code: 
  
  
  '''simple docstring'''
  
  
  
  import math
  from dataclasses import dataclass
  from typing import List, Optional, Tuple, Union
  
  import numpy as np
  import torch
  
  from ..configuration_utils import ConfigMixin, register_to_config
  from ..utils import BaseOutput, logging, randn_tensor
  from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin
  
  
  __lowercase			:		Tuple =			logging.get_logger(__name__)  # pylint: disable=invalid-name
  
  
  
  
  
  @dataclass
  # Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->EulerDiscrete
  class 							__UpperCamelCase       (	lowerCAmelCase_      ):
  	A_			    =					42
  	A_			    =					None
  
  
  
  
  
  
  def 						lowerCamelCase							(_SCREAMING_SNAKE_CASE   :     Any		,   _SCREAMING_SNAKE_CASE   :     Tuple=0.9_9_9		,   _SCREAMING_SNAKE_CASE   :     Optional[int]="cosine"		,   ):
  					if alpha_transform_type == "cosine":
  
  										def alpha_bar_fn(_SCREAMING_SNAKE_CASE   :     List[str]     ):
  															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(_SCREAMING_SNAKE_CASE   :     Optional[Any]     ):
  															return math.exp(t * -1_2.0     )
  
  					else:
  										raise ValueError(F"""Unsupported alpha_tranform_type: {alpha_transform_type}"""     )
  
  					__a    : Any      					=						[]
  					for i in range(_SCREAMING_SNAKE_CASE     ):
  										__a    : Optional[int]      					=						i / num_diffusion_timesteps
  										__a    : List[str]      					=						(i + 1) / num_diffusion_timesteps
  										betas.append(min(1 - alpha_bar_fn(_SCREAMING_SNAKE_CASE     ) / alpha_bar_fn(_SCREAMING_SNAKE_CASE     )		,   _SCREAMING_SNAKE_CASE     )     )
  					return torch.tensor(_SCREAMING_SNAKE_CASE		,   dtype=torch.floataa     )
  
  
  
  
  
  class 							__UpperCamelCase       (	lowerCAmelCase_	,   lowerCAmelCase_      ):
  	A_			    =					[e.name for e in KarrasDiffusionSchedulers]
  	A_			    =					1
  
  
  
  
  	@register_to_config
  	def __init__(     self		,       __a = 1000		,       __a = 0.0001		,       __a = 0.02		,       __a = "linear"		,       __a = None		,       __a = "epsilon"		,       __a = "linear"		,       __a = False		,       __a = "linspace"		,       __a = 0		,       ):
  
  						'''simple docstring'''
  
  
  
  
  
  						if trained_betas is not None:
  											__a    : Any      					=						torch.tensor(__a		,       dtype=torch.floataa							)
  						elif beta_schedule == "linear":
  											__a    : Dict      					=						torch.linspace(__a		,       __a		,       __a		,       dtype=torch.floataa							)
  						elif beta_schedule == "scaled_linear":
  											# this schedule is very specific to the latent diffusion model.
  											__a    : Any      					=						(
  											    torch.linspace(beta_start**0.5		,       beta_end**0.5		,       __a		,       dtype=torch.floataa							) ** 2
  											)
  						elif beta_schedule == "squaredcos_cap_v2":
  											# Glide cosine schedule
  											__a    : int      					=						betas_for_alpha_bar(__a							)
  						else:
  											raise NotImplementedError(f"""{beta_schedule} does is not implemented for {self.__class__}"""							)
  
  						__a    : List[Any]      					=						1.0 - self.betas
  						__a    : int      					=						torch.cumprod(self.alphas		,       dim=0							)
  
  						__a    : List[Any]      					=						np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5							)
  						__a    : List[Any]      					=						np.concatenate([sigmas[::-1], [0.0]]							).astype(np.floataa							)
  						__a    : Union[str, Any]      					=						torch.from_numpy(__a							)
  
  						# setable values
  						__a    : List[str]      					=						None
  						__a    : List[Any]      					=						np.linspace(0		,       num_train_timesteps - 1		,       __a		,       dtype=__a							)[::-1].copy()
  						__a    : Optional[Any]      					=						torch.from_numpy(__a							)
  						__a    : Any      					=						False
  						__a    : str      					=						use_karras_sigmas
  
  
  
  
  	@property
  	def    __UpperCAmelCase							(     self							):
  
  						'''simple docstring'''
  
  
  
  
  
  						if self.config.timestep_spacing in ["linspace", "trailing"]:
  											return self.sigmas.max()
  
  						return (self.sigmas.max() ** 2 + 1) ** 0.5
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a							):
  
  						'''simple docstring'''
  
  
  
  
  
  						if isinstance(__a		,       torch.Tensor							):
  											__a    : Optional[int]      					=						timestep.to(self.timesteps.device							)
  						__a    : Optional[Any]      					=						(self.timesteps == timestep).nonzero().item()
  						__a    : List[str]      					=						self.sigmas[step_index]
  
  						__a    : int      					=						sample / ((sigma**2 + 1) ** 0.5)
  
  						__a    : Union[str, Any]      					=						True
  						return sample
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a = None							):
  
  						'''simple docstring'''
  
  
  
  
  
  						__a    : Dict      					=						num_inference_steps
  
  						# "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
  						if self.config.timestep_spacing == "linspace":
  											__a    : Optional[Any]      					=						np.linspace(0		,       self.config.num_train_timesteps - 1		,       __a		,       dtype=__a							)[
  											    ::-1
  											].copy()
  						elif self.config.timestep_spacing == "leading":
  											__a    : str      					=						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
  											__a    : int      					=						(np.arange(0		,       __a							) * step_ratio).round()[::-1].copy().astype(__a							)
  											timesteps += self.config.steps_offset
  						elif self.config.timestep_spacing == "trailing":
  											__a    : Optional[int]      					=						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
  											__a    : str      					=						(np.arange(self.config.num_train_timesteps		,       0		,       -step_ratio							)).round().copy().astype(__a							)
  											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    : Tuple      					=						np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5							)
  						__a    : Tuple      					=						np.log(__a							)
  
  						if self.config.interpolation_type == "linear":
  											__a    : Optional[Any]      					=						np.interp(__a		,       np.arange(0		,       len(__a							)							)		,       __a							)
  						elif self.config.interpolation_type == "log_linear":
  											__a    : Optional[Any]      					=						torch.linspace(np.log(sigmas[-1]							)		,       np.log(sigmas[0]							)		,       num_inference_steps + 1							).exp()
  						else:
  											raise ValueError(
  											    f"""{self.config.interpolation_type} is not implemented. Please specify interpolation_type to either"""
  											    ' \'linear\' or \'log_linear\''							)
  
  						if self.use_karras_sigmas:
  											__a    : Any      					=						self._convert_to_karras(in_sigmas=__a		,       num_inference_steps=self.num_inference_steps							)
  											__a    : Tuple      					=						np.array([self._sigma_to_t(__a		,       __a							) for sigma in sigmas]							)
  
  						__a    : str      					=						np.concatenate([sigmas, [0.0]]							).astype(np.floataa							)
  						__a    : List[Any]      					=						torch.from_numpy(__a							).to(device=__a							)
  						if str(__a							).startswith('mps'							):
  											# mps does not support float64
  											__a    : int      					=						torch.from_numpy(__a							).to(__a		,       dtype=torch.floataa							)
  						else:
  											__a    : int      					=						torch.from_numpy(__a							).to(device=__a							)
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a							):
  
  						'''simple docstring'''
  
  
  
  
  
  						__a    : Any      					=						np.log(__a							)
  
  						# get distribution
  						__a    : Dict      					=						log_sigma - log_sigmas[:, np.newaxis]
  
  						# get sigmas range
  						__a    : Optional[Any]      					=						np.cumsum((dists >= 0)		,       axis=0							).argmax(axis=0							).clip(max=log_sigmas.shape[0] - 2							)
  						__a    : int      					=						low_idx + 1
  
  						__a    : Tuple      					=						log_sigmas[low_idx]
  						__a    : Any      					=						log_sigmas[high_idx]
  
  						# interpolate sigmas
  						__a    : Dict      					=						(low - log_sigma) / (low - high)
  						__a    : List[Any]      					=						np.clip(__a		,       0		,       1							)
  
  						# transform interpolation to time range
  						__a    : Union[str, Any]      					=						(1 - w) * low_idx + w * high_idx
  						__a    : Optional[Any]      					=						t.reshape(sigma.shape							)
  						return t
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a							):
  
  						'''simple docstring'''
  
  
  
  
  
  						__a    : float      					=						in_sigmas[-1].item()
  						__a    : float      					=						in_sigmas[0].item()
  
  						__a    : Union[str, Any]      					=						7.0  # 7.0 is the value used in the paper
  						__a    : int      					=						np.linspace(0		,       1		,       __a							)
  						__a    : int      					=						sigma_min ** (1 / rho)
  						__a    : Optional[Any]      					=						sigma_max ** (1 / rho)
  						__a    : Optional[Any]      					=						(max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
  						return sigmas
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a		,       __a		,       __a = 0.0		,       __a = 0.0		,       __a = float('inf'							)		,       __a = 1.0		,       __a = None		,       __a = True		,       ):
  
  						'''simple docstring'''
  
  
  
  
  
  						if (
  						    isinstance(__a		,       __a							)
  						    or isinstance(__a		,       torch.IntTensor							)
  						    or isinstance(__a		,       torch.LongTensor							)
  						):
  											raise ValueError(
  											    (
  											        'Passing integer indices (e.g. from `enumerate(timesteps)`) as timesteps to'
  											        ' `EulerDiscreteScheduler.step()` is not supported. Make sure to pass'
  											        ' one of the `scheduler.timesteps` as a timestep.'
  											    )		,       )
  
  						if not self.is_scale_input_called:
  											logger.warning(
  											    'The `scale_model_input` function should be called before `step` to ensure correct denoising. '
  											    'See `StableDiffusionPipeline` for a usage example.'							)
  
  						if isinstance(__a		,       torch.Tensor							):
  											__a    : Optional[int]      					=						timestep.to(self.timesteps.device							)
  
  						__a    : str      					=						(self.timesteps == timestep).nonzero().item()
  						__a    : List[Any]      					=						self.sigmas[step_index]
  
  						__a    : List[Any]      					=						min(s_churn / (len(self.sigmas							) - 1)		,       2**0.5 - 1							) if s_tmin <= sigma <= s_tmax else 0.0
  
  						__a    : Optional[Any]      					=						randn_tensor(
  						    model_output.shape		,       dtype=model_output.dtype		,       device=model_output.device		,       generator=__a							)
  
  						__a    : Optional[Any]      					=						noise * s_noise
  						__a    : int      					=						sigma * (gamma + 1)
  
  						if gamma > 0:
  											__a    : Optional[Any]      					=						sample + eps * (sigma_hat**2 - sigma**2) ** 0.5
  
  						# 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
  						# NOTE: "original_sample" should not be an expected prediction_type but is left in for
  						# backwards compatibility
  						if self.config.prediction_type == "original_sample" or self.config.prediction_type == "sample":
  											__a    : Optional[Any]      					=						model_output
  						elif self.config.prediction_type == "epsilon":
  											__a    : Tuple      					=						sample - sigma_hat * model_output
  						elif self.config.prediction_type == "v_prediction":
  											# * c_out + input * c_skip
  											__a    : List[Any]      					=						model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
  						else:
  											raise ValueError(
  											    f"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"""							)
  
  						# 2. Convert to an ODE derivative
  						__a    : List[Any]      					=						(sample - pred_original_sample) / sigma_hat
  
  						__a    : Union[str, Any]      					=						self.sigmas[step_index + 1] - sigma_hat
  
  						__a    : Tuple      					=						sample + derivative * dt
  
  						if not return_dict:
  											return (prev_sample,)
  
  						return EulerDiscreteSchedulerOutput(prev_sample=__a		,       pred_original_sample=__a							)
  
  
  
  
  	def    __UpperCAmelCase							(     self		,       __a		,       __a		,       __a		,       ):
  
  						'''simple docstring'''
  
  
  
  
  
  						__a    : Optional[int]      					=						self.sigmas.to(device=original_samples.device		,       dtype=original_samples.dtype							)
  						if original_samples.device.type == "mps" and torch.is_floating_point(__a							):
  											# mps does not support float64
  											__a    : int      					=						self.timesteps.to(original_samples.device		,       dtype=torch.floataa							)
  											__a    : int      					=						timesteps.to(original_samples.device		,       dtype=torch.floataa							)
  						else:
  											__a    : Optional[Any]      					=						self.timesteps.to(original_samples.device							)
  											__a    : Any      					=						timesteps.to(original_samples.device							)
  
  						__a    : Tuple      					=						[(schedule_timesteps == t).nonzero().item() for t in timesteps]
  
  						__a    : str      					=						sigmas[step_indices].flatten()
  						while len(sigma.shape							) < len(original_samples.shape							):
  											__a    : str      					=						sigma.unsqueeze(-1							)
  
  						__a    : Optional[Any]      					=						original_samples + noise * sigma
  						return noisy_samples
  
  
  
  
  
  	def __len__(     self							):
  
  						'''simple docstring'''
  
  
  
  
  
  						return self.config.num_train_timesteps
  
  
  
  
style_context: 
  
  
  '''simple docstring'''
  
  
  
  import itertools
  
  
  
  
  
  
  def 						lowerCamelCase							(_SCREAMING_SNAKE_CASE   :     int = 1_000_000     ):
  					__a    : str      					=						[1]
  
  					for i in itertools.count(len(_SCREAMING_SNAKE_CASE     )     ):
  										__a    : Dict      					=						0
  										for j in itertools.count(1     ):
  															__a    : int      					=						-1 if j % 2 == 0 else +1
  															__a    : Optional[int]      					=						(j * j * 3 - j) // 2
  															if index > i:
  																				break
  															item += partitions[i - index] * sign
  															item %= number
  															index += j
  															if index > i:
  																				break
  															item += partitions[i - index] * sign
  															item %= number
  
  										if item == 0:
  															return i
  										partitions.append(_SCREAMING_SNAKE_CASE     )
  
  					return 0
  
  
  if __name__ == "__main__":
  	import doctest
  
  	doctest.testmod()
  
  	print(f'''{solution() = }''')
  
  
  
  