

'''simple docstring'''






# DISCLAIMER: This code is strongly influenced by https://github.com/pesser/pytorch_diffusion
# and https://github.com/hojonathanho/diffusion
import math
from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from diffusers.configuration_utils import ConfigMixin, register_to_config
from diffusers.schedulers.scheduling_utils import SchedulerMixin
from diffusers.utils import BaseOutput, deprecate






@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->DDIM
class     lowercase_			(lowerCamelCase__   ):




  """simple docstring"""




  SCREAMING_SNAKE_CASE					:						torch.FloatTensor
  SCREAMING_SNAKE_CASE					:						Optional[torch.FloatTensor]           =    None


def 	_A  (					A__  ,       A__=0.9_9_9  ,       A__="cosine"  ,       ):






 """simple docstring"""



 if alpha_transform_type == "cosine":

  def alpha_bar_fn(A__   ):
   return math.cos((t + 0.0_0_8) / 1.0_0_8 * math.pi / 2   ) ** 2

 elif alpha_transform_type == "exp":

  def alpha_bar_fn(A__   ):
   return math.exp(t * -1_2.0   )

 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     lowercase_			(lowerCamelCase__       , lowerCamelCase__   ):




  """simple docstring"""




  SCREAMING_SNAKE_CASE					:						str           =    1






  @register_to_config
  def __init__(   self     :  Union[str, Any]      ,lowercase__     :  int = 1_0_0_0      ,lowercase__     :  float = 0.0_0_0_1      ,lowercase__     :  float = 0.0_2      ,lowercase__     :  str = "linear"      ,lowercase__     :  Optional[Union[np.ndarray, List[float]]] = None      ,lowercase__     :  bool = True      ,lowercase__     :  bool = True      ,lowercase__     :  int = 0      ,lowercase__     :  str = "epsilon"      ,lowercase__     :  float = 1.0      ,**lowercase__     :  Optional[int]      ,):
   if kwargs.get('''set_alpha_to_one'''      ,lowercase__    ) is not None:
    __lowercase													=				(
        '''The `set_alpha_to_one` argument is deprecated. Please use `set_alpha_to_zero` instead.'''
    )
    deprecate('''set_alpha_to_one'''      ,'''1.0.0'''      ,lowercase__      ,standard_warn=lowercase__    )
    __lowercase													=				kwargs['''set_alpha_to_one''']
   if trained_betas is not None:
    __lowercase													=				torch.tensor(lowercase__      ,dtype=torch.floataa    )
   elif beta_schedule == "linear":
    __lowercase													=				torch.linspace(lowercase__      ,lowercase__      ,lowercase__      ,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      ,lowercase__      ,dtype=torch.floataa    ) ** 2
    )
   elif beta_schedule == "squaredcos_cap_v2":
    # Glide cosine schedule
    __lowercase													=				betas_for_alpha_bar(lowercase__    )
   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    )

   # At every step in inverted ddim, we are looking into the next alphas_cumprod
   # For the final step, there is no next alphas_cumprod, and the index is out of bounds
   # `set_alpha_to_zero` decides whether we set this parameter simply to zero
   # in this case, self.step() just output the predicted noise
   # or whether we use the final alpha of the "non-previous" one.
   __lowercase													=				torch.tensor(0.0    ) if set_alpha_to_zero else self.alphas_cumprod[-1]

   # standard deviation of the initial noise distribution
   __lowercase													=				1.0

   # setable values
   __lowercase													=				None
   __lowercase													=				torch.from_numpy(np.arange(0      ,lowercase__    ).copy().astype(np.intaa    )    )






  def       SCREAMING_SNAKE_CASE   (   self     :  str      ,lowercase__     :  torch.FloatTensor      ,lowercase__     :  Optional[int] = None    ):
   return sample






  def       SCREAMING_SNAKE_CASE   (   self     :  str      ,lowercase__     :  int      ,lowercase__     :  Union[str, torch.device] = None    ):

   if num_inference_steps > self.config.num_train_timesteps:
    raise ValueError(
        F"`num_inference_steps`: {num_inference_steps} cannot be larger than `self.config.train_timesteps`:"
        F" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
        F" maximal {self.config.num_train_timesteps} timesteps."    )

   __lowercase													=				num_inference_steps
   __lowercase													=				self.config.num_train_timesteps // self.num_inference_steps
   # creates integer timesteps by multiplying by ratio
   # casting to int to avoid issues when num_inference_step is power of 3
   __lowercase													=				(np.arange(0      ,lowercase__    ) * step_ratio).round().copy().astype(np.intaa    )
   __lowercase													=				torch.from_numpy(lowercase__    ).to(lowercase__    )
   self.timesteps += self.config.steps_offset






  def       SCREAMING_SNAKE_CASE   (   self     :  Optional[Any]      ,lowercase__     :  torch.FloatTensor      ,lowercase__     :  int      ,lowercase__     :  torch.FloatTensor      ,lowercase__     :  float = 0.0      ,lowercase__     :  bool = False      ,lowercase__     :  Optional[torch.FloatTensor] = None      ,lowercase__     :  bool = True      ,):
   # 1. get previous step value (=t+1)
   __lowercase													=				timestep + self.config.num_train_timesteps // self.num_inference_steps

   # 2. compute alphas, betas
   # change original implementation to exactly match noise levels for analogous forward process
   __lowercase													=				self.alphas_cumprod[timestep]
   __lowercase													=				(
       self.alphas_cumprod[prev_timestep]
       if prev_timestep < self.config.num_train_timesteps
       else self.final_alpha_cumprod
   )

   __lowercase													=				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":
    __lowercase													=				(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
    __lowercase													=				model_output
   elif self.config.prediction_type == "sample":
    __lowercase													=				model_output
    __lowercase													=				(sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5
   elif self.config.prediction_type == "v_prediction":
    __lowercase													=				(alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
    __lowercase													=				(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. Clip or threshold "predicted x_0"
   if self.config.clip_sample:
    __lowercase													=				pred_original_sample.clamp(
        -self.config.clip_sample_range      ,self.config.clip_sample_range    )

   # 5. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
   __lowercase													=				(1 - alpha_prod_t_prev) ** 0.5 * pred_epsilon

   # 6. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
   __lowercase													=				alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction

   if not return_dict:
    return (prev_sample, pred_original_sample)
   return DDIMSchedulerOutput(prev_sample=lowercase__      ,pred_original_sample=lowercase__    )






  def __len__(   self     :  List[Any]    ):
   return self.config.num_train_timesteps




'''simple docstring'''






def 	_A  (					A__ = 1000000   ):






 """simple docstring"""



 __lowercase													=				set(range(3  ,       A__  ,       2   )   )
 primes.add(2   )
 for p in range(3  ,       A__  ,       2   ):
  if p not in primes:
   continue
  primes.difference_update(set(range(p * p  ,       A__  ,       A__   )   )   )

 __lowercase													=				[float(A__   ) for n in range(limit + 1   )]

 for p in primes:
  for n in range(A__  ,       limit + 1  ,       A__   ):
   phi[n] *= 1 - 1 / p

 return int(sum(phi[2:]   )   )


if __name__ == "__main__":
 print(f'{solution() = }')

