






# DISCLAIMER: This code is strongly influenced by https://github.com/pesser/pytorch_diffusion
# and https://github.com/hojonathanho/diffusion
import math
from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from diffusers.configuration_utils import ConfigMixin, register_to_config
from diffusers.schedulers.scheduling_utils import SchedulerMixin
from diffusers.utils import BaseOutput, deprecate




@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->DDIM
class     _lowerCAmelCase(	UpperCamelCase__ ):





							"""simple docstring"""







							a    :		Optional[int]									=42
							a    :		List[str]									=None







def   snake_case (UpperCAmelCase__     ,   UpperCAmelCase__=0.999     ,   UpperCAmelCase__="cosine"     ,   )	->    Optional[Any]:
		if alpha_transform_type == "cosine":

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

		elif alpha_transform_type == "exp":

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

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

		UpperCamelCase_: Optional[int]   				=    []
		for i in range(UpperCamelCase__  ):
				UpperCamelCase_: Optional[Any]   				=    i / num_diffusion_timesteps
				UpperCamelCase_: str   				=    (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     _lowerCAmelCase(	UpperCamelCase__							,     UpperCamelCase__ ):





							"""simple docstring"""







							a    :		Optional[Any]									=1





							@register_to_config
							def __init__( self  ,      _lowerCamelCase = 1_0_0_0  ,      _lowerCamelCase = 0.0_0_0_1  ,      _lowerCamelCase = 0.0_2  ,      _lowerCamelCase = "linear"  ,      _lowerCamelCase = None  ,      _lowerCamelCase = True  ,      _lowerCamelCase = True  ,      _lowerCamelCase = 0  ,      _lowerCamelCase = "epsilon"  ,      _lowerCamelCase = 1.0  ,      **_lowerCamelCase  ,      ):
									if kwargs.get('set_alpha_to_one'  ,      __lowerCamelCase							) is not None:
											UpperCamelCase_: List[Any]   				=    (
											    'The `set_alpha_to_one` argument is deprecated. Please use `set_alpha_to_zero` instead.'
											)
											deprecate('set_alpha_to_one'  ,      '1.0.0'  ,      __lowerCamelCase  ,      standard_warn=__lowerCamelCase							)
											UpperCamelCase_: List[Any]   				=    kwargs['set_alpha_to_one']
									if trained_betas is not None:
											UpperCamelCase_: List[Any]   				=    torch.tensor(__lowerCamelCase  ,      dtype=torch.floataa							)
									elif beta_schedule == "linear":
											UpperCamelCase_: Tuple   				=    torch.linspace(__lowerCamelCase  ,      __lowerCamelCase  ,      __lowerCamelCase  ,      dtype=torch.floataa							)
									elif beta_schedule == "scaled_linear":
											# this schedule is very specific to the latent diffusion model.
											UpperCamelCase_: int   				=    (
											    torch.linspace(beta_start**0.5  ,      beta_end**0.5  ,      __lowerCamelCase  ,      dtype=torch.floataa							) ** 2
											)
									elif beta_schedule == "squaredcos_cap_v2":
											# Glide cosine schedule
											UpperCamelCase_: Dict   				=    betas_for_alpha_bar(__lowerCamelCase							)
									else:
											raise NotImplementedError(f'''{beta_schedule} does is not implemented for {self.__class__}'''							)

									UpperCamelCase_: Optional[int]   				=    1.0 - self.betas
									UpperCamelCase_: Dict   				=    torch.cumprod(self.alphas  ,      dim=0							)

									# At every step in inverted ddim, we are looking into the next alphas_cumprod
									# For the final step, there is no next alphas_cumprod, and the index is out of bounds
									# `set_alpha_to_zero` decides whether we set this parameter simply to zero
									# in this case, self.step() just output the predicted noise
									# or whether we use the final alpha of the "non-previous" one.
									UpperCamelCase_: str   				=    torch.tensor(0.0							) if set_alpha_to_zero else self.alphas_cumprod[-1]

									# standard deviation of the initial noise distribution
									UpperCamelCase_: Union[str, Any]   				=    1.0

									# setable values
									UpperCamelCase_: List[str]   				=    None
									UpperCamelCase_: List[Any]   				=    torch.from_numpy(np.arange(0  ,      __lowerCamelCase							).copy().astype(np.intaa							)							)





							def       _a   ( self  ,      _lowerCamelCase  ,      _lowerCamelCase = None							):
									return sample





							def       _a   ( self  ,      _lowerCamelCase  ,      _lowerCamelCase = None							):

									if num_inference_steps > self.config.num_train_timesteps:
											raise ValueError(
											    f'''`num_inference_steps`: {num_inference_steps} cannot be larger than `self.config.train_timesteps`:'''
											    f''' {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle'''
											    f''' maximal {self.config.num_train_timesteps} timesteps.'''							)

									UpperCamelCase_: Optional[Any]   				=    num_inference_steps
									UpperCamelCase_: Any   				=    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
									UpperCamelCase_: Union[str, Any]   				=    (np.arange(0  ,      __lowerCamelCase							) * step_ratio).round().copy().astype(np.intaa							)
									UpperCamelCase_: Dict   				=    torch.from_numpy(__lowerCamelCase							).to(__lowerCamelCase							)
									self.timesteps += self.config.steps_offset





							def       _a   ( self  ,      _lowerCamelCase  ,      _lowerCamelCase  ,      _lowerCamelCase  ,      _lowerCamelCase = 0.0  ,      _lowerCamelCase = False  ,      _lowerCamelCase = None  ,      _lowerCamelCase = True  ,      ):
									# 1. get previous step value (=t+1)
									UpperCamelCase_: Optional[int]   				=    timestep + self.config.num_train_timesteps // self.num_inference_steps

									# 2. compute alphas, betas
									# change original implementation to exactly match noise levels for analogous forward process
									UpperCamelCase_: Dict   				=    self.alphas_cumprod[timestep]
									UpperCamelCase_: str   				=    (
									    self.alphas_cumprod[prev_timestep]
									    if prev_timestep < self.config.num_train_timesteps
									    else self.final_alpha_cumprod
									)

									UpperCamelCase_: List[Any]   				=    1 - alpha_prod_t

									# 3. compute predicted original sample from predicted noise also called
									# "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
									if self.config.prediction_type == "epsilon":
											UpperCamelCase_: str   				=    (sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
											UpperCamelCase_: Union[str, Any]   				=    model_output
									elif self.config.prediction_type == "sample":
											UpperCamelCase_: Tuple   				=    model_output
											UpperCamelCase_: Optional[Any]   				=    (sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5
									elif self.config.prediction_type == "v_prediction":
											UpperCamelCase_: Any   				=    (alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
											UpperCamelCase_: List[Any]   				=    (alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
									else:
											raise ValueError(
											    f'''prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or'''
											    ' `v_prediction`'							)

									# 4. Clip or threshold "predicted x_0"
									if self.config.clip_sample:
											UpperCamelCase_: int   				=    pred_original_sample.clamp(
											    -self.config.clip_sample_range  ,      self.config.clip_sample_range							)

									# 5. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
									UpperCamelCase_: Optional[Any]   				=    (1 - alpha_prod_t_prev) ** 0.5 * pred_epsilon

									# 6. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
									UpperCamelCase_: Any   				=    alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction

									if not return_dict:
											return (prev_sample, pred_original_sample)
									return DDIMSchedulerOutput(prev_sample=__lowerCamelCase  ,      pred_original_sample=__lowerCamelCase							)





							def __len__( self							):
									return self.config.num_train_timesteps







def   snake_case (UpperCAmelCase__  )	->    int:
  assert isinstance(UpperCAmelCase__     ,   UpperCAmelCase__  ), F'''The input value of [n={number}] is not an integer'''

  if number == 1:
    return 2
  elif number < 1:
    UpperCamelCase_: List[Any]   				=    F'''The input value of [n={number}] has to be > 0'''
    raise ValueError(UpperCAmelCase__  )
  else:
    UpperCamelCase_: str   				=    sylvester(number - 1  )
    UpperCamelCase_: str   				=    num - 1
    UpperCamelCase_: Any   				=    num
    return lower * upper + 1


if __name__ == "__main__":
   print(F'''The 8th number in Sylvester\'s sequence: {sylvester(8)}''')