


def 	_UpperCamelCase							(				lowercase__   ,	lowercase__   ,	lowercase__				):
						if mass < 0:
												raise ValueError('''The mass of the body cannot be negative'''				)
						if radius <= 0:
												raise ValueError('''The radius is always a positive non zero integer'''				)
						return (mass * (velocity) ** 2) / radius


if __name__ == "__main__":
						import doctest

						doctest.testmod(verbose=True)




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


__lowerCAmelCase     :			str      =logging.get_logger(__name__)  # pylint: disable=invalid-name
@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->EulerDiscrete
class 	_lowercase				(      A__	):






				'''simple docstring'''



				SCREAMING_SNAKE_CASE__    :      torch.FloatTensor
				SCREAMING_SNAKE_CASE__    :      Optional[torch.FloatTensor]      =		None




def 	_UpperCamelCase							(				lowercase__   ,	lowercase__=0.999   ,	lowercase__="cosine"   ,	):
						if alpha_transform_type == "cosine":

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

						elif alpha_transform_type == "exp":

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

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

						__SCREAMING_SNAKE_CASE						:					List[str]		      =		[]
						for i in range(lowercase__				):
												__SCREAMING_SNAKE_CASE						:					List[Any]		      =		i / num_diffusion_timesteps
												__SCREAMING_SNAKE_CASE						:					Dict		      =		(i + 1) / num_diffusion_timesteps
												betas.append(min(1 - alpha_bar_fn(lowercase__				) / alpha_bar_fn(lowercase__				)   ,	lowercase__				)				)
						return torch.tensor(lowercase__   ,	dtype=torch.floataa				)
class 	_lowercase				(      A__						,      A__	):






				'''simple docstring'''



				SCREAMING_SNAKE_CASE__    :      int      =		[e.name for e in KarrasDiffusionSchedulers]
				SCREAMING_SNAKE_CASE__    :      Optional[int]      =		1


				@register_to_config
				def __init__(	self      :Tuple      ,      lowerCAmelCase__      :int = 1_000      ,      lowerCAmelCase__      :float = 0.0001      ,      lowerCAmelCase__      :float = 0.02      ,      lowerCAmelCase__      :str = "linear"      ,      lowerCAmelCase__      :Optional[Union[np.ndarray, List[float]]] = None      ,      lowerCAmelCase__      :str = "epsilon"      ,      lowerCAmelCase__      :str = "linear"      ,      lowerCAmelCase__      :Optional[bool] = False      ,      lowerCAmelCase__      :str = "linspace"      ,      lowerCAmelCase__      :int = 0      ,      )		->      Optional[int]:
										if trained_betas is not None:
																__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		torch.tensor(lowerCAmelCase__      ,      dtype=torch.floataa     )
										elif beta_schedule == "linear":
																__SCREAMING_SNAKE_CASE						:					str		      =		torch.linspace(lowerCAmelCase__      ,      lowerCAmelCase__      ,      lowerCAmelCase__      ,      dtype=torch.floataa     )
										elif beta_schedule == "scaled_linear":
																# this schedule is very specific to the latent diffusion model.
																__SCREAMING_SNAKE_CASE						:					List[Any]		      =		(
																    torch.linspace(beta_start**0.5      ,      beta_end**0.5      ,      lowerCAmelCase__      ,      dtype=torch.floataa     ) ** 2
																)
										elif beta_schedule == "squaredcos_cap_v2":
																# Glide cosine schedule
																__SCREAMING_SNAKE_CASE						:					Dict		      =		betas_for_alpha_bar(lowerCAmelCase__     )
										else:
																raise NotImplementedError(f'''{beta_schedule} does is not implemented for {self.__class__}'''     )

										__SCREAMING_SNAKE_CASE						:					Dict		      =		1.0 - self.betas
										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		torch.cumprod(self.alphas      ,      dim=0     )

										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5     )
										__SCREAMING_SNAKE_CASE						:					Dict		      =		np.concatenate([sigmas[::-1], [0.0]]     ).astype(np.floataa     )
										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		torch.from_numpy(lowerCAmelCase__     )

										# setable values
										__SCREAMING_SNAKE_CASE						:					str		      =		None
										__SCREAMING_SNAKE_CASE						:					int		      =		np.linspace(0      ,      num_train_timesteps - 1      ,      lowerCAmelCase__      ,      dtype=lowerCAmelCase__     )[::-1].copy()
										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		torch.from_numpy(lowerCAmelCase__     )
										__SCREAMING_SNAKE_CASE						:					str		      =		False
										__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		use_karras_sigmas


				@property
				def 		__magic_name__(	self      :int     )		->      Union[str, Any]:
										# 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 		__magic_name__(	self      :Dict      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :Union[float, torch.FloatTensor]     )		->      torch.FloatTensor:
										if isinstance(lowerCAmelCase__      ,      torch.Tensor     ):
																__SCREAMING_SNAKE_CASE						:					Any		      =		timestep.to(self.timesteps.device     )
										__SCREAMING_SNAKE_CASE						:					Tuple		      =		(self.timesteps == timestep).nonzero().item()
										__SCREAMING_SNAKE_CASE						:					Any		      =		self.sigmas[step_index]

										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		sample / ((sigma**2 + 1) ** 0.5)

										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		True
										return sample


				def 		__magic_name__(	self      :Dict      ,      lowerCAmelCase__      :int      ,      lowerCAmelCase__      :Union[str, torch.device] = None     )		->      int:
										__SCREAMING_SNAKE_CASE						:					List[str]		      =		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":
																__SCREAMING_SNAKE_CASE						:					List[Any]		      =		np.linspace(0      ,      self.config.num_train_timesteps - 1      ,      lowerCAmelCase__      ,      dtype=lowerCAmelCase__     )[
																    ::-1
																].copy()
										elif self.config.timestep_spacing == "leading":
																__SCREAMING_SNAKE_CASE						:					List[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
																__SCREAMING_SNAKE_CASE						:					List[str]		      =		(np.arange(0      ,      lowerCAmelCase__     ) * step_ratio).round()[::-1].copy().astype(lowerCAmelCase__     )
																timesteps += self.config.steps_offset
										elif self.config.timestep_spacing == "trailing":
																__SCREAMING_SNAKE_CASE						:					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
																__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		(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\'.'''     )

										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5     )
										__SCREAMING_SNAKE_CASE						:					int		      =		np.log(lowerCAmelCase__     )

										if self.config.interpolation_type == "linear":
																__SCREAMING_SNAKE_CASE						:					str		      =		np.interp(lowerCAmelCase__      ,      np.arange(0      ,      len(lowerCAmelCase__     )     )      ,      lowerCAmelCase__     )
										elif self.config.interpolation_type == "log_linear":
																__SCREAMING_SNAKE_CASE						:					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:
																__SCREAMING_SNAKE_CASE						:					Tuple		      =		self._convert_to_karras(in_sigmas=lowerCAmelCase__      ,      num_inference_steps=self.num_inference_steps     )
																__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		np.array([self._sigma_to_t(lowerCAmelCase__      ,      lowerCAmelCase__     ) for sigma in sigmas]     )

										__SCREAMING_SNAKE_CASE						:					Tuple		      =		np.concatenate([sigmas, [0.0]]     ).astype(np.floataa     )
										__SCREAMING_SNAKE_CASE						:					Any		      =		torch.from_numpy(lowerCAmelCase__     ).to(device=lowerCAmelCase__     )
										if str(lowerCAmelCase__     ).startswith('''mps'''     ):
																# mps does not support float64
																__SCREAMING_SNAKE_CASE						:					Tuple		      =		torch.from_numpy(lowerCAmelCase__     ).to(lowerCAmelCase__      ,      dtype=torch.floataa     )
										else:
																__SCREAMING_SNAKE_CASE						:					Tuple		      =		torch.from_numpy(lowerCAmelCase__     ).to(device=lowerCAmelCase__     )


				def 		__magic_name__(	self      :Optional[int]      ,      lowerCAmelCase__      :Optional[Any]      ,      lowerCAmelCase__      :Union[str, Any]     )		->      int:
										# get log sigma
										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		np.log(lowerCAmelCase__     )

										# get distribution
										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		log_sigma - log_sigmas[:, np.newaxis]

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

										__SCREAMING_SNAKE_CASE						:					List[str]		      =		log_sigmas[low_idx]
										__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		log_sigmas[high_idx]

										# interpolate sigmas
										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		(low - log_sigma) / (low - high)
										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		np.clip(lowerCAmelCase__      ,      0      ,      1     )

										# transform interpolation to time range
										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		(1 - w) * low_idx + w * high_idx
										__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		t.reshape(sigma.shape     )
										return t


				def 		__magic_name__(	self      :Optional[int]      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :Union[str, Any]     )		->      torch.FloatTensor:
										__SCREAMING_SNAKE_CASE						:					float		      =		in_sigmas[-1].item()
										__SCREAMING_SNAKE_CASE						:					float		      =		in_sigmas[0].item()

										__SCREAMING_SNAKE_CASE						:					Tuple		      =		7.0  # 7.0 is the value used in the paper
										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		np.linspace(0      ,      1      ,      lowerCAmelCase__     )
										__SCREAMING_SNAKE_CASE						:					str		      =		sigma_min ** (1 / rho)
										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		sigma_max ** (1 / rho)
										__SCREAMING_SNAKE_CASE						:					List[str]		      =		(max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
										return sigmas


				def 		__magic_name__(	self      :Union[str, Any]      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :Union[float, torch.FloatTensor]      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :float = 0.0      ,      lowerCAmelCase__      :float = 0.0      ,      lowerCAmelCase__      :float = float('''inf'''     )      ,      lowerCAmelCase__      :float = 1.0      ,      lowerCAmelCase__      :Optional[torch.Generator] = None      ,      lowerCAmelCase__      :bool = True      ,      )		->      Union[EulerDiscreteSchedulerOutput, Tuple]:

										if (
										    isinstance(lowerCAmelCase__      ,      lowerCAmelCase__     )
										    or isinstance(lowerCAmelCase__      ,      torch.IntTensor     )
										    or isinstance(lowerCAmelCase__      ,      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(lowerCAmelCase__      ,      torch.Tensor     ):
																__SCREAMING_SNAKE_CASE						:					List[str]		      =		timestep.to(self.timesteps.device     )

										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		(self.timesteps == timestep).nonzero().item()
										__SCREAMING_SNAKE_CASE						:					str		      =		self.sigmas[step_index]

										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		min(s_churn / (len(self.sigmas     ) - 1)      ,      2**0.5 - 1     ) if s_tmin <= sigma <= s_tmax else 0.0

										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		randn_tensor(
										    model_output.shape      ,      dtype=model_output.dtype      ,      device=model_output.device      ,      generator=lowerCAmelCase__     )

										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		noise * s_noise
										__SCREAMING_SNAKE_CASE						:					Optional[int]		      =		sigma * (gamma + 1)

										if gamma > 0:
																__SCREAMING_SNAKE_CASE						:					str		      =		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":
																__SCREAMING_SNAKE_CASE						:					Any		      =		model_output
										elif self.config.prediction_type == "epsilon":
																__SCREAMING_SNAKE_CASE						:					Any		      =		sample - sigma_hat * model_output
										elif self.config.prediction_type == "v_prediction":
																# * c_out + input * c_skip
																__SCREAMING_SNAKE_CASE						:					Union[str, 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
										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		(sample - pred_original_sample) / sigma_hat

										__SCREAMING_SNAKE_CASE						:					Tuple		      =		self.sigmas[step_index + 1] - sigma_hat

										__SCREAMING_SNAKE_CASE						:					List[Any]		      =		sample + derivative * dt

										if not return_dict:
																return (prev_sample,)

										return EulerDiscreteSchedulerOutput(prev_sample=lowerCAmelCase__      ,      pred_original_sample=lowerCAmelCase__     )


				def 		__magic_name__(	self      :str      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :torch.FloatTensor      ,      lowerCAmelCase__      :torch.FloatTensor      ,      )		->      torch.FloatTensor:
										# Make sure sigmas and timesteps have the same device and dtype as original_samples
										__SCREAMING_SNAKE_CASE						:					Optional[Any]		      =		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
																__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		self.timesteps.to(original_samples.device      ,      dtype=torch.floataa     )
																__SCREAMING_SNAKE_CASE						:					int		      =		timesteps.to(original_samples.device      ,      dtype=torch.floataa     )
										else:
																__SCREAMING_SNAKE_CASE						:					Tuple		      =		self.timesteps.to(original_samples.device     )
																__SCREAMING_SNAKE_CASE						:					Dict		      =		timesteps.to(original_samples.device     )

										__SCREAMING_SNAKE_CASE						:					Dict		      =		[(schedule_timesteps == t).nonzero().item() for t in timesteps]

										__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		sigmas[step_indices].flatten()
										while len(sigma.shape     ) < len(original_samples.shape     ):
																__SCREAMING_SNAKE_CASE						:					Union[str, Any]		      =		sigma.unsqueeze(-1     )

										__SCREAMING_SNAKE_CASE						:					Any		      =		original_samples + noise * sigma
										return noisy_samples



				def __len__(	self      :Optional[Any]     )		->      Union[str, Any]:
										return self.config.num_train_timesteps
