"""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 					UpperCamelCase   (							lowerCAmelCase__  ):
       SCREAMING_SNAKE_CASE_						      = 42
       SCREAMING_SNAKE_CASE_						      = None







def    UpperCAmelCase       (						UpperCAmelCase		,   UpperCAmelCase=0.999		,   UpperCAmelCase="cosine"		,   )							-> str:
     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}'   )

     snake_case_   			=   []
     for i in range(UpperCAmelCase   ):
          snake_case_   			=   i / num_diffusion_timesteps
          snake_case_   			=   (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 					UpperCamelCase   (							lowerCAmelCase__       ,						lowerCAmelCase__  ):
       SCREAMING_SNAKE_CASE_						      = [e.name for e in KarrasDiffusionSchedulers]
       SCREAMING_SNAKE_CASE_						      = 1







       @register_to_config
       def __init__(					self,				lowerCAmelCase__ = 1000,				lowerCAmelCase__ = 0.0001,				lowerCAmelCase__ = 0.02,				lowerCAmelCase__ = "linear",				lowerCAmelCase__ = None,				lowerCAmelCase__ = False,				lowerCAmelCase__ = "epsilon",				lowerCAmelCase__ = "linspace",				lowerCAmelCase__ = 0,				) -> Optional[Any]:
            if trained_betas is not None:
                 snake_case_   			=   torch.tensor(lowerCAmelCase__,				dtype=torch.floataa)
            elif beta_schedule == "linear":
                 snake_case_   			=   torch.linspace(lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__,				dtype=torch.floataa)
            elif beta_schedule == "scaled_linear":
                 # this schedule is very specific to the latent diffusion model.
                 snake_case_   			=   (
                     torch.linspace(beta_start**0.5,				beta_end**0.5,				lowerCAmelCase__,				dtype=torch.floataa) ** 2
                 )
            elif beta_schedule == "squaredcos_cap_v2":
                 # Glide cosine schedule
                 snake_case_   			=   betas_for_alpha_bar(lowerCAmelCase__)
            else:
                 raise NotImplementedError(f'{beta_schedule} does is not implemented for {self.__class__}')

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

            snake_case_   			=   np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5)
            snake_case_   			=   np.concatenate([sigmas[::-1], [0.0]]).astype(np.floataa)
            snake_case_   			=   torch.from_numpy(lowerCAmelCase__)

            # setable values
            snake_case_   			=   None
            snake_case_   			=   use_karras_sigmas
            self.set_timesteps(lowerCAmelCase__,				lowerCAmelCase__)
            snake_case_   			=   []
            snake_case_   			=   False







       @property
       def 						a_		(					self) -> List[str]:
            # 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,				lowerCAmelCase__,				lowerCAmelCase__) -> torch.FloatTensor:
            if isinstance(lowerCAmelCase__,				torch.Tensor):
                 snake_case_   			=   timestep.to(self.timesteps.device)
            snake_case_   			=   (self.timesteps == timestep).nonzero().item()
            snake_case_   			=   self.sigmas[step_index]
            snake_case_   			=   sample / ((sigma**2 + 1) ** 0.5)
            snake_case_   			=   True
            return sample







       def 						a_		(					self,				lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__) -> Union[str, Any]:

            def lms_derivative(lowerCAmelCase__):
                 snake_case_   			=   1.0
                 for k in range(lowerCAmelCase__):
                      if current_order == k:
                           continue
                      prod *= (tau - self.sigmas[t - k]) / (self.sigmas[t - current_order] - self.sigmas[t - k])
                 return prod

            snake_case_   			=   integrate.quad(lowerCAmelCase__,				self.sigmas[t],				self.sigmas[t + 1],				epsrel=1e-4)[0]

            return integrated_coeff







       def 						a_		(					self,				lowerCAmelCase__,				lowerCAmelCase__ = None) -> List[Any]:
            snake_case_   			=   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":
                 snake_case_   			=   np.linspace(0,				self.config.num_train_timesteps - 1,				lowerCAmelCase__,				dtype=lowerCAmelCase__)[
                     ::-1
                 ].copy()
            elif self.config.timestep_spacing == "leading":
                 snake_case_   			=   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
                 snake_case_   			=   (np.arange(0,				lowerCAmelCase__) * step_ratio).round()[::-1].copy().astype(lowerCAmelCase__)
                 timesteps += self.config.steps_offset
            elif self.config.timestep_spacing == "trailing":
                 snake_case_   			=   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
                 snake_case_   			=   (np.arange(self.config.num_train_timesteps,				0,				-step_ratio)).round().copy().astype(lowerCAmelCase__)
                 timesteps -= 1
            else:
                 raise ValueError(
                     f'{self.config.timestep_spacing} is not supported. Please make sure to choose one of \'linspace\', \'leading\' or \'trailing\'.')

            snake_case_   			=   np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5)
            snake_case_   			=   np.log(lowerCAmelCase__)
            snake_case_   			=   np.interp(lowerCAmelCase__,				np.arange(0,				len(lowerCAmelCase__)),				lowerCAmelCase__)

            if self.use_karras_sigmas:
                 snake_case_   			=   self._convert_to_karras(in_sigmas=lowerCAmelCase__)
                 snake_case_   			=   np.array([self._sigma_to_t(lowerCAmelCase__,				lowerCAmelCase__) for sigma in sigmas])

            snake_case_   			=   np.concatenate([sigmas, [0.0]]).astype(np.floataa)

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

            snake_case_   			=   []







       def 						a_		(					self,				lowerCAmelCase__,				lowerCAmelCase__) -> Tuple:
            # get log sigma
            snake_case_   			=   np.log(lowerCAmelCase__)

            # get distribution
            snake_case_   			=   log_sigma - log_sigmas[:, np.newaxis]

            # get sigmas range
            snake_case_   			=   np.cumsum((dists >= 0),				axis=0).argmax(axis=0).clip(max=log_sigmas.shape[0] - 2)
            snake_case_   			=   low_idx + 1

            snake_case_   			=   log_sigmas[low_idx]
            snake_case_   			=   log_sigmas[high_idx]

            # interpolate sigmas
            snake_case_   			=   (low - log_sigma) / (low - high)
            snake_case_   			=   np.clip(lowerCAmelCase__,				0,				1)

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







       def 						a_		(					self,				lowerCAmelCase__) -> torch.FloatTensor:
            snake_case_   			=   in_sigmas[-1].item()
            snake_case_   			=   in_sigmas[0].item()

            snake_case_   			=   7.0  # 7.0 is the value used in the paper
            snake_case_   			=   np.linspace(0,				1,				self.num_inference_steps)
            snake_case_   			=   sigma_min ** (1 / rho)
            snake_case_   			=   sigma_max ** (1 / rho)
            snake_case_   			=   (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
            return sigmas







       def 						a_		(					self,				lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__ = 4,				lowerCAmelCase__ = True,				) -> Union[LMSDiscreteSchedulerOutput, Tuple]:
            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(lowerCAmelCase__,				torch.Tensor):
                 snake_case_   			=   timestep.to(self.timesteps.device)
            snake_case_   			=   (self.timesteps == timestep).nonzero().item()
            snake_case_   			=   self.sigmas[step_index]

            # 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
            if self.config.prediction_type == "epsilon":
                 snake_case_   			=   sample - sigma * model_output
            elif self.config.prediction_type == "v_prediction":
                 # * c_out + input * c_skip
                 snake_case_   			=   model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
            elif self.config.prediction_type == "sample":
                 snake_case_   			=   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
            snake_case_   			=   (sample - pred_original_sample) / sigma
            self.derivatives.append(lowerCAmelCase__)
            if len(self.derivatives) > order:
                 self.derivatives.pop(0)

            # 3. Compute linear multistep coefficients
            snake_case_   			=   min(step_index + 1,				lowerCAmelCase__)
            snake_case_   			=   [self.get_lms_coefficient(lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__) for curr_order in range(lowerCAmelCase__)]

            # 4. Compute previous sample based on the derivatives path
            snake_case_   			=   sample + sum(
                coeff * derivative for coeff, derivative in zip(lowerCAmelCase__,				reversed(self.derivatives)))

            if not return_dict:
                 return (prev_sample,)

            return LMSDiscreteSchedulerOutput(prev_sample=lowerCAmelCase__,				pred_original_sample=lowerCAmelCase__)







       def 						a_		(					self,				lowerCAmelCase__,				lowerCAmelCase__,				lowerCAmelCase__,				) -> torch.FloatTensor:
            # Make sure sigmas and timesteps have the same device and dtype as original_samples
            snake_case_   			=   self.sigmas.to(device=original_samples.device,				dtype=original_samples.dtype)
            if original_samples.device.type == "mps" and torch.is_floating_point(lowerCAmelCase__):
                 # mps does not support float64
                 snake_case_   			=   self.timesteps.to(original_samples.device,				dtype=torch.floataa)
                 snake_case_   			=   timesteps.to(original_samples.device,				dtype=torch.floataa)
            else:
                 snake_case_   			=   self.timesteps.to(original_samples.device)
                 snake_case_   			=   timesteps.to(original_samples.device)

            snake_case_   			=   [(schedule_timesteps == t).nonzero().item() for t in timesteps]

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

            snake_case_   			=   original_samples + noise * sigma
            return noisy_samples







       def __len__(					self) -> List[str]:
            return self.config.num_train_timesteps


"""simple docstring"""


from math import factorial







def    UpperCAmelCase       (						UpperCAmelCase		,   UpperCAmelCase   )							-> List[str]:
     return factorial(UpperCAmelCase   ) / (factorial(UpperCAmelCase   ) * factorial(n - r   ))







def    UpperCAmelCase       (						)							-> Optional[Any]:
     snake_case_   			=   0

     for i in range(1		,   101   ):
          for j in range(1		,   i + 1   ):
               if combinations(UpperCAmelCase		,   UpperCAmelCase   ) > 1e6:
                    total += 1
     return total


if __name__ == "__main__":
    print(solution())

