





'''simple docstring'''


from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403


def 	__a(SCREAMING_SNAKE_CASE_  :					str		,       SCREAMING_SNAKE_CASE_  :					complex		,       SCREAMING_SNAKE_CASE_  :					str = "x"		,       SCREAMING_SNAKE_CASE_  :					float = 10**-10		,       SCREAMING_SNAKE_CASE_  :					int = 1		,       ):



    '''simple docstring'''





    _lowerCAmelCase				=      symbols(SCREAMING_SNAKE_CASE_    )
    _lowerCAmelCase				=      lambdify(SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    )
    _lowerCAmelCase				=      lambdify(SCREAMING_SNAKE_CASE_		,       diff(SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    )    )

    _lowerCAmelCase				=      starting_point

    while True:
        if diff_function(SCREAMING_SNAKE_CASE_    ) != 0:
            _lowerCAmelCase				=      prev_guess - multiplicity * func(SCREAMING_SNAKE_CASE_    ) / diff_function(
                SCREAMING_SNAKE_CASE_    )
        else:
            raise ZeroDivisionError("Could not find root"    ) from None

        # Precision is checked by comparing the difference of consecutive guesses
        if abs(next_guess - prev_guess    ) < precision:
            return next_guess

        _lowerCAmelCase				=      next_guess


# 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
       # Find fourth Root of 5
       print(f'''The root of x**4 - 5 = 0 is {newton_raphson('x**4 -5', 0.4 +5J)}''')

       # Find value of e
       print(
           "The root of log(y) - 1 = 0 is ",
           f'''{newton_raphson('log(y) - 1', 2, variable='y')}''',
       )

       # Exponential Roots
       print(
           "The root of exp(x) - 1 = 0 is",
           f'''{newton_raphson('exp(x) - 1', 10, precision=0.005)}''',
       )

       # Find root of cos(x)
       print(f'''The root of cos(x) = 0 is {newton_raphson('cos(x)', 0)}''')









'''simple docstring'''


from math import isqrt, loga


def 	__a(SCREAMING_SNAKE_CASE_  :					int    ):



    '''simple docstring'''





    _lowerCAmelCase				=      [True] * max_number
    for i in range(2		,       isqrt(max_number - 1    ) + 1    ):
        if is_prime[i]:
            for j in range(i**2		,       SCREAMING_SNAKE_CASE_		,       SCREAMING_SNAKE_CASE_    ):
                _lowerCAmelCase				=      False

    return [i for i in range(2		,       SCREAMING_SNAKE_CASE_    ) if is_prime[i]]


def 	__a(SCREAMING_SNAKE_CASE_  :					int = 800800		,       SCREAMING_SNAKE_CASE_  :					int = 800800    ):



    '''simple docstring'''





    _lowerCAmelCase				=      degree * loga(SCREAMING_SNAKE_CASE_    )
    _lowerCAmelCase				=      int(SCREAMING_SNAKE_CASE_    )
    _lowerCAmelCase				=      calculate_prime_numbers(SCREAMING_SNAKE_CASE_    )

    _lowerCAmelCase				=      0
    _lowerCAmelCase				=      0
    _lowerCAmelCase				=      len(SCREAMING_SNAKE_CASE_    ) - 1
    while left < right:
        while (
            prime_numbers[right] * loga(prime_numbers[left]    )
            + prime_numbers[left] * loga(prime_numbers[right]    )
            > upper_bound
        ):
            right -= 1
        hybrid_integers_count += right - left
        left += 1

    return hybrid_integers_count


if __name__ == "__main__":
       print(f'''{solution() = }''')


