




'''simple docstring'''





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


def 					UpperCamelCase_(     snake_case							:	str							,    snake_case							:	complex							,    snake_case							:	str = "x"							,    snake_case							:	float = 1_0**-1_0							,    snake_case							:	int = 1							,    ):

							'''simple docstring'''






							snake_case_      =				symbols(snake_case					)
							snake_case_      =				lambdify(snake_case							,    snake_case					)
							snake_case_      =				lambdify(snake_case							,    diff(snake_case							,    snake_case					)					)

							snake_case_      =				starting_point

							while True:
														if diff_function(snake_case					) != 0:
																					snake_case_      =				prev_guess - multiplicity * func(snake_case					) / diff_function(
																					    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

														snake_case_      =				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=0.0_0_5)}",
		)

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










'''simple docstring'''





def 					UpperCamelCase_(     snake_case							:	int = 1_0_0_0_0_0_0					):

							'''simple docstring'''






							snake_case_      =				set(range(3							,    snake_case							,    2					)					)
							primes.add(2					)
							for p in range(3							,    snake_case							,    2					):
														if p not in primes:
																					continue
														primes.difference_update(set(range(p * p							,    snake_case							,    snake_case					)					)					)

							snake_case_      =				[float(snake_case					) for n in range(limit + 1					)]

							for p in primes:
														for n in range(snake_case							,    limit + 1							,    snake_case					):
																					phi[n] *= 1 - 1 / p

							return int(sum(phi[2:]					)					)


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




