code: 
  
  
  """simple docstring"""
  
  
  
  
  import math
  import warnings
  from dataclasses import dataclass
  from typing import List, Optional, Tuple, Union
  
  import numpy as np
  import torch
  from scipy import integrate
  
  from ..configuration_utils import ConfigMixin, register_to_config
  from ..utils import BaseOutput
  from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin
  
  
  
  
  @dataclass
  # Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->LMSDiscrete
  class  __snake_case      (lowerCamelCase     ):
  							__a				   =					42
  							__a				   =					None
  
  
  
  
  
  def 			a_   (  lowercase__   :List[Any], lowercase__   :str=0.999, lowercase__   :Tuple="cosine", ):
  
  
  
  
  
  
  			if alpha_transform_type == "cosine":
  
  						def alpha_bar_fn(lowercase__   :str    ):
  									return math.cos((t + 0.008) / 1.008 * math.pi / 2    ) ** 2
  
  			elif alpha_transform_type == "exp":
  
  						def alpha_bar_fn(lowercase__   :Tuple    ):
  									return math.exp(t * -12.0    )
  
  			else:
  						raise ValueError(f'Unsupported alpha_tranform_type: {alpha_transform_type}'    )
  
  			__lowerCamelCase        =			[]
  			for i in range(_UpperCAmelCase    ):
  						__lowerCamelCase        =			i / num_diffusion_timesteps
  						__lowerCamelCase        =			(i + 1) / num_diffusion_timesteps
  						betas.append(min(1 - alpha_bar_fn(_UpperCAmelCase    ) / alpha_bar_fn(_UpperCAmelCase    ), _UpperCAmelCase    )    )
  			return torch.tensor(_UpperCAmelCase, dtype=torch.floataa    )
  
  
  
  
  class  __snake_case      (lowerCamelCase ,		lowerCamelCase     ):
  							__a				   =					[e.name for e in KarrasDiffusionSchedulers]
  							__a				   =					1
  
  
  
  
  							@register_to_config
  							def __init__(    self:						List[str]    ,       A_:						int = 10_00    ,       A_:						float = 0.0_001    ,       A_:						float = 0.02    ,       A_:						str = "linear"    ,       A_:						Optional[Union[np.ndarray, List[float]]] = None    ,       A_:						Optional[bool] = False    ,       A_:						str = "epsilon"    ,       A_:						str = "linspace"    ,       A_:						int = 0    ,       ):
  										if trained_betas is not None:
  													__lowerCamelCase        =			torch.tensor(UpperCAmelCase_    ,       dtype=torch.floataa      )
  										elif beta_schedule == "linear":
  													__lowerCamelCase        =			torch.linspace(UpperCAmelCase_    ,       UpperCAmelCase_    ,       UpperCAmelCase_    ,       dtype=torch.floataa      )
  										elif beta_schedule == "scaled_linear":
  													# this schedule is very specific to the latent diffusion model.
  													__lowerCamelCase        =			(
  													    torch.linspace(beta_start**0.5    ,       beta_end**0.5    ,       UpperCAmelCase_    ,       dtype=torch.floataa      ) ** 2
  													)
  										elif beta_schedule == "squaredcos_cap_v2":
  													# Glide cosine schedule
  													__lowerCamelCase        =			betas_for_alpha_bar(UpperCAmelCase_      )
  										else:
  													raise NotImplementedError(f'{beta_schedule} does is not implemented for {self.__class__}'      )
  
  										__lowerCamelCase        =			1.0 - self.betas
  										__lowerCamelCase        =			torch.cumprod(self.alphas    ,       dim=0      )
  
  										__lowerCamelCase        =			np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5      )
  										__lowerCamelCase        =			np.concatenate([sigmas[::-1], [0.0]]      ).astype(np.floataa      )
  										__lowerCamelCase        =			torch.from_numpy(UpperCAmelCase_      )
  
  										# setable values
  										__lowerCamelCase        =			None
  										__lowerCamelCase        =			use_karras_sigmas
  										self.set_timesteps(UpperCAmelCase_    ,       UpperCAmelCase_      )
  										__lowerCamelCase        =			[]
  										__lowerCamelCase        =			False
  
  
  
  
  							@property
  							def      __a      (    self:						Dict      ):
  										# standard deviation of the initial noise distribution
  										if self.config.timestep_spacing in ["linspace", "trailing"]:
  													return self.sigmas.max()
  
  										return (self.sigmas.max() ** 2 + 1) ** 0.5
  
  
  
  
  							def      __a      (    self:						Any    ,       A_:						torch.FloatTensor    ,       A_:						Union[float, torch.FloatTensor]      ):
  										if isinstance(UpperCAmelCase_    ,       torch.Tensor      ):
  													__lowerCamelCase        =			timestep.to(self.timesteps.device      )
  										__lowerCamelCase        =			(self.timesteps == timestep).nonzero().item()
  										__lowerCamelCase        =			self.sigmas[step_index]
  										__lowerCamelCase        =			sample / ((sigma**2 + 1) ** 0.5)
  										__lowerCamelCase        =			True
  										return sample
  
  
  
  
  							def      __a      (    self:						Any    ,       A_:						Any    ,       A_:						Tuple    ,       A_:						Optional[Any]      ):
  
  										def lms_derivative(A_:						int      ):
  													__lowerCamelCase        =			1.0
  													for k in range(UpperCAmelCase_      ):
  																if current_order == k:
  																			continue
  																prod *= (tau - self.sigmas[t - k]) / (self.sigmas[t - current_order] - self.sigmas[t - k])
  													return prod
  
  										__lowerCamelCase        =			integrate.quad(UpperCAmelCase_    ,       self.sigmas[t]    ,       self.sigmas[t + 1]    ,       epsrel=1E-4      )[0]
  
  										return integrated_coeff
  
  
  
  
  							def      __a      (    self:						int    ,       A_:						int    ,       A_:						Union[str, torch.device] = None      ):
  										__lowerCamelCase        =			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":
  													__lowerCamelCase        =			np.linspace(0    ,       self.config.num_train_timesteps - 1    ,       UpperCAmelCase_    ,       dtype=UpperCAmelCase_      )[
  													    ::-1
  													].copy()
  										elif self.config.timestep_spacing == "leading":
  													__lowerCamelCase        =			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
  													__lowerCamelCase        =			(np.arange(0    ,       UpperCAmelCase_      ) * step_ratio).round()[::-1].copy().astype(UpperCAmelCase_      )
  													timesteps += self.config.steps_offset
  										elif self.config.timestep_spacing == "trailing":
  													__lowerCamelCase        =			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
  													__lowerCamelCase        =			(np.arange(self.config.num_train_timesteps    ,       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\'.'      )
  
  										__lowerCamelCase        =			np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5      )
  										__lowerCamelCase        =			np.log(UpperCAmelCase_      )
  										__lowerCamelCase        =			np.interp(UpperCAmelCase_    ,       np.arange(0    ,       len(UpperCAmelCase_      )      )    ,       UpperCAmelCase_      )
  
  										if self.use_karras_sigmas:
  													__lowerCamelCase        =			self._convert_to_karras(in_sigmas=UpperCAmelCase_      )
  													__lowerCamelCase        =			np.array([self._sigma_to_t(UpperCAmelCase_    ,       UpperCAmelCase_      ) for sigma in sigmas]      )
  
  										__lowerCamelCase        =			np.concatenate([sigmas, [0.0]]      ).astype(np.floataa      )
  
  										__lowerCamelCase        =			torch.from_numpy(UpperCAmelCase_      ).to(device=UpperCAmelCase_      )
  										if str(UpperCAmelCase_      ).startswith("""mps"""      ):
  													# mps does not support float64
  													__lowerCamelCase        =			torch.from_numpy(UpperCAmelCase_      ).to(UpperCAmelCase_    ,       dtype=torch.floataa      )
  										else:
  													__lowerCamelCase        =			torch.from_numpy(UpperCAmelCase_      ).to(device=UpperCAmelCase_      )
  
  										__lowerCamelCase        =			[]
  
  
  
  
  							def      __a      (    self:						str    ,       A_:						Optional[Any]    ,       A_:						List[str]      ):
  										# get log sigma
  										__lowerCamelCase        =			np.log(UpperCAmelCase_      )
  
  										# get distribution
  										__lowerCamelCase        =			log_sigma - log_sigmas[:, np.newaxis]
  
  										# get sigmas range
  										__lowerCamelCase        =			np.cumsum((dists >= 0)    ,       axis=0      ).argmax(axis=0      ).clip(max=log_sigmas.shape[0] - 2      )
  										__lowerCamelCase        =			low_idx + 1
  
  										__lowerCamelCase        =			log_sigmas[low_idx]
  										__lowerCamelCase        =			log_sigmas[high_idx]
  
  										# interpolate sigmas
  										__lowerCamelCase        =			(low - log_sigma) / (low - high)
  										__lowerCamelCase        =			np.clip(UpperCAmelCase_    ,       0    ,       1      )
  
  										# transform interpolation to time range
  										__lowerCamelCase        =			(1 - w) * low_idx + w * high_idx
  										__lowerCamelCase        =			t.reshape(sigma.shape      )
  										return t
  
  
  
  
  							def      __a      (    self:						Any    ,       A_:						torch.FloatTensor      ):
  										__lowerCamelCase        =			in_sigmas[-1].item()
  										__lowerCamelCase        =			in_sigmas[0].item()
  
  										__lowerCamelCase        =			7.0  # 7.0 is the value used in the paper
  										__lowerCamelCase        =			np.linspace(0    ,       1    ,       self.num_inference_steps      )
  										__lowerCamelCase        =			sigma_min ** (1 / rho)
  										__lowerCamelCase        =			sigma_max ** (1 / rho)
  										__lowerCamelCase        =			(max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
  										return sigmas
  
  
  
  
  							def      __a      (    self:						Optional[int]    ,       A_:						torch.FloatTensor    ,       A_:						Union[float, torch.FloatTensor]    ,       A_:						torch.FloatTensor    ,       A_:						int = 4    ,       A_:						bool = True    ,       ):
  										if not self.is_scale_input_called:
  													warnings.warn(
  													    """The `scale_model_input` function should be called before `step` to ensure correct denoising. """
  													    """See `StableDiffusionPipeline` for a usage example."""      )
  
  										if isinstance(UpperCAmelCase_    ,       torch.Tensor      ):
  													__lowerCamelCase        =			timestep.to(self.timesteps.device      )
  										__lowerCamelCase        =			(self.timesteps == timestep).nonzero().item()
  										__lowerCamelCase        =			self.sigmas[step_index]
  
  										# 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
  										if self.config.prediction_type == "epsilon":
  													__lowerCamelCase        =			sample - sigma * model_output
  										elif self.config.prediction_type == "v_prediction":
  													# * c_out + input * c_skip
  													__lowerCamelCase        =			model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
  										elif self.config.prediction_type == "sample":
  													__lowerCamelCase        =			model_output
  										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
  										__lowerCamelCase        =			(sample - pred_original_sample) / sigma
  										self.derivatives.append(UpperCAmelCase_      )
  										if len(self.derivatives      ) > order:
  													self.derivatives.pop(0      )
  
  										# 3. Compute linear multistep coefficients
  										__lowerCamelCase        =			min(step_index + 1    ,       UpperCAmelCase_      )
  										__lowerCamelCase        =			[self.get_lms_coefficient(UpperCAmelCase_    ,       UpperCAmelCase_    ,       UpperCAmelCase_      ) for curr_order in range(UpperCAmelCase_      )]
  
  										# 4. Compute previous sample based on the derivatives path
  										__lowerCamelCase        =			sample + sum(
  										    coeff * derivative for coeff, derivative in zip(UpperCAmelCase_    ,       reversed(self.derivatives      )      )      )
  
  										if not return_dict:
  													return (prev_sample,)
  
  										return LMSDiscreteSchedulerOutput(prev_sample=UpperCAmelCase_    ,       pred_original_sample=UpperCAmelCase_      )
  
  
  
  
  							def      __a      (    self:						Optional[int]    ,       A_:						torch.FloatTensor    ,       A_:						torch.FloatTensor    ,       A_:						torch.FloatTensor    ,       ):
  										# Make sure sigmas and timesteps have the same device and dtype as original_samples
  										__lowerCamelCase        =			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
  													__lowerCamelCase        =			self.timesteps.to(original_samples.device    ,       dtype=torch.floataa      )
  													__lowerCamelCase        =			timesteps.to(original_samples.device    ,       dtype=torch.floataa      )
  										else:
  													__lowerCamelCase        =			self.timesteps.to(original_samples.device      )
  													__lowerCamelCase        =			timesteps.to(original_samples.device      )
  
  										__lowerCamelCase        =			[(schedule_timesteps == t).nonzero().item() for t in timesteps]
  
  										__lowerCamelCase        =			sigmas[step_indices].flatten()
  										while len(sigma.shape      ) < len(original_samples.shape      ):
  													__lowerCamelCase        =			sigma.unsqueeze(-1      )
  
  										__lowerCamelCase        =			original_samples + noise * sigma
  										return noisy_samples
  
  
  
  
  							def __len__(    self:						Dict      ):
  										return self.config.num_train_timesteps
  
  
  
  
  
style_context: 
  
  
  """simple docstring"""
  
  
  
  
  def 			a_   (  lowercase__   :str, lowercase__   :int = 13    ):
  			__lowerCamelCase        =			""""""
  			for c in s:
  						if "A" <= c <= "Z":
  									out += chr(ord("""A"""    ) + (ord(lowercase__    ) - ord("""A"""    ) + n) % 26    )
  						elif "a" <= c <= "z":
  									out += chr(ord("""a"""    ) + (ord(lowercase__    ) - ord("""a"""    ) + n) % 26    )
  						else:
  									out += c
  			return out
  
  
  
  
  
  def 			a_   (  ):
  			__lowerCamelCase        =			input("""Enter message: """    )
  
  			__lowerCamelCase        =			dencrypt(lowercase__, 13    )
  			print("""Encryption:""", lowercase__    )
  
  			__lowerCamelCase        =			dencrypt(lowercase__, 13    )
  			print("""Decryption: """, lowercase__    )
  
  
  if __name__ == "__main__":
  					import doctest
  
  					doctest.testmod()
  					main()
  
  
  
  
  