





from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean





def      lowerCAmelCase_ (       __a		)     ->		Union[str, Any]:







  """simple docstring"""





  def is_in_circle(__a	,    __a		) -> bool:
    lowerCamelCase__:      Tuple			  =sqrt((x**2) + (y**2)		)
    # Our circle has a radius of 1, so a distance
    # greater than 1 would land outside the circle.
    return distance_from_centre <= 1

  # The proportion of guesses that landed in the circle
  lowerCamelCase__:      List[Any]			  =mean(
      int(is_in_circle(uniform(-1.0	,    1.0		)	,    uniform(-1.0	,    1.0		)		)		)
      for _ in range(__a		)		)
  # The ratio of the area for circle to square is pi/4.
  lowerCamelCase__:      Dict			  =proportion * 4
  print(F"""The estimated value of pi is {pi_estimate}"""		)
  print(F"""The numpy value of pi is {pi}"""		)
  print(F"""The total error is {abs(pi - pi_estimate		)}"""		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a = 0.0	,    __a = 1.0	,    )     ->		float:







  """simple docstring"""





  return mean(
      function_to_integrate(uniform(__a	,    __a		)		) for _ in range(__a		)		) * (max_value - min_value)





def      lowerCAmelCase_ (       __a	,    __a = 0.0	,    __a = 1.0		)     ->		None:







  """simple docstring"""





  def identity_function(__a		) -> float:
    return x

  lowerCamelCase__:      Dict			  =area_under_curve_estimator(
      __a	,    __a	,    __a	,    __a		)
  lowerCamelCase__:      Union[str, Any]			  =(max_value * max_value - min_value * min_value) / 2

  print("******************"		)
  print(F"""Estimating area under y=x where x varies from {min_value} to {max_value}"""		)
  print(F"""Estimated value is {estimated_value}"""		)
  print(F"""Expected value is {expected_value}"""		)
  print(F"""Total error is {abs(estimated_value - expected_value		)}"""		)
  print("******************"		)





def      lowerCAmelCase_ (       __a		)     ->		None:







  """simple docstring"""





  def function_to_integrate(__a		) -> float:
    return sqrt(4.0 - x * x		)

  lowerCamelCase__:      List[Any]			  =area_under_curve_estimator(
      __a	,    __a	,    0.0	,    2.0		)

  print("******************"		)
  print("Estimating pi using area_under_curve_estimator"		)
  print(F"""Estimated value is {estimated_value}"""		)
  print(F"""Expected value is {pi}"""		)
  print(F"""Total error is {abs(estimated_value - pi		)}"""		)
  print("******************"		)


if __name__ == "__main__":
     import doctest

     doctest.testmod()








import itertools
import math





def      lowerCAmelCase_ (       __a		)     ->		bool:







  """simple docstring"""





  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5	,    int(math.sqrt(__a		) + 1		)	,    6		):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True





def      lowerCAmelCase_ (       )     ->		str:







  """simple docstring"""





  lowerCamelCase__:      Optional[int]			  =2
  while True:
    if is_prime(__a		):
      yield num
    num += 1





def      lowerCAmelCase_ (       __a = 10001		)     ->		int:







  """simple docstring"""





  return next(itertools.islice(prime_generator()	,    nth - 1	,    __a		)		)


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

