

"""simple docstring"""






lowerCamelCase__				:  Union[str, Any]													=  10_00
def       UpperCamelCase					(			_lowerCAmelCase     :						float )				->			float:
	return x * x
def       UpperCamelCase					(			_lowerCAmelCase     :						Optional[int],  _lowerCAmelCase     :						float,  _lowerCAmelCase     :						float,  _lowerCAmelCase     :						int = 4 )				->			float:
	assert callable(
	    _lowerCAmelCase ), f'''the function(object) passed should be callable your input : {function}'''
	assert isinstance(_lowerCAmelCase,  (float, int) ), f'''a should be float or integer your input : {a}'''
	assert isinstance(function(_lowerCAmelCase ),  (float, int) ), (
	    "the function should return integer or float return type of your function, "
	    f'''{type(_lowerCAmelCase )}'''
	)
	assert isinstance(_lowerCAmelCase,  (float, int) ), f'''b should be float or integer your input : {b}'''
	assert (
	    isinstance(_lowerCAmelCase,  _lowerCAmelCase ) 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

	_UpperCAmelCase    :							Any									=					(b - a) / N_STEPS
	_UpperCAmelCase    :							List[Any]									=					function(_lowerCAmelCase ) + function(_lowerCAmelCase )

	for i in range(1,  _lowerCAmelCase ):
		_UpperCAmelCase    :							int									=					a + h * i
		result += function(_lowerCAmelCase ) * (4 if i % 2 else 2)

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


if __name__ == "__main__":
						import doctest

						doctest.testmod()



"""simple docstring"""






from __future__ import annotations
def       UpperCamelCase					(			_lowerCAmelCase     :						list,  _lowerCAmelCase     :						list )				->			list:
	_UpperCAmelCase    :							List[Any]									=					[None] * (len(_lowerCAmelCase ) + len(_lowerCAmelCase ))

	_UpperCAmelCase    :							Union[str, Any]									=					0  # pointer to current index for left Half
	_UpperCAmelCase    :							str									=					0  # pointer to current index for the right Half
	_UpperCAmelCase    :							Optional[int]									=					0  # pointer to current index for the sorted array Half

	while pointera < len(_lowerCAmelCase ) and pointera < len(_lowerCAmelCase ):
		if left_half[pointera] < right_half[pointera]:
			_UpperCAmelCase    :							Tuple									=					left_half[pointera]
			pointera += 1
			index += 1
		else:
			_UpperCAmelCase    :							int									=					right_half[pointera]
			pointera += 1
			index += 1
	while pointera < len(_lowerCAmelCase ):
		_UpperCAmelCase    :							Tuple									=					left_half[pointera]
		pointera += 1
		index += 1

	while pointera < len(_lowerCAmelCase ):
		_UpperCAmelCase    :							Union[str, Any]									=					right_half[pointera]
		pointera += 1
		index += 1

	return sorted_array
def       UpperCamelCase					(			_lowerCAmelCase     :						list )				->			list:
	if len(_lowerCAmelCase ) <= 1:
		return array
	# the actual formula to calculate the middle element = left + (right - left) // 2
	# this avoids integer overflow in case of large N
	_UpperCAmelCase    :							Any									=					0 + (len(_lowerCAmelCase ) - 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
	_UpperCAmelCase    :							Optional[Any]									=					array[:middle]
	_UpperCAmelCase    :							Optional[int]									=					array[middle:]

	return merge(merge_sort(_lowerCAmelCase ),  merge_sort(_lowerCAmelCase ) )


if __name__ == "__main__":
						import doctest

						doctest.testmod()





