code: 
  
  
  '''simple docstring'''
  
  import math
  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 						__lowerCAmelCase		(lowercase_  ):
  
  
  
  
  
        '''simple docstring'''
  
  
  
  
  
  
        lowerCAmelCase__      :  torch.FloatTensor
        lowerCAmelCase__      :  torch.FloatTensor
  
  
  
  
  
  
  def _SCREAMING_SNAKE_CASE   (A						,				A=0.999						,				A="cosine"						,				) -> Dict:
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if alpha_transform_type == "cosine":
  
                def alpha_bar_fn(A ):
                       return math.cos((t + 0.008) / 1.008 * math.pi / 2 ) ** 2
  
         elif alpha_transform_type == "exp":
  
                def alpha_bar_fn(A ):
                       return math.exp(t * -12.0 )
  
         else:
                raise ValueError(f"Unsupported alpha_tranform_type: {alpha_transform_type}" )
  
         lowercase__       =				[]
         for i in range(A ):
                lowercase__       =				i / num_diffusion_timesteps
                lowercase__       =				(i + 1) / num_diffusion_timesteps
                betas.append(min(1 - alpha_bar_fn(A ) / alpha_bar_fn(A )						,				A ) )
         return torch.tensor(A						,				dtype=torch.floataa )
  
  class 						__lowerCAmelCase		(lowercase_						,  lowercase_  ):
  
  
  
  
  
        '''simple docstring'''
  
  
  
  
  
  
        lowerCAmelCase__      :  List[str]						      =							1
  
  
  
        @register_to_config
        def __init__(self					:					Tuple     ,		UpperCamelCase					:					int = 1000     ,		UpperCamelCase					:					float = 0.00_01     ,		UpperCamelCase					:					float = 0.02     ,		UpperCamelCase					:					str = "linear"     ,		UpperCamelCase					:					float = 0.0     ,		UpperCamelCase					:					Optional[np.ndarray] = None     ,		UpperCamelCase					:					bool = True     ,		):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               if trained_betas is not None:
                      lowercase__       =				torch.from_numpy(UpperCamelCase				)
               elif beta_schedule == "linear":
                      lowercase__       =				torch.linspace(UpperCamelCase     ,		UpperCamelCase     ,		UpperCamelCase     ,		dtype=torch.floataa				)
               elif beta_schedule == "scaled_linear":
                      # this schedule is very specific to the latent diffusion model.
                      lowercase__       =				(
                          torch.linspace(beta_start**0.5     ,		beta_end**0.5     ,		UpperCamelCase     ,		dtype=torch.floataa				) ** 2
                      )
               elif beta_schedule == "squaredcos_cap_v2":
                      # Glide cosine schedule
                      lowercase__       =				betas_for_alpha_bar(UpperCamelCase				)
               elif beta_schedule == "sigmoid":
                      # GeoDiff sigmoid schedule
                      lowercase__       =				torch.linspace(-6     ,		6     ,		UpperCamelCase				)
                      lowercase__       =				torch.sigmoid(UpperCamelCase				) * (beta_end - beta_start) + beta_start
               else:
                      raise NotImplementedError(f"{beta_schedule} does is not implemented for {self.__class__}"				)
  
               lowercase__       =				1.0 - self.betas
               lowercase__       =				torch.cumprod(self.alphas     ,		dim=0				)
               lowercase__       =				torch.tensor(1.0				)
  
               lowercase__       =				torch.tensor(1.0				)
  
               # standard deviation of the initial noise distribution
               lowercase__       =				1.0
  
               # setable values
               lowercase__       =				None
               lowercase__       =				torch.from_numpy(np.arange(0     ,		UpperCamelCase				)[::-1].copy()				)
  
               lowercase__       =				eta
  
  
  
        def    UpperCamelCase__	(self					:					int     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					Optional[int] = None				):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               return sample
  
  
  
        def    UpperCamelCase__	(self					:					str     ,		UpperCamelCase					:					int     ,		UpperCamelCase					:					int = 10     ,		UpperCamelCase					:					int = 10     ,		UpperCamelCase					:					Union[str, torch.device] = None     ,		):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               lowercase__       =				min(self.config.num_train_timesteps     ,		UpperCamelCase				)
               lowercase__       =				num_inference_steps
  
               lowercase__       =				[]
  
               lowercase__       =				{}
               for j in range(0     ,		num_inference_steps - jump_length     ,		UpperCamelCase				):
                      lowercase__       =				jump_n_sample - 1
  
               lowercase__       =				num_inference_steps
               while t >= 1:
                      lowercase__       =				t - 1
                      timesteps.append(UpperCamelCase				)
  
                      if jumps.get(UpperCamelCase     ,		0				) > 0:
                             lowercase__       =				jumps[t] - 1
                             for _ in range(UpperCamelCase				):
                                    lowercase__       =				t + 1
                                    timesteps.append(UpperCamelCase				)
  
               lowercase__       =				np.array(UpperCamelCase				) * (self.config.num_train_timesteps // self.num_inference_steps)
               lowercase__       =				torch.from_numpy(UpperCamelCase				).to(UpperCamelCase				)
  
  
  
        def    UpperCamelCase__	(self					:					str     ,		UpperCamelCase					:					Optional[int]				):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               lowercase__       =				t - self.config.num_train_timesteps // self.num_inference_steps
  
               lowercase__       =				self.alphas_cumprod[t]
               lowercase__       =				self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
               lowercase__       =				1 - alpha_prod_t
               lowercase__       =				1 - alpha_prod_t_prev
  
               # For t > 0, compute predicted variance βt (see formula (6) and (7) from
               # https://arxiv.org/pdf/2006.11239.pdf) and sample from it to get
               # previous sample x_{t-1} ~ N(pred_prev_sample, variance) == add
               # variance to pred_sample
               # Is equivalent to formula (16) in https://arxiv.org/pdf/2010.02502.pdf
               # without eta.
               # variance = (1 - alpha_prod_t_prev) / (1 - alpha_prod_t) * self.betas[t]
               lowercase__       =				(beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)
  
               return variance
  
  
  
        def    UpperCamelCase__	(self					:					int     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					int     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					Optional[torch.Generator] = None     ,		UpperCamelCase					:					bool = True     ,		):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               lowercase__       =				timestep
               lowercase__       =				timestep - self.config.num_train_timesteps // self.num_inference_steps
  
               # 1. compute alphas, betas
               lowercase__       =				self.alphas_cumprod[t]
               lowercase__       =				self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
               lowercase__       =				1 - alpha_prod_t
  
               # 2. compute predicted original sample from predicted noise also called
               # "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
               lowercase__       =				(sample - beta_prod_t**0.5 * model_output) / alpha_prod_t**0.5
  
               # 3. Clip "predicted x_0"
               if self.config.clip_sample:
                      lowercase__       =				torch.clamp(UpperCamelCase     ,		-1     ,		1				)
  
               # We choose to follow RePaint Algorithm 1 to get x_{t-1}, however we
               # substitute formula (7) in the algorithm coming from DDPM paper
               # (formula (4) Algorithm 2 - Sampling) with formula (12) from DDIM paper.
               # DDIM schedule gives the same results as DDPM with eta = 1.0
               # Noise is being reused in 7. and 8., but no impact on quality has
               # been observed.
  
               # 5. Add noise
               lowercase__       =				model_output.device
               lowercase__       =				randn_tensor(model_output.shape     ,		generator=UpperCamelCase     ,		device=UpperCamelCase     ,		dtype=model_output.dtype				)
               lowercase__       =				self.eta * self._get_variance(UpperCamelCase				) ** 0.5
  
               lowercase__       =				0
               if t > 0 and self.eta > 0:
                      lowercase__       =				std_dev_t * noise
  
               # 6. compute "direction pointing to x_t" of formula (12)
               # from https://arxiv.org/pdf/2010.02502.pdf
               lowercase__       =				(1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * model_output
  
               # 7. compute x_{t-1} of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
               lowercase__       =				alpha_prod_t_prev**0.5 * pred_original_sample + pred_sample_direction + variance
  
               # 8. Algorithm 1 Line 5 https://arxiv.org/pdf/2201.09865.pdf
               lowercase__       =				(alpha_prod_t_prev**0.5) * original_image + ((1 - alpha_prod_t_prev) ** 0.5) * noise
  
               # 9. Algorithm 1 Line 8 https://arxiv.org/pdf/2201.09865.pdf
               lowercase__       =				mask * prev_known_part + (1.0 - mask) * prev_unknown_part
  
               if not return_dict:
                      return (
                          pred_prev_sample,
                          pred_original_sample,
                      )
  
               return RePaintSchedulerOutput(prev_sample=UpperCamelCase     ,		pred_original_sample=UpperCamelCase				)
  
  
  
        def    UpperCamelCase__	(self					:					Tuple     ,		UpperCamelCase					:					int     ,		UpperCamelCase					:					Any     ,		UpperCamelCase					:					List[str]=None				):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               lowercase__       =				self.config.num_train_timesteps // self.num_inference_steps
  
               for i in range(UpperCamelCase				):
                      lowercase__       =				self.betas[timestep + i]
                      if sample.device.type == "mps":
                             # randn does not work reproducibly on mps
                             lowercase__       =				randn_tensor(sample.shape     ,		dtype=sample.dtype     ,		generator=UpperCamelCase				)
                             lowercase__       =				noise.to(sample.device				)
                      else:
                             lowercase__       =				randn_tensor(sample.shape     ,		generator=UpperCamelCase     ,		device=sample.device     ,		dtype=sample.dtype				)
  
                      # 10. Algorithm 1 Line 10 https://arxiv.org/pdf/2201.09865.pdf
                      lowercase__       =				(1 - beta) ** 0.5 * sample + beta**0.5 * noise
  
               return sample
  
  
  
        def    UpperCamelCase__	(self					:					str     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					torch.FloatTensor     ,		UpperCamelCase					:					torch.IntTensor     ,		):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               raise NotImplementedError('''Use `DDPMScheduler.add_noise()` to train for sampling with RePaint.'''				)
  
  
  
  
  
  
        def __len__(self					:					Dict				):
  
               '''simple docstring'''
  
  
  
  
  
  
  
               return self.config.num_train_timesteps
  
  
  
  
  
style_context: 
  
  
  '''simple docstring'''
  
  from __future__ import annotations
  
  from collections import Counter
  
  def _SCREAMING_SNAKE_CASE   (A ) -> list[int]:
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         lowercase__       =				[True] * n
         lowercase__       =				False
         lowercase__       =				False
         lowercase__       =				True
  
         for i in range(3						,				int(n**0.5 + 1 )						,				2 ):
                lowercase__       =				i * 2
                while index < n:
                       lowercase__       =				False
                       lowercase__       =				index + i
  
         lowercase__       =				[2]
  
         for i in range(3						,				A						,				2 ):
                if is_prime[i]:
                       primes.append(A )
  
         return primes
  
  def _SCREAMING_SNAKE_CASE   (A ) -> list[list[int]]:
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         lowercase__       =				str(A )
         lowercase__       =				[]
         lowercase__       =				['''0''', '''1''', '''2''', '''3''', '''4''', '''5''', '''6''', '''7''', '''8''', '''9''']
  
         for duplicate in Counter(A ) - Counter(set(A ) ):
                lowercase__       =				[int(number_str.replace(A						,				A ) ) for digit in digits]
                replacements.append(A )
  
         return replacements
  
  def _SCREAMING_SNAKE_CASE   (A = 8 ) -> int:
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         lowercase__       =				set()
  
         # Filter primes with less than 3 replaceable digits
         lowercase__       =				{
             x for x in set(prime_sieve(1_000_000 ) ) if len(str(A ) ) - len(set(str(A ) ) ) >= 3
         }
  
         for prime in primes:
                if prime in numbers_checked:
                       continue
  
                lowercase__       =				digit_replacements(A )
  
                for family in replacements:
                       numbers_checked.update(A )
                       lowercase__       =				primes.intersection(A )
  
                       if len(A ) != family_length:
                              continue
  
                       return min(A )
  
         return -1
  
  
  if __name__ == "__main__":
   print(solution())
  
  
  
  
  