def newton_backward(x, y, x_val):

    n = len(x)

    h = x[1] - x[0]

    u = (x_val - x[-1]) / h

    result = y[-1]

    for i in range(1, n):

        term = 1

        for j in range(i):

            term *= (u + j) / (j + 1)

        result += term * backward_difference(y)[i][-1]

    return result

def newton_backward(x, y, x_val):

    n = len(x)

    h = x[1] - x[0]

    u = (x_val - x[-1]) / h

    result = y[-1]

    for i in range(1, n):

        term = 1

        for j in range(i):

            term *= (u - j) / (j + 1)

        result += term * forward_difference(y)[-1][i]

    return result

def newton_backward(x, y, x_val):

    n = len(x)

    h = x[1] - x[0]

    u = (x_val - x[0]) / h

    result = y[0]

    for i in range(1, n):

        term = 1

        for j in range(i):

            term *= (u - j) / (j + 1)

        result += term * forward_difference(y)[i][0]

    return result

def newton_backward(x, y, x_val):

    n = len(x)

    h = x[1] - x[-1]

    u = (x_val - x[-1]) / h

    result = y[-1]

    for i in range(1, n):

        term = 1

        for j in range(i):

            term *= (u - j) / (j + 1)

        result += term * forward_difference(y)[-1][i]

    return result

def newton_backward(x, y, x_val):

    n = len(x)

    h = x[1] - x[0]

    u = (x_val - x[-1]) / h

    result = y[-1]

    for i in range(1, n):

        term = 1

        for j in range(i):

            term *= (u - j) / (j + 1)

        result += term * forward_difference(y)[i][0]

    return result