






"""simple docstring"""


# 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		(							_lowerCAmelCase						):


   '''simple docstring'''

   A_						:	torch.FloatTensor
   A_						:	torch.FloatTensor





class 					snake_case		(							_lowerCAmelCase   ,	_lowerCAmelCase						):


   '''simple docstring'''

   A_						:	List[str]						 = 1


   @register_to_config
   def __init__(	self :     str,       _lowerCamelCase :     int = 20_00,       _lowerCamelCase :     float = 0.15,       _lowerCamelCase :     float = 0.01,       _lowerCamelCase :     float = 13_48.0,       _lowerCamelCase :     float = 1e-5,       _lowerCamelCase :     int = 1,       ):

     '''simple docstring'''
     # standard deviation of the initial noise distribution
     __A									= sigma_max

     # setable values
     __A									= None

     self.set_sigmas(_lowerCamelCase,       _lowerCamelCase,       _lowerCamelCase,       _lowerCamelCase  )


   def 				_SCREAMING_SNAKE_CASE				(	self :     Any,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     Optional[int] = None  ):

     '''simple docstring'''
     return sample


   def 				_SCREAMING_SNAKE_CASE				(	self :     str,       _lowerCamelCase :     int,       _lowerCamelCase :     float = None,       _lowerCamelCase :     Union[str, torch.device] = None  ):

     '''simple docstring'''
     __A									= sampling_eps if sampling_eps is not None else self.config.sampling_eps

     __A									= torch.linspace(1,       _lowerCamelCase,       _lowerCamelCase,       device=_lowerCamelCase  )


   def 				_SCREAMING_SNAKE_CASE				(	self :     List[str],       _lowerCamelCase :     int,       _lowerCamelCase :     float = None,       _lowerCamelCase :     float = None,       _lowerCamelCase :     float = None  ):

     '''simple docstring'''
     __A									= sigma_min if sigma_min is not None else self.config.sigma_min
     __A									= sigma_max if sigma_max is not None else self.config.sigma_max
     __A									= sampling_eps if sampling_eps is not None else self.config.sampling_eps
     if self.timesteps is None:
       self.set_timesteps(_lowerCamelCase,       _lowerCamelCase  )

     __A									= sigma_min * (sigma_max / sigma_min) ** (self.timesteps / sampling_eps)
     __A									= torch.exp(torch.linspace(math.log(_lowerCamelCase  ),       math.log(_lowerCamelCase  ),       _lowerCamelCase  )  )
     __A									= torch.tensor([sigma_min * (sigma_max / sigma_min) ** t for t in self.timesteps]  )


   def 				_SCREAMING_SNAKE_CASE				(	self :     str,       _lowerCamelCase :     Union[str, Any],       _lowerCamelCase :     Union[str, Any]  ):

     '''simple docstring'''
     return torch.where(
         timesteps == 0,       torch.zeros_like(t.to(timesteps.device  )  ),       self.discrete_sigmas[timesteps - 1].to(timesteps.device  ),       )


   def 				_SCREAMING_SNAKE_CASE				(	self :     Tuple,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     int,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     Optional[torch.Generator] = None,       _lowerCamelCase :     bool = 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'''  )

     __A									= timestep * torch.ones(
         sample.shape[0],       device=sample.device  )  # torch.repeat_interleave(timestep, sample.shape[0])
     __A									= (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
     __A									= timesteps.to(self.discrete_sigmas.device  )

     __A									= self.discrete_sigmas[timesteps].to(sample.device  )
     __A									= self.get_adjacent_sigma(_lowerCamelCase,       _lowerCamelCase  ).to(sample.device  )
     __A									= torch.zeros_like(_lowerCamelCase  )
     __A									= (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
     __A									= diffusion.flatten()
     while len(diffusion.shape  ) < len(sample.shape  ):
       __A									= diffusion.unsqueeze(-1  )
     __A									= drift - diffusion**2 * model_output

     #  equation 6: sample noise for the diffusion term of
     __A									= randn_tensor(
         sample.shape,       layout=sample.layout,       generator=_lowerCamelCase,       device=sample.device,       dtype=sample.dtype  )
     __A									= sample - drift  # subtract because `dt` is a small negative timestep
     # TODO is the variable diffusion the correct scaling term for the noise?
     __A									= 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=_lowerCamelCase,       prev_sample_mean=_lowerCamelCase  )


   def 				_SCREAMING_SNAKE_CASE				(	self :     Optional[int],       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     Optional[torch.Generator] = None,       _lowerCamelCase :     bool = 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
     __A									= randn_tensor(sample.shape,       layout=sample.layout,       generator=_lowerCamelCase  ).to(sample.device  )

     # compute step size from the model_output, the noise, and the snr
     __A									= torch.norm(model_output.reshape(model_output.shape[0],       -1  ),       dim=-1  ).mean()
     __A									= torch.norm(noise.reshape(noise.shape[0],       -1  ),       dim=-1  ).mean()
     __A									= (self.config.snr * noise_norm / grad_norm) ** 2 * 2
     __A									= 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
     __A									= step_size.flatten()
     while len(step_size.shape  ) < len(sample.shape  ):
       __A									= step_size.unsqueeze(-1  )
     __A									= sample + step_size * model_output
     __A									= prev_sample_mean + ((step_size * 2) ** 0.5) * noise

     if not return_dict:
       return (prev_sample,)

     return SchedulerOutput(prev_sample=_lowerCamelCase  )


   def 				_SCREAMING_SNAKE_CASE				(	self :     Any,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     torch.FloatTensor,       _lowerCamelCase :     torch.FloatTensor,       ):

     '''simple docstring'''
     # Make sure sigmas and timesteps have the same device and dtype as original_samples
     __A									= timesteps.to(original_samples.device  )
     __A									= self.discrete_sigmas.to(original_samples.device  )[timesteps]
     __A									= (
         noise * sigmas[:, None, None, None]
         if noise is not None
         else torch.randn_like(_lowerCamelCase  ) * sigmas[:, None, None, None]
     )
     __A									= noise + original_samples
     return noisy_samples






   def __len__(	self :     Optional[Any]  ):

     '''simple docstring'''
     return self.config.num_train_timesteps












"""simple docstring"""


from collections import defaultdict
from math import ceil, sqrt



def 					lowerCAmelCase				(    __UpperCamelCase = 1_0_0_0_0_0_0		,       __UpperCamelCase = 1_0 ):




  """simple docstring"""







  __A									= defaultdict(__UpperCamelCase )

  for outer_width in range(3		,       (t_limit // 4) + 2 ):
    if outer_width * outer_width > t_limit:
      __A									= max(
          ceil(sqrt(outer_width * outer_width - t_limit ) )		,       1 )
    else:
      __A									= 1

    hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

    for hole_width in range(__UpperCamelCase		,       outer_width - 1		,       2 ):
      count[outer_width * outer_width - hole_width * hole_width] += 1

  return sum(1 for n in count.values() if 1 <= n <= 1_0 )


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




