


"""simple docstring"""
from __future__ import annotations







def     _a					(   _SCREAMING_SNAKE_CASE					,			_SCREAMING_SNAKE_CASE					)     ->					list:
							snake_case_     							=		[None] * (len(_snake_case					) + len(_snake_case					))

							snake_case_     							=		0  # pointer to current index for left Half
							snake_case_     							=		0  # pointer to current index for the right Half
							snake_case_     							=		0  # pointer to current index for the sorted array Half

							while pointera < len(_snake_case					) and pointera < len(_snake_case					):
														if left_half[pointera] < right_half[pointera]:
																					snake_case_     							=		left_half[pointera]
																					pointera += 1
																					index += 1
														else:
																					snake_case_     							=		right_half[pointera]
																					pointera += 1
																					index += 1
							while pointera < len(_snake_case					):
														snake_case_     							=		left_half[pointera]
														pointera += 1
														index += 1

							while pointera < len(_snake_case					):
														snake_case_     							=		right_half[pointera]
														pointera += 1
														index += 1

							return sorted_array







def     _a					(   _SCREAMING_SNAKE_CASE					)     ->					list:





							if len(_snake_case					) <= 1:
														return array
							# the actual formula to calculate the middle element = left + (right - left) // 2
							# this avoids integer overflow in case of large N
							snake_case_     							=		0 + (len(_snake_case					) - 0) // 2

							# Split the array into halves till the array length becomes equal to One
							# merge the arrays of single length returned by mergeSort function and
							# pass them into the merge arrays function which merges the array
							snake_case_     							=		array[:middle]
							snake_case_     							=		array[middle:]

							return merge(merge_sort(_snake_case					)					,			merge_sort(_snake_case					)					)


if __name__ == "__main__":
				import doctest

				doctest.testmod()







"""simple docstring"""
SCREAMING_SNAKE_CASE					:       List[Any]							   =							1000






def 						lowercase				(	_snake_case		: float	)	->float:
	"""simple docstring"""





	return x * x






def 						lowercase				(	_snake_case		: Union[str, Any]	,  _snake_case		: float	,  _snake_case		: float	,  _snake_case		: int = 4	)	->float:
	"""simple docstring"""





	assert callable(
	    _snake_case	), f"""the function(object) passed should be callable your input : {function}"""
	assert isinstance(_snake_case	,  (float, int)	), f"""a should be float or integer your input : {a}"""
	assert isinstance(function(_snake_case	)	,  (float, int)	), (
	    "the function should return integer or float return type of your function, "
	    f"""{type(_snake_case	)}"""
	)
	assert isinstance(_snake_case	,  (float, int)	), f"""b should be float or integer your input : {b}"""
	assert (
	    isinstance(_snake_case	,  _snake_case	) and precision > 0
	), f"""precision should be positive integer your input : {precision}"""

	# just applying the formula of simpson for approximate integration written in
	# mentioned article in first comment of this file and above this function

	__snake_case      :		Tuple         =							(b - a) / N_STEPS
	__snake_case      :		Union[str, Any]         =							function(_snake_case	) + function(_snake_case	)

	for i in range(1	,  _snake_case	):
		__snake_case      :		List[str]         =							a + h * i
		result += function(_snake_case	) * (4 if i % 2 else 2)

	result *= h / 3
	return round(_snake_case	,  _snake_case	)


if __name__ == "__main__":
			import doctest

			doctest.testmod()





