




'''simple docstring'''





def UpperCAmelCase						(					UpperCAmelCase_	):
    if len(lowerCAmelCase__	) == 0 or not all(len(lowerCAmelCase__	) == 2 for pair in coordinates	):
        raise ValueError("""The program cannot work out a fitting polynomial."""	)

    if len({tuple(lowerCAmelCase__	) for pair in coordinates}	) != len(lowerCAmelCase__	):
        raise ValueError("""The program cannot work out a fitting polynomial."""	)

    _a							:       Dict       					=			{x for x, _ in coordinates}
    if len(lowerCAmelCase__	) == 1:
        return F'''x={coordinates[0][0]}'''

    if len(lowerCAmelCase__	) != len(lowerCAmelCase__	):
        raise ValueError("""The program cannot work out a fitting polynomial."""	)

    _a							:       Dict       					=			len(lowerCAmelCase__	)

    _a							:       List[str]       					=			0
    _a							:       list[list[float]]       					=			[]
    # put the x and x to the power values in a matrix
    while count_of_line < x:
        _a							:       Union[str, Any]       					=			0
        _a							:       Optional[Any]       					=			coordinates[count_of_line][0]
        _a							:       list[float]       					=			[]
        while count_in_line < x:
            count_line.append(a ** (x - (count_in_line + 1))	)
            count_in_line += 1
        matrix.append(lowerCAmelCase__	)
        count_of_line += 1

    _a							:       Any       					=			0
    # put the y values into a vector
    _a							:       list[float]       					=			[]
    while count_of_line < x:
        vector.append(coordinates[count_of_line][1]	)
        count_of_line += 1

    _a							:       List[str]       					=			0

    while count < x:
        _a							:       Union[str, Any]       					=			0
        while zahlen < x:
            if count == zahlen:
                zahlen += 1
            if zahlen == x:
                break
            _a							:       Optional[Any]       					=			matrix[zahlen][count] / matrix[count][count]
            for counting_columns, item in enumerate(matrix[count]	):
                # manipulating all the values in the matrix
                matrix[zahlen][counting_columns] -= item * bruch
            # manipulating the values in the vector
            vector[zahlen] -= vector[count] * bruch
            zahlen += 1
        count += 1

    _a							:       str       					=			0
    # make solutions
    _a							:       list[str]       					=			[]
    while count < x:
        solution.append(str(vector[count] / matrix[count][count]	)	)
        count += 1

    _a							:       Optional[int]       					=			0
    _a							:       Union[str, Any]       					=			'f(x)='

    while count < x:
        _a							:       list[str]       					=			solution[count].split("""E"""	)
        if len(lowerCAmelCase__	) > 1:
            _a							:       List[str]       					=			F'''{remove_e[0]}*10^{remove_e[1]}'''
        solved += F'''x^{x - (count + 1)}*{solution[count]}'''
        if count + 1 != x:
            solved += "+"
        count += 1

    return solved


if __name__ == "__main__":
  print(points_to_polynomial([]))
  print(points_to_polynomial([[]]))
  print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
  print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
  print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
  print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))








'''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					(  snake_case__					):
  a    :torch.FloatTensor
  a    :Optional[torch.FloatTensor]      					=				None






def UpperCAmelCase						(					UpperCAmelCase_      ,  UpperCAmelCase_=0.999      ,  UpperCAmelCase_="cosine"      ,  ):
    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}'''	)

    _a							:       Union[str, Any]       					=			[]
    for i in range(UpperCAmelCase_	):
        _a							:       Optional[int]       					=			i / num_diffusion_timesteps
        _a							:       Optional[Any]       					=			(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					(  snake_case__			,		snake_case__					):
  a    :Any      					=				[e.name for e in KarrasDiffusionSchedulers]
  a    :Tuple      					=				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		,    ):







      '''simple docstring'''


      if trained_betas is not None:
          _a							:       str       					=			torch.tensor(__lowerCAmelCase		,    dtype=torch.floataa   )
      elif beta_schedule == "linear":
          _a							:       int       					=			torch.linspace(__lowerCAmelCase		,    __lowerCAmelCase		,    __lowerCAmelCase		,    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		,    __lowerCAmelCase		,    dtype=torch.floataa   ) ** 2
          )
      elif beta_schedule == "squaredcos_cap_v2":
          # Glide cosine schedule
          _a							:       Optional[Any]       					=			betas_for_alpha_bar(__lowerCAmelCase   )
      else:
          raise NotImplementedError(f'''{beta_schedule} does is not implemented for {self.__class__}'''   )

      _a							:       Dict       					=			1.0 - self.betas
      _a							:       Any       					=			torch.cumprod(self.alphas		,    dim=0   )

      _a							:       List[Any]       					=			np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5   )
      _a							:       List[str]       					=			np.concatenate([sigmas[::-1], [0.0]]   ).astype(np.floataa   )
      _a							:       Optional[Any]       					=			torch.from_numpy(__lowerCAmelCase   )

      # setable values
      _a							:       Tuple       					=			None
      _a							:       List[Any]       					=			use_karras_sigmas
      self.set_timesteps(__lowerCAmelCase		,    __lowerCAmelCase   )
      _a							:       Dict       					=			[]
      _a							:       int       					=			False



  @property
  def 						a_       ( self   ):







      '''simple docstring'''


      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   ):







      '''simple docstring'''


      if isinstance(__lowerCAmelCase		,    torch.Tensor   ):
          _a							:       int       					=			timestep.to(self.timesteps.device   )
      _a							:       Optional[Any]       					=			(self.timesteps == timestep).nonzero().item()
      _a							:       int       					=			self.sigmas[step_index]
      _a							:       Optional[int]       					=			sample / ((sigma**2 + 1) ** 0.5)
      _a							:       str       					=			True
      return sample



  def 						a_       ( self		,    __lowerCAmelCase		,    __lowerCAmelCase		,    __lowerCAmelCase   ):







      '''simple docstring'''


      def lms_derivative(__lowerCAmelCase   ):
          _a							:       Optional[Any]       					=			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

      _a							:       Union[str, Any]       					=			integrate.quad(__lowerCAmelCase		,    self.sigmas[t]		,    self.sigmas[t + 1]		,    epsrel=1e-4   )[0]

      return integrated_coeff



  def 						a_       ( self		,    __lowerCAmelCase		,    __lowerCAmelCase = None   ):







      '''simple docstring'''


      _a							:       Optional[int]       					=			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							:       str       					=			np.linspace(0		,    self.config.num_train_timesteps - 1		,    __lowerCAmelCase		,    dtype=__lowerCAmelCase   )[
              ::-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							:       List[str]       					=			(np.arange(0		,    __lowerCAmelCase   ) * step_ratio).round()[::-1].copy().astype(__lowerCAmelCase   )
          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							:       List[Any]       					=			(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\'.'''   )

      _a							:       Optional[int]       					=			np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5   )
      _a							:       Optional[int]       					=			np.log(__lowerCAmelCase   )
      _a							:       Any       					=			np.interp(__lowerCAmelCase		,    np.arange(0		,    len(__lowerCAmelCase   )   )		,    __lowerCAmelCase   )

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

      _a							:       List[Any]       					=			np.concatenate([sigmas, [0.0]]   ).astype(np.floataa   )

      _a							:       int       					=			torch.from_numpy(__lowerCAmelCase   ).to(device=__lowerCAmelCase   )
      if str(__lowerCAmelCase   ).startswith("""mps"""   ):
          # mps does not support float64
          _a							:       List[str]       					=			torch.from_numpy(__lowerCAmelCase   ).to(__lowerCAmelCase		,    dtype=torch.floataa   )
      else:
          _a							:       int       					=			torch.from_numpy(__lowerCAmelCase   ).to(device=__lowerCAmelCase   )

      _a							:       Any       					=			[]



  def 						a_       ( self		,    __lowerCAmelCase		,    __lowerCAmelCase   ):







      '''simple docstring'''


      _a							:       Any       					=			np.log(__lowerCAmelCase   )

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

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

      _a							:       Union[str, Any]       					=			log_sigmas[low_idx]
      _a							:       str       					=			log_sigmas[high_idx]

      # interpolate sigmas
      _a							:       Union[str, Any]       					=			(low - log_sigma) / (low - high)
      _a							:       Tuple       					=			np.clip(__lowerCAmelCase		,    0		,    1   )

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



  def 						a_       ( self		,    __lowerCAmelCase   ):







      '''simple docstring'''


      _a							:       float       					=			in_sigmas[-1].item()
      _a							:       float       					=			in_sigmas[0].item()

      _a							:       Tuple       					=			7.0  # 7.0 is the value used in the paper
      _a							:       Dict       					=			np.linspace(0		,    1		,    self.num_inference_steps   )
      _a							:       Optional[int]       					=			sigma_min ** (1 / rho)
      _a							:       Tuple       					=			sigma_max ** (1 / rho)
      _a							:       Dict       					=			(max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
      return sigmas



  def 						a_       ( self		,    __lowerCAmelCase		,    __lowerCAmelCase		,    __lowerCAmelCase		,    __lowerCAmelCase = 4		,    __lowerCAmelCase = True		,    ):







      '''simple docstring'''


      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   ):
          _a							:       Tuple       					=			timestep.to(self.timesteps.device   )
      _a							:       List[str]       					=			(self.timesteps == timestep).nonzero().item()
      _a							:       str       					=			self.sigmas[step_index]

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

      # 3. Compute linear multistep coefficients
      _a							:       Optional[Any]       					=			min(step_index + 1		,    __lowerCAmelCase   )
      _a							:       int       					=			[self.get_lms_coefficient(__lowerCAmelCase		,    __lowerCAmelCase		,    __lowerCAmelCase   ) for curr_order in range(__lowerCAmelCase   )]

      # 4. Compute previous sample based on the derivatives path
      _a							:       Optional[Any]       					=			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		,    ):







      '''simple docstring'''


      _a							:       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
          _a							:       Any       					=			self.timesteps.to(original_samples.device		,    dtype=torch.floataa   )
          _a							:       Optional[Any]       					=			timesteps.to(original_samples.device		,    dtype=torch.floataa   )
      else:
          _a							:       int       					=			self.timesteps.to(original_samples.device   )
          _a							:       Optional[Any]       					=			timesteps.to(original_samples.device   )

      _a							:       Any       					=			[(schedule_timesteps == t).nonzero().item() for t in timesteps]

      _a							:       List[str]       					=			sigmas[step_indices].flatten()
      while len(sigma.shape   ) < len(original_samples.shape   ):
          _a							:       int       					=			sigma.unsqueeze(-1   )

      _a							:       List[str]       					=			original_samples + noise * sigma
      return noisy_samples



  def __len__( self   ):







      '''simple docstring'''


      return self.config.num_train_timesteps


