



"""simple docstring"""
def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		list							):
    _enforce_args(UpperCamelCase__     ,       UpperCamelCase__							)
    if n == 0:
        return 0
    _UpperCAmelCase							:  List[str]											=    float('''-inf'''							)
    for i in range(1     ,       n + 1							):
        _UpperCAmelCase							:  List[Any]											=    max(
            UpperCamelCase__     ,       prices[i - 1] + naive_cut_rod_recursive(n - i     ,       UpperCamelCase__							)							)

    return max_revue





def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		list							):
    _enforce_args(UpperCamelCase__     ,       UpperCamelCase__							)
    _UpperCAmelCase							:  int											=    [float('''-inf'''							) for _ in range(n + 1							)]
    return _top_down_cut_rod_recursive(UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)





def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		list     ,       UpperCamelCase__		:		list							):
    if max_rev[n] >= 0:
        return max_rev[n]
    elif n == 0:
        return 0
    else:
        _UpperCAmelCase							:  str											=    float('''-inf'''							)
        for i in range(1     ,       n + 1							):
            _UpperCAmelCase							:  List[str]											=    max(
                UpperCamelCase__     ,       prices[i - 1] + _top_down_cut_rod_recursive(n - i     ,       UpperCamelCase__     ,       UpperCamelCase__							)     ,       )

        _UpperCAmelCase							:  Union[str, Any]											=    max_revenue

    return max_rev[n]





def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		list							):
    _enforce_args(UpperCamelCase__     ,       UpperCamelCase__							)

    # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
    # length 0.
    _UpperCAmelCase							:  Any											=    [float('''-inf'''							) for _ in range(n + 1							)]
    _UpperCAmelCase							:  int											=    0

    for i in range(1     ,       n + 1							):
        _UpperCAmelCase							:  Dict											=    max_rev[i]
        for j in range(1     ,       i + 1							):
            _UpperCAmelCase							:  str											=    max(UpperCamelCase__     ,       prices[j - 1] + max_rev[i - j]							)

        _UpperCAmelCase							:  Optional[int]											=    max_revenue_i

    return max_rev[n]





def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		list							):
    if n < 0:
        _UpperCAmelCase							:  Optional[int]											=    F'n must be greater than or equal to 0. Got n = {n}'
        raise ValueError(UpperCamelCase__							)

    if n > len(UpperCamelCase__							):
        _UpperCAmelCase							:  int											=    (
            '''Each integral piece of rod must have a corresponding price. '''
            F'Got n = {n} but length of prices = {len(UpperCamelCase__							)}'
        )
        raise ValueError(UpperCamelCase__							)





def  lowerCamelCase_						():
    _UpperCAmelCase							:  Union[str, Any]											=    [6, 10, 12, 15, 20, 23]
    _UpperCAmelCase							:  str											=    len(UpperCamelCase__							)

    # the best revenue comes from cutting the rod into 6 pieces, each
    # of length 1 resulting in a revenue of 6 * 6 = 36.
    _UpperCAmelCase							:  List[str]											=    36

    _UpperCAmelCase							:  Dict											=    top_down_cut_rod(UpperCamelCase__     ,       UpperCamelCase__							)
    _UpperCAmelCase							:  Optional[Any]											=    bottom_up_cut_rod(UpperCamelCase__     ,       UpperCamelCase__							)
    _UpperCAmelCase							:  List[str]											=    naive_cut_rod_recursive(UpperCamelCase__     ,       UpperCamelCase__							)

    assert expected_max_revenue == max_rev_top_down
    assert max_rev_top_down == max_rev_bottom_up
    assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
  main()







"""simple docstring"""
import math

from numpy import inf
from scipy.integrate import quad





def  lowerCamelCase_						(UpperCamelCase__		:		float							):
    if num <= 0:
        raise ValueError('''math domain error'''							)

    return quad(UpperCamelCase__     ,       0     ,       UpperCamelCase__     ,       args=(UpperCamelCase__)							)[0]





def  lowerCamelCase_						(UpperCamelCase__		:		float     ,       UpperCamelCase__		:		float							):
    return math.pow(UpperCamelCase__     ,       z - 1							) * math.exp(-x							)


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

  testmod()


