return "".join([initial_word, *capitalized_words]    )


if __name__ == "__main__":
       from doctest import testmod

       testmod()


