



'''simple docstring'''





__lowerCamelCase         =			range(2, 20 + 1)
__lowerCamelCase         =			[10**k for k in range(ks[-1] + 1)]
__lowerCamelCase         =			{}







def    _snake_case (       __snake_case					,  __snake_case					,  __snake_case					,  __snake_case    )     ->     Union[str, Any]:






      '''simple docstring'''







      UpperCAmelCase_  :	int				      =		sum(a_i[j] for j in range(__snake_case					,  len(__snake_case    )    )    )
      UpperCAmelCase_  :	List[Any]				      =		sum(a_i[j] * base[j] for j in range(min(len(__snake_case    )					,  __snake_case    )    )    )

      UpperCAmelCase_  :	Optional[int]				      =		0, 0
      UpperCAmelCase_  :	Any				      =		n - i

      UpperCAmelCase_  :	Optional[int]				      =		memo.get(__snake_case    )

      if sub_memo is not None:
            UpperCAmelCase_  :	Union[str, Any]				      =		sub_memo.get(__snake_case    )

            if jumps is not None and len(__snake_case    ) > 0:
                  # find and make the largest jump without going over
                  UpperCAmelCase_  :	List[str]				      =		-1
                  for _k in range(len(__snake_case    ) - 1					,  -1					,  -1    ):
                        if jumps[_k][2] <= k and jumps[_k][1] <= max_dn:
                              UpperCAmelCase_  :	str				      =		_k
                              break

                  if max_jump >= 0:
                        UpperCAmelCase_  :	Any				      =		jumps[max_jump]
                        # since the difference between jumps is cached, add c
                        UpperCAmelCase_  :	Union[str, Any]				      =		diff + c
                        for j in range(min(__snake_case					,  len(__snake_case    )    )    ):
                              UpperCAmelCase_  :	Any				      =		divmod(__snake_case					,  1_0    )
                        if new_c > 0:
                              add(__snake_case					,  __snake_case					,  __snake_case    )

            else:
                  UpperCAmelCase_  :	int				      =		[]
      else:
            UpperCAmelCase_  :	Dict				      =		{c: []}
            UpperCAmelCase_  :	Optional[int]				      =		sub_memo

      if dn >= max_dn or c + diff >= base[k]:
            return diff, dn

      if k > ks[0]:
            while True:
                  # keep doing smaller jumps
                  UpperCAmelCase_  :	Tuple				      =		next_term(__snake_case					,  k - 1					,  i + dn					,  __snake_case    )
                  diff += _diff
                  dn += terms_jumped

                  if dn >= max_dn or c + diff >= base[k]:
                        break
      else:
            # would be too small a jump, just compute sequential terms instead
            UpperCAmelCase_  :	List[Any]				      =		compute(__snake_case					,  __snake_case					,  i + dn					,  __snake_case    )
            diff += _diff
            dn += terms_jumped

      UpperCAmelCase_  :	Optional[int]				      =		sub_memo[c]

      # keep jumps sorted by # of terms skipped
      UpperCAmelCase_  :	str				      =		0
      while j < len(__snake_case    ):
            if jumps[j][1] > dn:
                  break
            j += 1

      # cache the jump for this value digitsum(b) and c
      sub_memo[c].insert(__snake_case					,  (diff, dn, k)    )
      return (diff, dn)







def    _snake_case (       __snake_case					,  __snake_case					,  __snake_case					,  __snake_case    )     ->     List[str]:






      '''simple docstring'''







      if i >= n:
            return 0, i
      if k > len(__snake_case    ):
            a_i.extend([0 for _ in range(k - len(__snake_case    )    )]    )

      # note: a_i -> b * 10^k + c
      # ds_b -> digitsum(b)
      # ds_c -> digitsum(c)
      UpperCAmelCase_  :	str				      =		i
      UpperCAmelCase_  :	int				      =		0, 0, 0
      for j in range(len(__snake_case    )    ):
            if j >= k:
                  ds_b += a_i[j]
            else:
                  ds_c += a_i[j]

      while i < n:
            i += 1
            UpperCAmelCase_  :	Tuple				      =		ds_c + ds_b
            diff += addend
            UpperCAmelCase_  :	Union[str, Any]				      =		0
            for j in range(__snake_case    ):
                  UpperCAmelCase_  :	Optional[Any]				      =		a_i[j] + addend
                  UpperCAmelCase_  :	Optional[Any]				      =		divmod(__snake_case					,  1_0    )

                  ds_c += a_i[j]

            if addend > 0:
                  break

      if addend > 0:
            add(__snake_case					,  __snake_case					,  __snake_case    )
      return diff, i - start_i







def    _snake_case (       __snake_case					,  __snake_case					,  __snake_case    )     ->     Any:






      '''simple docstring'''







      for j in range(__snake_case					,  len(__snake_case    )    ):
            UpperCAmelCase_  :	int				      =		digits[j] + addend
            if s >= 1_0:
                  UpperCAmelCase_  :	str				      =		divmod(__snake_case					,  1_0    )
                  UpperCAmelCase_  :	int				      =		addend // 1_0 + quotient
            else:
                  UpperCAmelCase_  :	Optional[int]				      =		s
                  UpperCAmelCase_  :	Any				      =		addend // 1_0

            if addend == 0:
                  break

      while addend > 0:
            UpperCAmelCase_  :	Tuple				      =		divmod(__snake_case					,  1_0    )
            digits.append(__snake_case    )







def    _snake_case (       __snake_case = 1_0**1_5    )     ->     int:






      '''simple docstring'''







      UpperCAmelCase_  :	Optional[Any]				      =		[1]
      UpperCAmelCase_  :	int				      =		1
      UpperCAmelCase_  :	Union[str, Any]				      =		0
      while True:
            UpperCAmelCase_  :	Optional[int]				      =		next_term(__snake_case					,  2_0					,  i + dn					,  __snake_case    )
            dn += terms_jumped
            if dn == n - i:
                  break

      UpperCAmelCase_  :	Tuple				      =		0
      for j in range(len(__snake_case    )    ):
            a_n += digits[j] * 1_0**j
      return a_n


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





# DISCLAIMER: This file is strongly influenced by https://github.com/yang-song/score_sde_pytorch

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

import torch

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


@dataclass
class   snake_case_ (lowercase__  ):




       """simple docstring"""






       _lowerCamelCase			    =				42
       _lowerCamelCase			    =				42


class   snake_case_ (lowercase__ , lowercase__  ):




       """simple docstring"""






       _lowerCamelCase			    =				1



       @register_to_config
       def __init__(     self     ,lowercase = 2000     ,lowercase = 0.15     ,lowercase = 0.01     ,lowercase = 1348.0     ,lowercase = 1E-5     ,lowercase = 1     ,):

             """simple docstring"""





             UpperCAmelCase_  :	Optional[int]				      =		sigma_max

             # setable values
             UpperCAmelCase_  :	Optional[int]				      =		None

             self.set_sigmas(lowercase     ,lowercase     ,lowercase     ,lowercase)



       def  A_   (     self     ,lowercase     ,lowercase = None):

             """simple docstring"""





             return sample



       def  A_   (     self     ,lowercase     ,lowercase = None     ,lowercase = None):

             """simple docstring"""





             UpperCAmelCase_  :	int				      =		sampling_eps if sampling_eps is not None else self.config.sampling_eps

             UpperCAmelCase_  :	List[Any]				      =		torch.linspace(1     ,lowercase     ,lowercase     ,device=lowercase)



       def  A_   (     self     ,lowercase     ,lowercase = None     ,lowercase = None     ,lowercase = None):

             """simple docstring"""





             UpperCAmelCase_  :	Any				      =		sigma_min if sigma_min is not None else self.config.sigma_min
             UpperCAmelCase_  :	int				      =		sigma_max if sigma_max is not None else self.config.sigma_max
             UpperCAmelCase_  :	Union[str, Any]				      =		sampling_eps if sampling_eps is not None else self.config.sampling_eps
             if self.timesteps is None:
                   self.set_timesteps(lowercase     ,lowercase)

             UpperCAmelCase_  :	Union[str, Any]				      =		sigma_min * (sigma_max / sigma_min) ** (self.timesteps / sampling_eps)
             UpperCAmelCase_  :	Optional[int]				      =		torch.exp(torch.linspace(math.log(lowercase)     ,math.log(lowercase)     ,lowercase))
             UpperCAmelCase_  :	Any				      =		torch.tensor([sigma_min * (sigma_max / sigma_min) ** t for t in self.timesteps])



       def  A_   (     self     ,lowercase     ,lowercase):

             """simple docstring"""





             return torch.where(
                 timesteps == 0     ,torch.zeros_like(t.to(timesteps.device))     ,self.discrete_sigmas[timesteps - 1].to(timesteps.device)     ,)



       def  A_   (     self     ,lowercase     ,lowercase     ,lowercase     ,lowercase = None     ,lowercase = True     ,):

             """simple docstring"""





             if self.timesteps is None:
                   raise ValueError(
                       "`self.timesteps` is not set, you need to run 'set_timesteps' after creating the scheduler")

             UpperCAmelCase_  :	Optional[int]				      =		timestep * torch.ones(
                 sample.shape[0]     ,device=sample.device)  # torch.repeat_interleave(timestep, sample.shape[0])
             UpperCAmelCase_  :	Tuple				      =		(timestep * (len(self.timesteps) - 1)).long()

             # mps requires indices to be in the same device, so we use cpu as is the default with cuda
             UpperCAmelCase_  :	Optional[int]				      =		timesteps.to(self.discrete_sigmas.device)

             UpperCAmelCase_  :	Optional[Any]				      =		self.discrete_sigmas[timesteps].to(sample.device)
             UpperCAmelCase_  :	Optional[Any]				      =		self.get_adjacent_sigma(lowercase     ,lowercase).to(sample.device)
             UpperCAmelCase_  :	Any				      =		torch.zeros_like(lowercase)
             UpperCAmelCase_  :	Dict				      =		(sigma**2 - adjacent_sigma**2) ** 0.5

             # equation 6 in the paper: the model_output modeled by the network is grad_x log pt(x)
             # also equation 47 shows the analog from SDE models to ancestral sampling methods
             UpperCAmelCase_  :	Dict				      =		diffusion.flatten()
             while len(diffusion.shape) < len(sample.shape):
                   UpperCAmelCase_  :	List[str]				      =		diffusion.unsqueeze(-1)
             UpperCAmelCase_  :	List[Any]				      =		drift - diffusion**2 * model_output

             #  equation 6: sample noise for the diffusion term of
             UpperCAmelCase_  :	Union[str, Any]				      =		randn_tensor(
                 sample.shape     ,layout=sample.layout     ,generator=lowercase     ,device=sample.device     ,dtype=sample.dtype)
             UpperCAmelCase_  :	Any				      =		sample - drift  # subtract because `dt` is a small negative timestep
             # TODO is the variable diffusion the correct scaling term for the noise?
             UpperCAmelCase_  :	Tuple				      =		prev_sample_mean + diffusion * noise  # add impact of diffusion field g

             if not return_dict:
                   return (prev_sample, prev_sample_mean)

             return SdeVeOutput(prev_sample=lowercase     ,prev_sample_mean=lowercase)



       def  A_   (     self     ,lowercase     ,lowercase     ,lowercase = None     ,lowercase = True     ,):

             """simple docstring"""





             if self.timesteps is None:
                   raise ValueError(
                       "`self.timesteps` is not set, you need to run 'set_timesteps' after creating the scheduler")

             # For small batch sizes, the paper "suggest replacing norm(z) with sqrt(d), where d is the dim. of z"
             # sample noise for correction
             UpperCAmelCase_  :	int				      =		randn_tensor(sample.shape     ,layout=sample.layout     ,generator=lowercase).to(sample.device)

             # compute step size from the model_output, the noise, and the snr
             UpperCAmelCase_  :	Union[str, Any]				      =		torch.norm(model_output.reshape(model_output.shape[0]     ,-1)     ,dim=-1).mean()
             UpperCAmelCase_  :	Optional[Any]				      =		torch.norm(noise.reshape(noise.shape[0]     ,-1)     ,dim=-1).mean()
             UpperCAmelCase_  :	List[Any]				      =		(self.config.snr * noise_norm / grad_norm) ** 2 * 2
             UpperCAmelCase_  :	Optional[Any]				      =		step_size * torch.ones(sample.shape[0]).to(sample.device)
             # self.repeat_scalar(step_size, sample.shape[0])

             # compute corrected sample: model_output term and noise term
             UpperCAmelCase_  :	Any				      =		step_size.flatten()
             while len(step_size.shape) < len(sample.shape):
                   UpperCAmelCase_  :	Tuple				      =		step_size.unsqueeze(-1)
             UpperCAmelCase_  :	Dict				      =		sample + step_size * model_output
             UpperCAmelCase_  :	int				      =		prev_sample_mean + ((step_size * 2) ** 0.5) * noise

             if not return_dict:
                   return (prev_sample,)

             return SchedulerOutput(prev_sample=lowercase)



       def  A_   (     self     ,lowercase     ,lowercase     ,lowercase     ,):

             """simple docstring"""





             UpperCAmelCase_  :	Any				      =		timesteps.to(original_samples.device)
             UpperCAmelCase_  :	List[str]				      =		self.discrete_sigmas.to(original_samples.device)[timesteps]
             UpperCAmelCase_  :	Tuple				      =		(
                 noise * sigmas[:, None, None, None]
                 if noise is not None
                 else torch.randn_like(lowercase) * sigmas[:, None, None, None]
             )
             UpperCAmelCase_  :	Tuple				      =		noise + original_samples
             return noisy_samples



       def __len__(     self):

             """simple docstring"""





             return self.config.num_train_timesteps
