
def 		_UpperCAmelCase (						UpperCamelCase:  int      ,				UpperCamelCase:  bool = False	):






	"""simple docstring"""


	if n == 2:
		return True
	if not n % 2 or n < 2:
		return False
	if n > 5 and n % 1_0 not in (1, 3, 7, 9):  # can quickly check last digit
		return False
	if n > 3_3_1_7_0_4_4_0_6_4_6_7_9_8_8_7_3_8_5_9_6_1_9_8_1 and not allow_probable:
		raise ValueError(
		    "Warning: upper bound of deterministic test is exceeded. "
		    "Pass allow_probable=True to allow probabilistic test. "
		    "A return value of True indicates a probable prime."	)
	# array bounds provided by analysis
	__lowerCAmelCase    =				[
	    2_0_4_7,
	    1_3_7_3_6_5_3,
	    2_5_3_2_6_0_0_1,
	    3_2_1_5_0_3_1_7_5_1,
	    2_1_5_2_3_0_2_8_9_8_7_4_7,
	    3_4_7_4_7_4_9_6_6_0_3_8_3,
	    3_4_1_5_5_0_0_7_1_7_2_8_3_2_1,
	    1,
	    3_8_2_5_1_2_3_0_5_6_5_4_6_4_1_3_0_5_1,
	    1,
	    1,
	    3_1_8_6_6_5_8_5_7_8_3_4_0_3_1_1_5_1_1_6_7_4_6_1,
	    3_3_1_7_0_4_4_0_6_4_6_7_9_8_8_7_3_8_5_9_6_1_9_8_1,
	]

	__lowerCAmelCase    =				[2, 3, 5, 7, 1_1, 1_3, 1_7, 1_9, 2_3, 2_9, 3_1, 3_7, 4_1]
	for idx, _p in enumerate(_lowercase      ,				1	):
		if n < _p:
			# then we have our last prime to check
			__lowerCAmelCase    =				primes[:idx]
			break
	__lowerCAmelCase    =				n - 1, 0
	# break up n -1 into a power of 2 (s) and
	# remaining odd component
	# essentially, solve for d * 2 ** s == n - 1
	while d % 2 == 0:
		d //= 2
		s += 1
	for prime in plist:
		__lowerCAmelCase    =				False
		for r in range(_lowercase	):
			__lowerCAmelCase    =				pow(_lowercase      ,				d * 2**r      ,				_lowercase	)
			# see article for analysis explanation for m
			if (r == 0 and m == 1) or ((m + 1) % n == 0):
				__lowerCAmelCase    =				True
				# this loop will not determine compositeness
				break
		if pr:
			continue
		# if pr is False, then the above loop never evaluated to true,
		# and the n MUST be composite
		return False
	return True






def 		_UpperCAmelCase (						):






	"""simple docstring"""


	assert not miller_rabin(5_6_1	)
	assert miller_rabin(5_6_3	)
	# 2047

	assert not miller_rabin(8_3_8_2_0_1	)
	assert miller_rabin(8_3_8_2_0_7	)
	# 1_373_653

	assert not miller_rabin(1_7_3_1_6_0_0_1	)
	assert miller_rabin(1_7_3_1_6_0_1_7	)
	# 25_326_001

	assert not miller_rabin(3_0_7_8_3_8_6_6_4_1	)
	assert miller_rabin(3_0_7_8_3_8_6_6_5_3	)
	# 3_215_031_751

	assert not miller_rabin(1_7_1_3_0_4_5_5_7_4_8_0_1	)
	assert miller_rabin(1_7_1_3_0_4_5_5_7_4_8_1_9	)
	# 2_152_302_898_747

	assert not miller_rabin(2_7_7_9_7_9_9_7_2_8_3_0_7	)
	assert miller_rabin(2_7_7_9_7_9_9_7_2_8_3_2_7	)
	# 3_474_749_660_383

	assert not miller_rabin(1_1_3_8_5_0_0_2_3_9_0_9_4_4_1	)
	assert miller_rabin(1_1_3_8_5_0_0_2_3_9_0_9_5_2_7	)
	# 341_550_071_728_321

	assert not miller_rabin(1_2_7_5_0_4_1_0_1_8_8_4_8_8_0_4_3_5_1	)
	assert miller_rabin(1_2_7_5_0_4_1_0_1_8_8_4_8_8_0_4_3_9_1	)
	# 3_825_123_056_546_413_051

	assert not miller_rabin(7_9_6_6_6_4_6_4_4_5_8_5_0_7_7_8_7_7_9_1_8_6_7	)
	assert miller_rabin(7_9_6_6_6_4_6_4_4_5_8_5_0_7_7_8_7_7_9_1_9_5_1	)
	# 318_665_857_834_031_151_167_461

	assert not miller_rabin(5_5_2_8_4_0_6_7_7_4_4_6_6_4_7_8_9_7_6_6_0_3_3_3	)
	assert miller_rabin(5_5_2_8_4_0_6_7_7_4_4_6_6_4_7_8_9_7_6_6_0_3_5_9	)
	# 3_317_044_064_679_887_385_961_981
	# upper limit for probabilistic test


if __name__ == "__main__":
					test_miller_rabin()













"""simple docstring"""




from numpy import exp, pi, sqrt






def  _SCREAMING_SNAKE_CASE						(      _lowercase		:       Tuple       ,			_lowercase		:       float = 0.0       ,			_lowercase		:       float = 1.0   )					->int:






	'''simple docstring'''






	return 1 / sqrt(2 * pi * sigma**2   ) * exp(-((x - mu) ** 2) / (2 * sigma**2)   )


if __name__ == "__main__":
		import doctest

		doctest.testmod()


