code: 
  
  
  
  """simple docstring"""
  
  
  
  
  # DISCLAIMER: This code is strongly influenced by https://github.com/pesser/pytorch_diffusion
  # and https://github.com/hojonathanho/diffusion
  
  from dataclasses import dataclass
  from typing import Optional, Tuple, Union
  
  import flax
  import jax.numpy as jnp
  
  from ..configuration_utils import ConfigMixin, register_to_config
  from .scheduling_utils_flax import (
      CommonSchedulerState,
      FlaxKarrasDiffusionSchedulers,
      FlaxSchedulerMixin,
      FlaxSchedulerOutput,
      add_noise_common,
      get_velocity_common,
  )
  
  
  
  
  
  
  @flax.struct.dataclass
  class 			UpperCAmelCase_						:
        __SCREAMING_SNAKE_CASE		:     Tuple				  =				4_2
        __SCREAMING_SNAKE_CASE		:     List[Any]				  =				4_2
  
        # setable values
        __SCREAMING_SNAKE_CASE		:     Dict				  =				4_2
        __SCREAMING_SNAKE_CASE		:     str				  =				4_2
        __SCREAMING_SNAKE_CASE		:     Dict				  =				None
  
  
  
        @classmethod
        def 		snake_case_      (  cls  :			Union[str, Any]  ,			A  :			Optional[int]  ,			A  :			str  ,			A  :			Union[str, Any]  ,			A  :			Union[str, Any]  ,			):
             return cls(
                 common=snake_case_  ,			final_alpha_cumprod=snake_case_  ,			init_noise_sigma=snake_case_  ,			timesteps=snake_case_  ,			)
  
  
  
  
  
  
  
  @dataclass
  class 			UpperCAmelCase_						(       UpperCamelCase__		):
        __SCREAMING_SNAKE_CASE		:     Dict				  =				4_2
  
  
  
  
  
  
  
  class 			UpperCAmelCase_						(       UpperCamelCase__ ,    UpperCamelCase__		):
        __SCREAMING_SNAKE_CASE		:     List[Any]				  =				[e.name for e in FlaxKarrasDiffusionSchedulers]
  
        __SCREAMING_SNAKE_CASE		:     Optional[int]				  =				4_2
  
  
  
        @property
        def 		snake_case_      (  self  :			str		):
             return True
  
  
  
        @register_to_config
        def __init__(  self  :			Any  ,			A  :			int = 1_0_0_0  ,			A  :			Optional[Any] = 0.0_001  ,			A  :			List[str] = 0.02  ,			A  :			Union[str, Any] = "linear"  ,			A  :			List[Any] = None  ,			A  :			List[Any] = True  ,			A  :			str = 0  ,			A  :			List[str] = "epsilon"  ,			A  :			List[Any] = jnp.floataa  ,			):
             _UpperCAmelCase    :			Union[str, Any]              =					dtype
  
  
  
        def 		snake_case_      (  self  :			Tuple  ,			A  :			Dict = None		):
             if common is None:
                  _UpperCAmelCase    :			List[str]              =					CommonSchedulerState.create(self		)
  
             # At every step in ddim, we are looking into the previous alphas_cumprod
             # For the final step, there is no previous alphas_cumprod because we are already at 0
             # `set_alpha_to_one` decides whether we set this parameter simply to one or
             # whether we use the final alpha of the "non-previous" one.
             _UpperCAmelCase    :			Optional[int]              =					(
                 jnp.array(1.0  ,			dtype=self.dtype		) if self.config.set_alpha_to_one else common.alphas_cumprod[0]
             )
  
             # standard deviation of the initial noise distribution
             _UpperCAmelCase    :			List[str]              =					jnp.array(1.0  ,			dtype=self.dtype		)
  
             _UpperCAmelCase    :			Optional[int]              =					jnp.arange(0  ,			self.config.num_train_timesteps		).round()[::-1]
  
             return DDIMSchedulerState.create(
                 common=snake_case_  ,			final_alpha_cumprod=snake_case_  ,			init_noise_sigma=snake_case_  ,			timesteps=snake_case_  ,			)
  
  
  
        def 		snake_case_      (  self  :			List[Any]  ,			A  :			Any  ,			A  :			Union[str, Any]  ,			A  :			List[str] = None		):
             return sample
  
  
  
        def 		snake_case_      (  self  :			Any  ,			A  :			Optional[int]  ,			A  :			Optional[Any]  ,			A  :			List[Any] = ()		):
             _UpperCAmelCase    :			Union[str, Any]              =					self.config.num_train_timesteps // num_inference_steps
             # creates integer timesteps by multiplying by ratio
             # rounding to avoid issues when num_inference_step is power of 3
             _UpperCAmelCase    :			Union[str, Any]              =					(jnp.arange(0  ,			snake_case_		) * step_ratio).round()[::-1] + self.config.steps_offset
  
             return state.replace(
                 num_inference_steps=snake_case_  ,			timesteps=snake_case_  ,			)
  
  
  
        def 		snake_case_      (  self  :			List[str]  ,			A  :			Any  ,			A  :			Any  ,			A  :			Tuple		):
             _UpperCAmelCase    :			Union[str, Any]              =					state.common.alphas_cumprod[timestep]
             _UpperCAmelCase    :			Any              =					jnp.where(
                 prev_timestep >= 0  ,			state.common.alphas_cumprod[prev_timestep]  ,			state.final_alpha_cumprod		)
             _UpperCAmelCase    :			Dict              =					1 - alpha_prod_t
             _UpperCAmelCase    :			Dict              =					1 - alpha_prod_t_prev
  
             _UpperCAmelCase    :			str              =					(beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)
  
             return variance
  
  
  
        def 		snake_case_      (  self  :			Any  ,			A  :			str  ,			A  :			Union[str, Any]  ,			A  :			Dict  ,			A  :			Dict  ,			A  :			Any = 0.0  ,			A  :			Dict = True  ,			):
             if state.num_inference_steps is None:
                  raise ValueError(
                      "Number of inference steps is \'None\', you need to run \'set_timesteps\' after creating the scheduler"		)
  
             # See formulas (12) and (16) of DDIM paper https://arxiv.org/pdf/2010.02502.pdf
             # Ideally, read DDIM paper in-detail understanding
  
             # Notation (<variable name> -> <name in paper>
             # - pred_noise_t -> e_theta(x_t, t)
             # - pred_original_sample -> f_theta(x_t, t) or x_0
             # - std_dev_t -> sigma_t
             # - eta -> η
             # - pred_sample_direction -> "direction pointing to x_t"
             # - pred_prev_sample -> "x_t-1"
  
             # 1. get previous step value (=t-1)
             _UpperCAmelCase    :			Optional[int]              =					timestep - self.config.num_train_timesteps // state.num_inference_steps
  
             _UpperCAmelCase    :			List[str]              =					state.common.alphas_cumprod
             _UpperCAmelCase    :			Optional[int]              =					state.final_alpha_cumprod
  
             # 2. compute alphas, betas
             _UpperCAmelCase    :			List[Any]              =					alphas_cumprod[timestep]
             _UpperCAmelCase    :			Any              =					jnp.where(prev_timestep >= 0  ,			alphas_cumprod[prev_timestep]  ,			snake_case_		)
  
             _UpperCAmelCase    :			List[str]              =					1 - alpha_prod_t
  
             # 3. compute predicted original sample from predicted noise also called
             # "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
             if self.config.prediction_type == "epsilon":
                  _UpperCAmelCase    :			Union[str, Any]              =					(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
                  _UpperCAmelCase    :			Any              =					model_output
             elif self.config.prediction_type == "sample":
                  _UpperCAmelCase    :			Tuple              =					model_output
                  _UpperCAmelCase    :			str              =					(sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5
             elif self.config.prediction_type == "v_prediction":
                  _UpperCAmelCase    :			Optional[Any]              =					(alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
                  _UpperCAmelCase    :			Tuple              =					(alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
             else:
                  raise ValueError(
                      f'prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or'
                      " `v_prediction`"		)
  
             # 4. compute variance: "sigma_t(η)" -> see formula (16)
             # σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
             _UpperCAmelCase    :			Optional[int]              =					self._get_variance(snake_case_  ,			snake_case_  ,			snake_case_		)
             _UpperCAmelCase    :			Dict              =					eta * variance ** 0.5
  
             # 5. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
             _UpperCAmelCase    :			Optional[Any]              =					(1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * pred_epsilon
  
             # 6. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
             _UpperCAmelCase    :			List[str]              =					alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction
  
             if not return_dict:
                  return (prev_sample, state)
  
             return FlaxDDIMSchedulerOutput(prev_sample=snake_case_  ,			state=snake_case_		)
  
  
  
        def 		snake_case_      (  self  :			Tuple  ,			A  :			Tuple  ,			A  :			Any  ,			A  :			List[Any]  ,			A  :			Optional[int]  ,			):
             return add_noise_common(state.common  ,			snake_case_  ,			snake_case_  ,			snake_case_		)
  
  
  
        def 		snake_case_      (  self  :			Optional[int]  ,			A  :			Tuple  ,			A  :			Union[str, Any]  ,			A  :			Any  ,			A  :			Union[str, Any]  ,			):
             return get_velocity_common(state.common  ,			snake_case_  ,			snake_case_  ,			snake_case_		)
  
  
  
        def __len__(  self  :			List[str]		):
             return self.config.num_train_timesteps
  
style_context: 
  
  
  
  """simple docstring"""
  
  
  
  
  _lowerCAmelCase       :      str							=							{
      "0": 1,
      "1": 1,
      "2": 2,
      "3": 6,
      "4": 24,
      "5": 1_20,
      "6": 7_20,
      "7": 50_40,
      "8": 4_03_20,
      "9": 36_28_80,
  }
  
  _lowerCAmelCase       :      Optional[int]							=							{1_45: 1_45}
  
  _lowerCAmelCase       :      int							=							{
      1_45: 0,
      1_69: 3,
      3_63_01: 3,
      14_54: 3,
      8_71: 2,
      4_53_61: 2,
      8_72: 2,
  }
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       if n in CACHE_SUM_DIGIT_FACTORIALS:
            return CACHE_SUM_DIGIT_FACTORIALS[n]
       _UpperCAmelCase    :			Dict              =					sum(DIGIT_FACTORIALS[let] for let in str(SCREAMING_SNAKE_CASE__       )       )
       _UpperCAmelCase    :			Any              =					ret
       return ret
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int		,						SCREAMING_SNAKE_CASE__			:		set | None = None       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       _UpperCAmelCase    :			Dict              =					previous or set()
       if n in CHAIN_LENGTH_CACHE:
            return CHAIN_LENGTH_CACHE[n]
       _UpperCAmelCase    :			Dict              =					sum_digit_factorials(SCREAMING_SNAKE_CASE__       )
       if next_number in previous:
            _UpperCAmelCase    :			List[str]              =					0
            return 0
       else:
            previous.add(SCREAMING_SNAKE_CASE__       )
            _UpperCAmelCase    :			str              =					1 + chain_length(SCREAMING_SNAKE_CASE__		,						SCREAMING_SNAKE_CASE__       )
            _UpperCAmelCase    :			str              =					ret
            return ret
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int = 60		,						SCREAMING_SNAKE_CASE__			:		int = 1_000_000       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       return sum(1 for i in range(1		,						SCREAMING_SNAKE_CASE__       ) if chain_length(SCREAMING_SNAKE_CASE__       ) == num_terms       )
  
  
  if __name__ == "__main__":
   print(F"{solution() = }")
  