




def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			str			,				__lowercase   :			Optional[Any]			,				__lowercase   :			Optional[Any]			,				__lowercase   :			Optional[Any]    )						->      Optional[Any]:





    """simple docstring"""

    global f  # a global dp table for knapsack
    if f[i][j] < 0:
        if j < wt[i - 1]:
            __A         =	mf_knapsack(i - 1			,				__lowercase			,				__lowercase			,				__lowercase    )
        else:
            __A         =	max(
                mf_knapsack(i - 1			,				__lowercase			,				__lowercase			,				__lowercase    )			,				mf_knapsack(i - 1			,				__lowercase			,				__lowercase			,				j - wt[i - 1]    ) + val[i - 1]			,				)
        __A         =	val
    return f[i][j]



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Optional[int]			,				__lowercase   :			List[str]			,				__lowercase   :			Tuple			,				__lowercase   :			Dict    )						->      Tuple:





    """simple docstring"""

    __A         =	[[0] * (w + 1) for _ in range(n + 1    )]

    for i in range(1			,				n + 1    ):
        for w_ in range(1			,				w + 1    ):
            if wt[i - 1] <= w_:
                __A         =	max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]			,				dp[i - 1][w_]    )
            else:
                __A         =	dp[i - 1][w_]

    return dp[n][w_], dp



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			int			,				__lowercase   :			list			,				__lowercase   :			list    )						->      Optional[int]:





    """simple docstring"""

    if not (isinstance(__lowercase			,				(list, tuple)    ) and isinstance(__lowercase			,				(list, tuple)    )):
        raise ValueError(
            """Both the weights and values vectors must be either lists or tuples"""    )

    __A         =	len(__lowercase    )
    if num_items != len(__lowercase    ):
        __A         =	(
            """The number of weights must be the same as the number of values.\n"""
            f"But got {num_items} weights and {len(__lowercase    )} values"
        )
        raise ValueError(__lowercase    )
    for i in range(__lowercase    ):
        if not isinstance(wt[i]			,				__lowercase    ):
            __A         =	(
                """All weights must be integers but got weight of """
                f"type {type(wt[i]    )} at index {i}"
            )
            raise TypeError(__lowercase    )

    __A    ,	__A         =	knapsack(__lowercase			,				__lowercase			,				__lowercase			,				__lowercase    )
    __A         =	set()
    _construct_solution(__lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				__lowercase    )

    return optimal_val, example_optional_set



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			list			,				__lowercase   :			list			,				__lowercase   :			int			,				__lowercase   :			int			,				__lowercase   :			set    )						->      List[Any]:





    """simple docstring"""

    if i > 0 and j > 0:
        if dp[i - 1][j] == dp[i][j]:
            _construct_solution(__lowercase			,				__lowercase			,				i - 1			,				__lowercase			,				__lowercase    )
        else:
            optimal_set.add(__lowercase    )
            _construct_solution(__lowercase			,				__lowercase			,				i - 1			,				j - wt[i - 1]			,				__lowercase    )


if __name__ == "__main__":
  __a      :       Dict				  =   [3, 2, 4, 4]
  __a      :       Optional[int]				  =   [4, 3, 2, 3]
  __a      :       Optional[int]				  =   4
  __a      :       List[Any]				  =   6
  __a      :       Tuple				  =   [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
  __a      :       Optional[Any]				  =   knapsack(w, wt, val, n)
  print(optimal_solution)
  print(mf_knapsack(n, wt, val, w))  # switched the n and w

  # testing the dynamic programming problem with example
  # the optimal subset for the above example are items 3 and 4
  __a      :       Union[str, Any]				  =   knapsack_with_example_solution(w, wt, val)
  assert optimal_solution == 8
  assert optimal_subset == {3, 4}
  print("optimal_value = ", optimal_solution)
  print("An optimal subset corresponding to the optimal value", optimal_subset)












from __future__ import annotations

from decimal import Decimal
from math import *  # noqa: F403

from sympy import diff



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			str			,				__lowercase   :			float | Decimal			,				__lowercase   :			float = 1_0**-1_0    )						->      float:





    """simple docstring"""

    __A         =	a
    while True:
        __A         =	Decimal(__lowercase    ) - (
            Decimal(eval(__lowercase    )    ) / Decimal(eval(str(diff(__lowercase    )    )    )    )  # noqa: S307
        )
        # This number dictates the accuracy of the answer
        if abs(eval(__lowercase    )    ) < precision:  # noqa: S307
            return float(__lowercase    )


# Let's Execute
if __name__ == "__main__":
  # Find root of trigonometric function
  # Find value of pi
  print(f"""The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}""")
  # Find root of polynomial
  print(f"""The root of x**2 - 5*x + 2 = 0 is {newton_raphson('x**2 - 5*x + 2', 0.4)}""")
  # Find Square Root of 5
  print(f"""The root of log(x) - 1 = 0 is {newton_raphson('log(x) - 1', 2)}""")
  # Exponential Roots
  print(f"""The root of exp(x) - 1 = 0 is {newton_raphson('exp(x) - 1', 0)}""")




