code: import functools
  from math import sqrt
  from time import time
  def 		snake_case__  (		lowercase       ,							*lowercase       ,							**lowercase				):
     lowerCAmelCase_:       List[str]         =  time()
     lowerCAmelCase_:       Any         =  func(*lowercase       ,							**lowercase				)
     lowerCAmelCase_:       Dict         =  time()
     if int(end - start				) > 0:
        print(F'''{func.__name__} runtime: {(end - start):0.4f} s'''				)
     else:
        print(F'''{func.__name__} runtime: {(end - start) * 1000:0.4f} ms'''				)
     return output
  def 		snake_case__  (		lowercase				):
     if n < 0:
        raise Exception("n is negative"				)
     if n == 0:
        return [0]
     lowerCAmelCase_:       Tuple         =  [0, 1]
     for _ in range(n - 1				):
        fib.append(fib[-1] + fib[-2]				)
     return fib
  def 		snake_case__  (		lowercase				):
  
     def fib_recursive_term(lowercase				) -> int:
        if i < 0:
           raise Exception("n is negative"				)
        if i < 2:
           return i
        return fib_recursive_term(i - 1				) + fib_recursive_term(i - 2				)
  
     if n < 0:
        raise Exception("n is negative"				)
     return [fib_recursive_term(lowercase				) for i in range(n + 1				)]
  def 		snake_case__  (		lowercase				):
  
     @functools.cache
     def fib_recursive_term(lowercase				) -> int:
        if i < 0:
           raise Exception("n is negative"				)
        if i < 2:
           return i
        return fib_recursive_term(i - 1				) + fib_recursive_term(i - 2				)
  
     if n < 0:
        raise Exception("n is negative"				)
     return [fib_recursive_term(lowercase				) for i in range(n + 1				)]
  def 		snake_case__  (		lowercase				):
     if n < 0:
        raise Exception("n is negative"				)
     # Cache must be outside recursuive function
     # other it will reset every time it calls itself.
     lowerCAmelCase_:       dict[int, int]         =  {0: 0, 1: 1, 2: 1}  # Prefilled cache
  
     def rec_fn_memoized(lowercase				) -> int:
        if num in cache:
           return cache[num]
  
        lowerCAmelCase_:       Dict         =  rec_fn_memoized(num - 1				) + rec_fn_memoized(num - 2				)
        lowerCAmelCase_:       Optional[int]         =  value
        return value
  
     return [rec_fn_memoized(lowercase				) for i in range(n + 1				)]
  def 		snake_case__  (		lowercase				):
     if n < 0:
        raise Exception("n is negative"				)
     if n >= 1475:
        raise Exception("n is too large"				)
     lowerCAmelCase_:       Any         =  sqrt(5				)
     lowerCAmelCase_:       List[str]         =  (1 + sqrt_a) / 2
     return [round(phi**i / sqrt_a				) for i in range(n + 1				)]
  
  
  if __name__ == "__main__":
        a     :			Dict					    =						3_0
        time_func(fib_iterative, num)
        time_func(fib_recursive, num)  # Around 3s runtime
        time_func(fib_recursive_cached, num)  # Around 0ms runtime
        time_func(fib_memoization, num)
        time_func(fib_binet, num)
style_context: import random
  
  class 				_lowercase						:
  
  
  
  
  
  
      '''simple docstring'''
  
  
  
  
      def __init__(						self					,	lowerCamelCase__					,	lowerCamelCase__   ):
         lowerCAmelCase_:       int         =  x
         lowerCAmelCase_:       List[str]         =  y
  
  
  
  
      def     _a						(						self   ):
         return (self.x**2 + self.y**2) <= 1
  
  
  
  
      @classmethod
      def     _a						(						cls   ):
         return cls(x=random.random()					,	y=random.random()   )
  
  
  
  
  
  def 		snake_case__  (		lowercase				):
     if number_of_simulations < 1:
        raise ValueError("At least one simulation is necessary to estimate PI."				)
  
     lowerCAmelCase_:       List[Any]         =  0
     for _ in range(lowercase				):
        lowerCAmelCase_:       Tuple         =  Point.random_unit_square()
  
        if random_point.is_in_unit_circle():
           number_in_unit_circle += 1
  
     return 4 * number_in_unit_circle / number_of_simulations
  
  
  if __name__ == "__main__":
   # import doctest
  
   # doctest.testmod()
   from math import pi
  
   a     :			Union[str, Any]					    =						"""Please enter the desired number of Monte Carlo simulations: """
   a     :			int					    =						estimate_pi(int(input(prompt).strip()))
   print(f'''An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}''')