
'''simple docstring'''



from __future__ import annotations

from math import gcd


def A (__lowerCamelCase :int					,      __lowerCamelCase :int = 2					,      __lowerCamelCase :int = 1					,      __lowerCamelCase :int = 3					,      ):
						# A value less than 2 can cause an infinite loop in the algorithm.
						if num < 2:
												raise ValueError("""The input value cannot be less than 2"""						)

						# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
						# algorithm struggles to find factors that are divisible by two.
						# As a workaround, we specifically check for two and even inputs.
						#   See: https://math.stackexchange.com/a/2856214/165820
						if num > 2 and num % 2 == 0:
												return 2

						# Pollard's Rho algorithm requires a function that returns pseudorandom
						# values between 0 <= X < ``num``.  It doesn't need to be random in the
						# sense that the output value is cryptographically secure or difficult
						# to calculate, it only needs to be random in the sense that all output
						# values should be equally likely to appear.
						# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
						# However, the success of Pollard's algorithm isn't guaranteed and is
						# determined in part by the initial seed and the chosen random function.
						# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
						# where ``C`` is a value that we can modify between each attempt.
						def rand_fn(__lowerCamelCase :int					,      __lowerCamelCase :int					,      __lowerCamelCase :int						) -> int:
												return (pow(__lowerCamelCase					,      2						) + step) % modulus

						for _ in range(__lowerCamelCase						):
												# These track the position within the cycle detection logic.
												_lowerCAmelCase    		=					seed
												_lowerCAmelCase    		=					seed

												while True:
																		# At each iteration, the tortoise moves one step and the hare moves two.
																		_lowerCAmelCase    		=					rand_fn(__lowerCamelCase					,      __lowerCamelCase					,      __lowerCamelCase						)
																		_lowerCAmelCase    		=					rand_fn(__lowerCamelCase					,      __lowerCamelCase					,      __lowerCamelCase						)
																		_lowerCAmelCase    		=					rand_fn(__lowerCamelCase					,      __lowerCamelCase					,      __lowerCamelCase						)

																		# At some point both the tortoise and the hare will enter a cycle whose
																		# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
																		# the tortoise and hare will end up on the same value modulo ``p``.
																		# We can detect when this happens because the position difference between
																		# the tortoise and the hare will share a common divisor with ``num``.
																		_lowerCAmelCase    		=					gcd(hare - tortoise					,      __lowerCamelCase						)

																		if divisor == 1:
																								# No common divisor yet, just keep searching.
																								continue
																		else:
																								# We found a common divisor!
																								if divisor == num:
																														# Unfortunately, the divisor is ``num`` itself and is useless.
																														break
																								else:
																														# The divisor is a nontrivial factor of ``num``!
																														return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
												_lowerCAmelCase    		=					hare

												# The new step value for the random function can just be incremented.
												# At first the results will be similar to what the old function would
												# have produced, but the value will quickly diverge after a bit.
												step += 1

						# We haven't found a divisor within the requested number of attempts.
						# We were unlucky or ``num`` itself is actually prime.
						return None


if __name__ == "__main__":
						import argparse

						_lowercase          =						argparse.ArgumentParser()
						parser.add_argument(
						    """num""",
						    type=int,
						    help="""The value to find a divisor of""",
						)
						parser.add_argument(
						    """--attempts""",
						    type=int,
						    default=3,
						    help="""The number of attempts before giving up""",
						)
						_lowercase          =						parser.parse_args()

						_lowercase          =						pollard_rho(args.num, attempts=args.attempts)
						if divisor is None:
												print(F"""{args.num} is probably prime""")
						else:
												_lowercase          =						args.num // divisor
												print(F"""{args.num} = {divisor} * {quotient}""")


'''simple docstring'''



from .testing import (
    are_the_same_tensors,
    execute_subprocess_async,
    require_bnb,
    require_cpu,
    require_cuda,
    require_huggingface_suite,
    require_mps,
    require_multi_gpu,
    require_multi_xpu,
    require_safetensors,
    require_single_gpu,
    require_single_xpu,
    require_torch_min_version,
    require_tpu,
    require_xpu,
    skip,
    slow,
)
from .training import RegressionDataset, RegressionModel, RegressionModelaXPU


from .scripts import test_script, test_sync, test_ops  # isort: skip
