





def 							_lowerCAmelCase (  _a   :					int					,   _a   :					bool = False			)	->			bool:
     if n == 2:
          return True
     if not n % 2 or n < 2:
          return False
     if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
          return False
     if n > 3_31_70_44_06_46_79_88_73_85_96_19_81 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_					:       Optional[int]    =		[
         20_47,
         1_37_36_53,
         25_32_60_01,
         32_15_03_17_51,
         2_15_23_02_89_87_47,
         3_47_47_49_66_03_83,
         3_41_55_00_71_72_83_21,
         1,
         3_82_51_23_05_65_46_41_30_51,
         1,
         1,
         31_86_65_85_78_34_03_11_51_16_74_61,
         3_31_70_44_06_46_79_88_73_85_96_19_81,
     ]

     lowerCAmelCase_					:       Dict    =		[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
     for idx, _p in enumerate(_a					,   1			):
          if n < _p:
               # then we have our last prime to check
               lowerCAmelCase_					:       str    =		primes[:idx]
               break
     lowerCAmelCase_   ,  lowerCAmelCase_					:       Optional[Any]    =		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_					:       Tuple    =		False
          for r in range(_a			):
               lowerCAmelCase_					:       Tuple    =		pow(_a					,   d * 2**r					,   _a			)
               # see article for analysis explanation for m
               if (r == 0 and m == 1) or ((m + 1) % n == 0):
                    lowerCAmelCase_					:       List[Any]    =		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 							_lowerCAmelCase (  )	->			None:
     assert not miller_rabin(5_61			)
     assert miller_rabin(5_63			)
     # 2047

     assert not miller_rabin(83_82_01			)
     assert miller_rabin(83_82_07			)
     # 1_373_653

     assert not miller_rabin(17_31_60_01			)
     assert miller_rabin(17_31_60_17			)
     # 25_326_001

     assert not miller_rabin(30_78_38_66_41			)
     assert miller_rabin(30_78_38_66_53			)
     # 3_215_031_751

     assert not miller_rabin(1_71_30_45_57_48_01			)
     assert miller_rabin(1_71_30_45_57_48_19			)
     # 2_152_302_898_747

     assert not miller_rabin(2_77_97_99_72_83_07			)
     assert miller_rabin(2_77_97_99_72_83_27			)
     # 3_474_749_660_383

     assert not miller_rabin(1_13_85_00_23_90_94_41			)
     assert miller_rabin(1_13_85_00_23_90_95_27			)
     # 341_550_071_728_321

     assert not miller_rabin(1_27_50_41_01_88_48_80_43_51			)
     assert miller_rabin(1_27_50_41_01_88_48_80_43_91			)
     # 3_825_123_056_546_413_051

     assert not miller_rabin(7_96_66_46_44_58_50_77_87_79_18_67			)
     assert miller_rabin(7_96_66_46_44_58_50_77_87_79_19_51			)
     # 318_665_857_834_031_151_167_461

     assert not miller_rabin(55_28_40_67_74_46_64_78_97_66_03_33			)
     assert miller_rabin(55_28_40_67_74_46_64_78_97_66_03_59			)
     # 3_317_044_064_679_887_385_961_981
     # upper limit for probabilistic test


if __name__ == "__main__":
      test_miller_rabin()







from dataclasses import dataclass
from typing import Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput, randn_tensor
from .scheduling_utils import SchedulerMixin





@dataclass
class      lowercase__    (							__A  ):
  __UpperCamelCase	     =  42
  __UpperCamelCase	     =  42
  __UpperCamelCase	     =  None





class      lowercase__    (							__A     ,					__A  ):
  __UpperCamelCase	     =  2







  @register_to_config
  def __init__(	self    ,						_lowercase = 0.02    ,						_lowercase = 100    ,						_lowercase = 1.007    ,						_lowercase = 80    ,						_lowercase = 0.05    ,						_lowercase = 50    ,						):
       # standard deviation of the initial noise distribution
       lowerCAmelCase_					:       int    =		sigma_max

       # setable values
       lowerCAmelCase_					:       int    =		None
       lowerCAmelCase_					:       np.IntTensor    =		None
       lowerCAmelCase_					:       torch.FloatTensor    =		None  # sigma(t_i)







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase = None						):
       return sample







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase = None						):
       lowerCAmelCase_					:       int    =		num_inference_steps
       lowerCAmelCase_					:       int    =		np.arange(0    ,						self.num_inference_steps						)[::-1].copy()
       lowerCAmelCase_					:       List[Any]    =		torch.from_numpy(_lowercase						).to(_lowercase						)
       lowerCAmelCase_					:       List[str]    =		[
           (
               self.config.sigma_max**2
               * (self.config.sigma_min**2 / self.config.sigma_max**2) ** (i / (num_inference_steps - 1))
           )
           for i in self.timesteps
       ]
       lowerCAmelCase_					:       Optional[int]    =		torch.tensor(_lowercase    ,						dtype=torch.floataa    ,						device=_lowercase						)







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase    ,						_lowercase = None						):
       if self.config.s_min <= sigma <= self.config.s_max:
            lowerCAmelCase_					:       List[Any]    =		min(self.config.s_churn / self.num_inference_steps    ,						2**0.5 - 1						)
       else:
            lowerCAmelCase_					:       List[str]    =		0

       # sample eps ~ N(0, S_noise^2 * I)
       lowerCAmelCase_					:       List[Any]    =		self.config.s_noise * randn_tensor(sample.shape    ,						generator=_lowercase						).to(sample.device						)
       lowerCAmelCase_					:       Optional[int]    =		sigma + gamma * sigma
       lowerCAmelCase_					:       int    =		sample + ((sigma_hat**2 - sigma**2) ** 0.5 * eps)

       return sample_hat, sigma_hat







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase = True    ,						):
       lowerCAmelCase_					:       Union[str, Any]    =		sample_hat + sigma_hat * model_output
       lowerCAmelCase_					:       Dict    =		(sample_hat - pred_original_sample) / sigma_hat
       lowerCAmelCase_					:       List[Any]    =		sample_hat + (sigma_prev - sigma_hat) * derivative

       if not return_dict:
            return (sample_prev, derivative)

       return KarrasVeOutput(
           prev_sample=_lowercase    ,						derivative=_lowercase    ,						pred_original_sample=_lowercase						)







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase    ,						_lowercase = True    ,						):
       lowerCAmelCase_					:       str    =		sample_prev + sigma_prev * model_output
       lowerCAmelCase_					:       List[Any]    =		(sample_prev - pred_original_sample) / sigma_prev
       lowerCAmelCase_					:       Tuple    =		sample_hat + (sigma_prev - sigma_hat) * (0.5 * derivative + 0.5 * derivative_corr)

       if not return_dict:
            return (sample_prev, derivative)

       return KarrasVeOutput(
           prev_sample=_lowercase    ,						derivative=_lowercase    ,						pred_original_sample=_lowercase						)







  def 	UpperCAmelCase__		(	self    ,						_lowercase    ,						_lowercase    ,						_lowercase						):
       raise NotImplementedError()
