



import random

from .binary_exp_mod import bin_exp_mod





def 							A			(  _UpperCAmelCase       :     Any							,     _UpperCAmelCase       :     Dict=1_000					)       ->				Any:







			'''simple docstring'''

			if n < 2:
						return False

			if n % 2 == 0:
						return n == 2

			# this means n is odd
			_UpperCAmelCase	 =     n - 1
			_UpperCAmelCase	 =     0
			while d % 2 == 0:
						d /= 2
						exp += 1

			# n - 1=d*(2**exp)
			_UpperCAmelCase	 =     0
			while count < prec:
						_UpperCAmelCase	 =     random.randint(2							,     n - 1					)
						_UpperCAmelCase	 =     bin_exp_mod(SCREAMING_SNAKE_CASE_							,     SCREAMING_SNAKE_CASE_							,     SCREAMING_SNAKE_CASE_					)
						if b != 1:
									_UpperCAmelCase	 =     True
									for _ in range(SCREAMING_SNAKE_CASE_					):
												if b == n - 1:
															_UpperCAmelCase	 =     False
															break
												_UpperCAmelCase	 =     b * b
												b %= n
									if flag:
												return False
									count += 1
			return True


if __name__ == "__main__":
							UpperCAmelCase__     :   List[Any]     = abs(int(input("Enter bound : ").strip()))
							print("Here\'s the list of primes:")
							print(", ".join(str(i) for i in range(n + 1) if is_prime_big(i)))





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





def 							A			(  _UpperCAmelCase       :     str							,     _UpperCAmelCase       :     complex							,     _UpperCAmelCase       :     str = "x"							,     _UpperCAmelCase       :     float = 10**-10							,     _UpperCAmelCase       :     int = 1							,     )       ->				complex:







			'''simple docstring'''
			_UpperCAmelCase	 =     symbols(_UpperCAmelCase					)
			_UpperCAmelCase	 =     lambdify(_UpperCAmelCase							,     _UpperCAmelCase					)
			_UpperCAmelCase	 =     lambdify(_UpperCAmelCase							,     diff(_UpperCAmelCase							,     _UpperCAmelCase					)					)

			_UpperCAmelCase	 =     starting_point

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

						_UpperCAmelCase	 =     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)}""")
