


'''simple docstring'''


from __future__ import annotations

from math import ceil, floor, sqrt




def      lowerCamelCase					(      lowerCamelCase  :      int = 200_0000):
      A_						:  list[int]          =				[0]
      A_						:  int

      for idx in range(1							,						ceil(sqrt(target * 2) * 1.1)):
            triangle_numbers.append(triangle_numbers[-1] + idx)

      # we want this to be as close as possible to target
      A_						:  int          =				0
      # the area corresponding to the grid that gives the product closest to target
      A_						:  int          =				0
      # an estimate of b, using the quadratic formula
      A_						:  float
      # the largest integer less than b_estimate
      A_						:  int
      # the largest integer less than b_estimate
      A_						:  int
      # the triangle number corresponding to b_floor
      A_						:  int
      # the triangle number corresponding to b_ceil
      A_						:  int

      for idx_a, triangle_a in enumerate(triangle_numbers[1:]							,						1):
            A_						:  Tuple          =				(-1 + sqrt(1 + 8 * target / triangle_a)) / 2
            A_						:  Tuple          =				floor(lowerCamelCase)
            A_						:  Optional[int]          =				ceil(lowerCamelCase)
            A_						:  Tuple          =				triangle_numbers[b_floor]
            A_						:  Optional[Any]          =				triangle_numbers[b_ceil]

            if abs(target - triangle_b_first_guess * triangle_a) < abs(
                target - best_product):
                  A_						:  str          =				triangle_b_first_guess * triangle_a
                  A_						:  List[Any]          =				idx_a * b_floor

            if abs(target - triangle_b_second_guess * triangle_a) < abs(
                target - best_product):
                  A_						:  Optional[int]          =				triangle_b_second_guess * triangle_a
                  A_						:  int          =				idx_a * b_ceil

      return area


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




'''simple docstring'''


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




def      lowerCamelCase					(      lowerCamelCase  :      str							,						lowerCamelCase  :      complex							,						lowerCamelCase  :      str = "x"							,						lowerCamelCase  :      float = 10**-10							,						lowerCamelCase  :      int = 1							,						):
      A_						:  int          =				symbols(lowerCamelCase)
      A_						:  List[Any]          =				lambdify(lowerCamelCase							,						lowerCamelCase)
      A_						:  List[str]          =				lambdify(lowerCamelCase							,						diff(lowerCamelCase							,						lowerCamelCase))

      A_						:  str          =				starting_point

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

            A_						:  Union[str, Any]          =				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)}""")
