






import math
from collections import defaultdict
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput
def 						_a    (      a :Any	,							a :Optional[int]=0.999	,							a :Optional[int]="cosine"	,							)							->			Union[str, Any]:
 if alpha_transform_type == "cosine":

  def alpha_bar_fn(a :int ):
   return math.cos((t + 0.008) / 1.008 * math.pi / 2 ) ** 2

 elif alpha_transform_type == "exp":

  def alpha_bar_fn(a :int ):
   return math.exp(t * -12.0 )

 else:
  raise ValueError(F"""Unsupported alpha_tranform_type: {alpha_transform_type}""" )

 a		  =      []
 for i in range(a ):
  a		  =      i / num_diffusion_timesteps
  a		  =      (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_					(	lowercase ,  lowercase   ):






      '''simple docstring'''




      __snake_case       =  [e.name for e in KarrasDiffusionSchedulers]
      __snake_case       =  2







      @register_to_config
      def __init__(      self      :						Tuple ,					__UpperCAmelCase      :						int = 1_000 ,					__UpperCAmelCase      :						float = 0.00085 ,					__UpperCAmelCase      :						float = 0.012 ,					__UpperCAmelCase      :						str = "linear" ,					__UpperCAmelCase      :						Optional[Union[np.ndarray, List[float]]] = None ,					__UpperCAmelCase      :						str = "epsilon" ,					__UpperCAmelCase      :						str = "linspace" ,					__UpperCAmelCase      :						int = 0 ,					)    ->str:




       """simple docstring"""


       if trained_betas is not None:
        a		  =      torch.tensor(__UpperCAmelCase ,					dtype=torch.floataa  )
       elif beta_schedule == "linear":
        a		  =      torch.linspace(__UpperCAmelCase ,					__UpperCAmelCase ,					__UpperCAmelCase ,					dtype=torch.floataa  )
       elif beta_schedule == "scaled_linear":
        # this schedule is very specific to the latent diffusion model.
        a		  =      (
            torch.linspace(beta_start**0.5 ,					beta_end**0.5 ,					__UpperCAmelCase ,					dtype=torch.floataa  ) ** 2
        )
       elif beta_schedule == "squaredcos_cap_v2":
        # Glide cosine schedule
        a		  =      betas_for_alpha_bar(__UpperCAmelCase  )
       else:
        raise NotImplementedError(F"""{beta_schedule} does is not implemented for {self.__class__}"""  )

       a		  =      1.0 - self.betas
       a		  =      torch.cumprod(self.alphas ,					dim=0  )

       #  set all values
       self.set_timesteps(__UpperCAmelCase ,					__UpperCAmelCase ,					__UpperCAmelCase  )







      def       __lowerCAmelCase					(      self      :						int ,					__UpperCAmelCase      :						int ,					__UpperCAmelCase      :						Tuple=None  )    ->Union[str, Any]:




       """simple docstring"""


       if schedule_timesteps is None:
        a		  =      self.timesteps

       a		  =      (schedule_timesteps == timestep).nonzero()

       # The sigma index that is taken for the **very** first `step`
       # is always the second index (or the last index if there is only 1)
       # This way we can ensure we don't accidentally skip a sigma in
       # case we start in the middle of the denoising schedule (e.g. for image-to-image)
       if len(self._index_counter  ) == 0:
        a		  =      1 if len(__UpperCAmelCase  ) > 1 else 0
       else:
        a		  =      timestep.cpu().item() if torch.is_tensor(__UpperCAmelCase  ) else timestep
        a		  =      self._index_counter[timestep_int]

       return indices[pos].item()







      @property
      def       __lowerCAmelCase					(      self      :						List[Any]  )    ->Tuple:




       """simple docstring"""


       if self.config.timestep_spacing in ["linspace", "trailing"]:
        return self.sigmas.max()

       return (self.sigmas.max() ** 2 + 1) ** 0.5







      def       __lowerCAmelCase					(      self      :						Optional[Any] ,					__UpperCAmelCase      :						torch.FloatTensor ,					__UpperCAmelCase      :						Union[float, torch.FloatTensor] ,					)    ->torch.FloatTensor:




       """simple docstring"""


       a		  =      self.index_for_timestep(__UpperCAmelCase  )

       if self.state_in_first_order:
        a		  =      self.sigmas[step_index]
       else:
        a		  =      self.sigmas_interpol[step_index]

       a		  =      sample / ((sigma**2 + 1) ** 0.5)
       return sample







      def       __lowerCAmelCase					(      self      :						int ,					__UpperCAmelCase      :						int ,					__UpperCAmelCase      :						Union[str, torch.device] = None ,					__UpperCAmelCase      :						Optional[int] = None ,					)    ->Any:




       """simple docstring"""


       a		  =      num_inference_steps

       a		  =      num_train_timesteps or self.config.num_train_timesteps

       # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
       if self.config.timestep_spacing == "linspace":
        a		  =      np.linspace(0 ,					num_train_timesteps - 1 ,					__UpperCAmelCase ,					dtype=__UpperCAmelCase  )[::-1].copy()
       elif self.config.timestep_spacing == "leading":
        a		  =      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
        a		  =      (np.arange(0 ,					__UpperCAmelCase  ) * step_ratio).round()[::-1].copy().astype(__UpperCAmelCase  )
        timesteps += self.config.steps_offset
       elif self.config.timestep_spacing == "trailing":
        a		  =      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
        a		  =      (np.arange(__UpperCAmelCase ,					0 ,					-step_ratio  )).round().copy().astype(__UpperCAmelCase  )
        timesteps -= 1
       else:
        raise ValueError(
            F"""{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."""  )

       a		  =      np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5  )
       a		  =      torch.from_numpy(np.log(__UpperCAmelCase  )  ).to(__UpperCAmelCase  )

       a		  =      np.interp(__UpperCAmelCase ,					np.arange(0 ,					len(__UpperCAmelCase  )  ) ,					__UpperCAmelCase  )
       a		  =      np.concatenate([sigmas, [0.0]]  ).astype(np.floataa  )
       a		  =      torch.from_numpy(__UpperCAmelCase  ).to(device=__UpperCAmelCase  )

       # interpolate sigmas
       a		  =      sigmas.log().lerp(sigmas.roll(1  ).log() ,					0.5  ).exp()

       a		  =      torch.cat([sigmas[:1], sigmas[1:].repeat_interleave(2  ), sigmas[-1:]]  )
       a		  =      torch.cat(
           [sigmas_interpol[:1], sigmas_interpol[1:].repeat_interleave(2  ), sigmas_interpol[-1:]]  )

       if str(__UpperCAmelCase  ).startswith('''mps'''  ):
        # mps does not support float64
        a		  =      torch.from_numpy(__UpperCAmelCase  ).to(__UpperCAmelCase ,					dtype=torch.floataa  )
       else:
        a		  =      torch.from_numpy(__UpperCAmelCase  ).to(__UpperCAmelCase  )

       # interpolate timesteps
       a		  =      self.sigma_to_t(__UpperCAmelCase  ).to(__UpperCAmelCase ,					dtype=timesteps.dtype  )
       a		  =      torch.stack((timesteps_interpol[1:-1, None], timesteps[1:, None]) ,					dim=-1  ).flatten()

       a		  =      torch.cat([timesteps[:1], interleaved_timesteps]  )

       a		  =      None

       # for exp beta schedules, such as the one for `pipeline_shap_e.py`
       # we need an index counter
       a		  =      defaultdict(__UpperCAmelCase  )







      def       __lowerCAmelCase					(      self      :						Dict ,					__UpperCAmelCase      :						Optional[Any]  )    ->List[str]:




       """simple docstring"""


       a		  =      sigma.log()

       # get distribution
       a		  =      log_sigma - self.log_sigmas[:, None]

       # get sigmas range
       a		  =      dists.ge(0  ).cumsum(dim=0  ).argmax(dim=0  ).clamp(max=self.log_sigmas.shape[0] - 2  )
       a		  =      low_idx + 1

       a		  =      self.log_sigmas[low_idx]
       a		  =      self.log_sigmas[high_idx]

       # interpolate sigmas
       a		  =      (low - log_sigma) / (low - high)
       a		  =      w.clamp(0 ,					1  )

       # transform interpolation to time range
       a		  =      (1 - w) * low_idx + w * high_idx
       a		  =      t.view(sigma.shape  )
       return t







      @property
      def       __lowerCAmelCase					(      self      :						List[Any]  )    ->Union[str, Any]:




       """simple docstring"""


       return self.sample is None







      def       __lowerCAmelCase					(      self      :						List[Any] ,					__UpperCAmelCase      :						Union[torch.FloatTensor, np.ndarray] ,					__UpperCAmelCase      :						Union[float, torch.FloatTensor] ,					__UpperCAmelCase      :						Union[torch.FloatTensor, np.ndarray] ,					__UpperCAmelCase      :						bool = True ,					)    ->Union[SchedulerOutput, Tuple]:




       """simple docstring"""


       a		  =      self.index_for_timestep(__UpperCAmelCase  )

       # advance index counter by 1
       a		  =      timestep.cpu().item() if torch.is_tensor(__UpperCAmelCase  ) else timestep
       self._index_counter[timestep_int] += 1

       if self.state_in_first_order:
        a		  =      self.sigmas[step_index]
        a		  =      self.sigmas_interpol[step_index + 1]
        a		  =      self.sigmas[step_index + 1]
       else:
        # 2nd order / KDPM2's method
        a		  =      self.sigmas[step_index - 1]
        a		  =      self.sigmas_interpol[step_index]
        a		  =      self.sigmas[step_index]

       # currently only gamma=0 is supported. This usually works best anyways.
       # We can support gamma in the future but then need to scale the timestep before
       # passing it to the model which requires a change in API
       a		  =      0
       a		  =      sigma * (gamma + 1)  # Note: sigma_hat == sigma for now

       # 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
       if self.config.prediction_type == "epsilon":
        a		  =      sigma_hat if self.state_in_first_order else sigma_interpol
        a		  =      sample - sigma_input * model_output
       elif self.config.prediction_type == "v_prediction":
        a		  =      sigma_hat if self.state_in_first_order else sigma_interpol
        a		  =      model_output * (-sigma_input / (sigma_input**2 + 1) ** 0.5) + (
            sample / (sigma_input**2 + 1)
        )
       elif self.config.prediction_type == "sample":
        raise NotImplementedError('''prediction_type not implemented yet: sample'''  )
       else:
        raise ValueError(
            F"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"""  )

       if self.state_in_first_order:
        # 2. Convert to an ODE derivative for 1st order
        a		  =      (sample - pred_original_sample) / sigma_hat
        # 3. delta timestep
        a		  =      sigma_interpol - sigma_hat

        # store for 2nd order step
        a		  =      sample
       else:
        # DPM-Solver-2
        # 2. Convert to an ODE derivative for 2nd order
        a		  =      (sample - pred_original_sample) / sigma_interpol

        # 3. delta timestep
        a		  =      sigma_next - sigma_hat

        a		  =      self.sample
        a		  =      None

       a		  =      sample + derivative * dt

       if not return_dict:
        return (prev_sample,)

       return SchedulerOutput(prev_sample=__UpperCAmelCase  )







      def       __lowerCAmelCase					(      self      :						Dict ,					__UpperCAmelCase      :						torch.FloatTensor ,					__UpperCAmelCase      :						torch.FloatTensor ,					__UpperCAmelCase      :						torch.FloatTensor ,					)    ->torch.FloatTensor:




       """simple docstring"""


       a		  =      self.sigmas.to(device=original_samples.device ,					dtype=original_samples.dtype  )
       if original_samples.device.type == "mps" and torch.is_floating_point(__UpperCAmelCase  ):
        # mps does not support float64
        a		  =      self.timesteps.to(original_samples.device ,					dtype=torch.floataa  )
        a		  =      timesteps.to(original_samples.device ,					dtype=torch.floataa  )
       else:
        a		  =      self.timesteps.to(original_samples.device  )
        a		  =      timesteps.to(original_samples.device  )

       a		  =      [self.index_for_timestep(__UpperCAmelCase ,					__UpperCAmelCase  ) for t in timesteps]

       a		  =      sigmas[step_indices].flatten()
       while len(sigma.shape  ) < len(original_samples.shape  ):
        a		  =      sigma.unsqueeze(-1  )

       a		  =      original_samples + noise * sigma
       return noisy_samples







      def __len__(      self      :						Dict  )    ->Any:




       """simple docstring"""


       return self.config.num_train_timesteps










from math import factorial

UpperCAmelCase__				   =		{str(digit): factorial(digit) for digit in range(10)}
def 						_a    (      a :int )							->			int:
 if not isinstance(a	,							a ):
  raise TypeError('''Parameter number must be int''' )

 if number < 0:
  raise ValueError('''Parameter number must be greater than or equal to 0''' )

 # Converts number in string to iterate on its digits and adds its factorial.
 return sum(DIGIT_FACTORIAL[digit] for digit in str(a ) )
def 						_a    (      a :int = 60	,							a :int = 1_000_000 )							->			int:

 if not isinstance(a	,							a ) or not isinstance(a	,							a ):
  raise TypeError('''Parameters chain_length and number_limit must be int''' )

 if chain_length <= 0 or number_limit <= 0:
  raise ValueError(
      '''Parameters chain_length and number_limit must be greater than 0''' )

 # the counter for the chains with the exact desired length
 a		  =      0
 # the cached sizes of the previous chains
 a		  =      {}

 for start_chain_element in range(1	,							a ):
  # The temporary set will contain the elements of the chain
  a		  =      set()
  a		  =      0

  # Stop computing the chain when you find a cached size, a repeating item or the
  # length is greater then the desired one.
  a		  =      start_chain_element
  while (
      chain_element not in chain_sets_lengths
      and chain_element not in chain_set
      and chain_set_length <= chain_length
  ):
   chain_set.add(a )
   chain_set_length += 1
   a		  =      digit_factorial_sum(a )

  if chain_element in chain_sets_lengths:
   chain_set_length += chain_sets_lengths[chain_element]

  a		  =      chain_set_length

  # If chain contains the exact amount of elements increase the counter
  if chain_set_length == chain_length:
   chains_counter += 1

 return chains_counter


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     print(f"""{solution()}""")


