
def     __lowerCAmelCase			(			a__						,    a__ = 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_3170_4406_4679_8873_8596_1981 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
				__a										=    [
				    2047,
				    137_3653,
				    2532_6001,
				    32_1503_1751,
				    2_1523_0289_8747,
				    3_4747_4966_0383,
				    341_5500_7172_8321,
				    1,
				    382_5123_0565_4641_3051,
				    1,
				    1,
				    3186_6585_7834_0311_5116_7461,
				    3_3170_4406_4679_8873_8596_1981,
				]

				__a										=    [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
												__a										=    primes[:idx]
												break
				__a							,     __a										=    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:
								__a										=    False
								for r in range(a__  ):
												__a										=    pow(a__						,    d * 2**r						,    a__  )
												# see article for analysis explanation for m
												if (r == 0 and m == 1) or ((m + 1) % n == 0):
																__a										=    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(561  )
				assert miller_rabin(563  )
				# 2047

				assert not miller_rabin(83_8201  )
				assert miller_rabin(83_8207  )
				# 1_373_653

				assert not miller_rabin(1731_6001  )
				assert miller_rabin(1731_6017  )
				# 25_326_001

				assert not miller_rabin(30_7838_6641  )
				assert miller_rabin(30_7838_6653  )
				# 3_215_031_751

				assert not miller_rabin(1_7130_4557_4801  )
				assert miller_rabin(1_7130_4557_4819  )
				# 2_152_302_898_747

				assert not miller_rabin(2_7797_9972_8307  )
				assert miller_rabin(2_7797_9972_8327  )
				# 3_474_749_660_383

				assert not miller_rabin(113_8500_2390_9441  )
				assert miller_rabin(113_8500_2390_9527  )
				# 341_550_071_728_321

				assert not miller_rabin(127_5041_0188_4880_4351  )
				assert miller_rabin(127_5041_0188_4880_4391  )
				# 3_825_123_056_546_413_051

				assert not miller_rabin(796_6646_4458_5077_8779_1867  )
				assert miller_rabin(796_6646_4458_5077_8779_1951  )
				# 318_665_857_834_031_151_167_461

				assert not miller_rabin(5528_4067_7446_6478_9766_0333  )
				assert miller_rabin(5528_4067_7446_6478_9766_0359  )
				# 3_317_044_064_679_887_385_961_981
				# upper limit for probabilistic test


if __name__ == "__main__":
					test_miller_rabin()





'''simple docstring'''







from math import acos, sin
from typing import List, Tuple, Union

import numpy as np
import torch
from PIL import Image

from ...models import AutoencoderKL, UNetaDConditionModel
from ...schedulers import DDIMScheduler, DDPMScheduler
from ...utils import randn_tensor
from ..pipeline_utils import AudioPipelineOutput, BaseOutput, DiffusionPipeline, ImagePipelineOutput
from .mel import Mel


class 			_a  (	__a ):
       __a				:     str   				=					["""vqvae"""]
       def __init__(	self						:	str   ,				lowercase						:	AutoencoderKL   ,				lowercase						:	UNetaDConditionModel   ,				lowercase						:	Mel   ,				lowercase						:	Union[DDIMScheduler, DDPMScheduler]   ,				):



           '''simple docstring'''

           super().__init__()
           self.register_modules(unet=lowercase   ,				scheduler=lowercase   ,				mel=lowercase   ,				vqvae=lowercase				)
       def     A      (	self						:	Optional[Any]				):



           '''simple docstring'''

           return 50 if isinstance(self.scheduler   ,				lowercase				) else 1_000
       @torch.no_grad()
       def __call__(	self						:	Optional[Any]   ,				lowercase						:	int = 1   ,				lowercase						:	str = None   ,				lowercase						:	np.ndarray = None   ,				lowercase						:	int = 0   ,				lowercase						:	int = 0   ,				lowercase						:	int = None   ,				lowercase						:	torch.Generator = None   ,				lowercase						:	float = 0   ,				lowercase						:	float = 0   ,				lowercase						:	torch.Generator = None   ,				lowercase						:	float = 0   ,				lowercase						:	torch.Tensor = None   ,				lowercase						:	torch.Tensor = None   ,				lowercase						:	Tuple=True   ,				):



           '''simple docstring'''

           UpperCAmelCase			       =		steps or self.get_default_steps()
           self.scheduler.set_timesteps(lowercase				)
           UpperCAmelCase			       =		step_generator or generator
           # For backwards compatibility
           if type(self.unet.config.sample_size				) == int:
               UpperCAmelCase			       =		(self.unet.config.sample_size, self.unet.config.sample_size)
           if noise is None:
               UpperCAmelCase			       =		randn_tensor(
                   (
                       batch_size,
                       self.unet.config.in_channels,
                       self.unet.config.sample_size[0],
                       self.unet.config.sample_size[1],
                   )   ,				generator=lowercase   ,				device=self.device   ,				)
           UpperCAmelCase			       =		noise
           UpperCAmelCase			       =		None

           if audio_file is not None or raw_audio is not None:
               self.mel.load_audio(lowercase   ,				lowercase				)
               UpperCAmelCase			       =		self.mel.audio_slice_to_image(lowercase				)
               UpperCAmelCase			       =		np.frombuffer(input_image.tobytes()   ,				dtype='''uint8'''				).reshape(
                   (input_image.height, input_image.width)				)
               UpperCAmelCase			       =		(input_image / 255) * 2 - 1
               UpperCAmelCase			       =		torch.tensor(input_image[np.newaxis, :, :]   ,				dtype=torch.float				).to(self.device				)

               if self.vqvae is not None:
                   UpperCAmelCase			       =		self.vqvae.encode(torch.unsqueeze(lowercase   ,				0				)				).latent_dist.sample(
                       generator=lowercase				)[0]
                   UpperCAmelCase			       =		self.vqvae.config.scaling_factor * input_images

               if start_step > 0:
                   UpperCAmelCase			       =		self.scheduler.add_noise(lowercase   ,				lowercase   ,				self.scheduler.timesteps[start_step - 1]				)

               UpperCAmelCase			       =		(
                   self.unet.config.sample_size[1] * self.mel.get_sample_rate() / self.mel.x_res / self.mel.hop_length
               )
               UpperCAmelCase			       =		int(mask_start_secs * pixels_per_second				)
               UpperCAmelCase			       =		int(mask_end_secs * pixels_per_second				)
               UpperCAmelCase			       =		self.scheduler.add_noise(lowercase   ,				lowercase   ,				torch.tensor(self.scheduler.timesteps[start_step:]				)				)

           for step, t in enumerate(self.progress_bar(self.scheduler.timesteps[start_step:]				)				):
               if isinstance(self.unet   ,				lowercase				):
                   UpperCAmelCase			       =		self.unet(lowercase   ,				lowercase   ,				lowercase				)['''sample''']
               else:
                   UpperCAmelCase			       =		self.unet(lowercase   ,				lowercase				)['''sample''']

               if isinstance(self.scheduler   ,				lowercase				):
                   UpperCAmelCase			       =		self.scheduler.step(
                       model_output=lowercase   ,				timestep=lowercase   ,				sample=lowercase   ,				eta=lowercase   ,				generator=lowercase   ,				)['''prev_sample''']
               else:
                   UpperCAmelCase			       =		self.scheduler.step(
                       model_output=lowercase   ,				timestep=lowercase   ,				sample=lowercase   ,				generator=lowercase   ,				)['''prev_sample''']

               if mask is not None:
                   if mask_start > 0:
                       UpperCAmelCase			       =		mask[:, step, :, :mask_start]
                   if mask_end > 0:
                       UpperCAmelCase			       =		mask[:, step, :, -mask_end:]

           if self.vqvae is not None:
               # 0.18215 was scaling factor used in training to ensure unit variance
               UpperCAmelCase			       =		1 / self.vqvae.config.scaling_factor * images
               UpperCAmelCase			       =		self.vqvae.decode(lowercase				)['''sample''']

           UpperCAmelCase			       =		(images / 2 + 0.5).clamp(0   ,				1				)
           UpperCAmelCase			       =		images.cpu().permute(0   ,				2   ,				3   ,				1				).numpy()
           UpperCAmelCase			       =		(images * 255).round().astype('''uint8'''				)
           UpperCAmelCase			       =		list(
               (Image.fromarray(_[:, :, 0]				) for _ in images)
               if images.shape[3] == 1
               else (Image.fromarray(lowercase   ,				mode='''RGB'''				).convert('''L'''				) for _ in images)				)

           UpperCAmelCase			       =		[self.mel.image_to_audio(lowercase				) for _ in images]
           if not return_dict:
               return images, (self.mel.get_sample_rate(), audios)

           return BaseOutput(**AudioPipelineOutput(np.array(lowercase				)[:, np.newaxis, :]				)   ,				**ImagePipelineOutput(lowercase				)				)
       @torch.no_grad()
       def     A      (	self						:	Dict   ,				lowercase						:	List[Image.Image]   ,				lowercase						:	int = 50				):



           '''simple docstring'''

           assert isinstance(self.scheduler   ,				lowercase				)
           self.scheduler.set_timesteps(lowercase				)
           UpperCAmelCase			       =		np.array(
               [np.frombuffer(image.tobytes()   ,				dtype='''uint8'''				).reshape((1, image.height, image.width)				) for image in images]				)
           UpperCAmelCase			       =		(sample / 255) * 2 - 1
           UpperCAmelCase			       =		torch.Tensor(lowercase				).to(self.device				)

           for t in self.progress_bar(torch.flip(self.scheduler.timesteps   ,				(0,)				)				):
               UpperCAmelCase			       =		t - self.scheduler.config.num_train_timesteps // self.scheduler.num_inference_steps
               UpperCAmelCase			       =		self.scheduler.alphas_cumprod[t]
               UpperCAmelCase			       =		(
                   self.scheduler.alphas_cumprod[prev_timestep]
                   if prev_timestep >= 0
                   else self.scheduler.final_alpha_cumprod
               )
               UpperCAmelCase			       =		1 - alpha_prod_t
               UpperCAmelCase			       =		self.unet(lowercase   ,				lowercase				)['''sample''']
               UpperCAmelCase			       =		(1 - alpha_prod_t_prev) ** 0.5 * model_output
               UpperCAmelCase			       =		(sample - pred_sample_direction) * alpha_prod_t_prev ** (-0.5)
               UpperCAmelCase			       =		sample * alpha_prod_t ** 0.5 + beta_prod_t ** 0.5 * model_output

           return sample







       @staticmethod
       def     A      (	lowercase						:	torch.Tensor   ,				lowercase						:	torch.Tensor   ,				lowercase						:	float				):



           '''simple docstring'''

           UpperCAmelCase			       =		acos(torch.dot(torch.flatten(lowercase				)   ,				torch.flatten(lowercase				)				) / torch.norm(lowercase				) / torch.norm(lowercase				)				)
           return sin((1 - alpha) * theta				) * xa / sin(lowercase				) + sin(alpha * theta				) * xa / sin(lowercase				)




