






'''simple docstring'''







def  lowerCAmelCase__		(					a_  :     list[list[int]]			)   -> str:
       if len(a_			) == 0 or not all(len(a_			) == 2 for pair in coordinates			):
              raise ValueError('''The program cannot work out a fitting polynomial.'''			)

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

       UpperCAmelCase__	:   Optional[Any]												=	{x for x, _ in coordinates}
       if len(a_			) == 1:
              return f"""x={coordinates[0][0]}"""

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

       UpperCAmelCase__	:   int												=	len(a_			)

       UpperCAmelCase__	:   str												=	0
       UpperCAmelCase__	:   list[list[float]]												=	[]
       # put the x and x to the power values in a matrix
       while count_of_line < x:
              UpperCAmelCase__	:   Tuple												=	0
              UpperCAmelCase__	:   Any												=	coordinates[count_of_line][0]
              UpperCAmelCase__	:   list[float]												=	[]
              while count_in_line < x:
                     count_line.append(a ** (x - (count_in_line + 1))			)
                     count_in_line += 1
              matrix.append(a_			)
              count_of_line += 1

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

       UpperCAmelCase__	:   List[Any]												=	0

       while count < x:
              UpperCAmelCase__	:   Union[str, Any]												=	0
              while zahlen < x:
                     if count == zahlen:
                            zahlen += 1
                     if zahlen == x:
                            break
                     UpperCAmelCase__	:   Tuple												=	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

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

       UpperCAmelCase__	:   List[Any]												=	0
       UpperCAmelCase__	:   List[Any]												=	'''f(x)='''

       while count < x:
              UpperCAmelCase__	:   list[str]												=	solution[count].split('''E'''			)
              if len(a_			) > 1:
                     UpperCAmelCase__	:   Any												=	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, 11]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))







'''simple docstring'''







from typing import Union

import torch
from PIL import Image
from torchvision import transforms as tfms
from tqdm.auto import tqdm
from transformers import CLIPTextModel, CLIPTokenizer

from diffusers import (
    AutoencoderKL,
    DDIMScheduler,
    DiffusionPipeline,
    LMSDiscreteScheduler,
    PNDMScheduler,
    UNetaDConditionModel,
)




class 					__UpperCAmelCase (		UpperCamelCase__	):

      '''simple docstring'''





      def __init__(					self  ,		_UpperCAmelCase  ,		_UpperCAmelCase  ,		_UpperCAmelCase  ,		_UpperCAmelCase  ,		_UpperCAmelCase  ,		):
             super().__init__()

             self.register_modules(vae=_UpperCAmelCase  ,		text_encoder=_UpperCAmelCase  ,		tokenizer=_UpperCAmelCase  ,		unet=_UpperCAmelCase  ,		scheduler=_UpperCAmelCase						)




      def 							lowerCamelCase     (					self  ,		_UpperCAmelCase						):
             with torch.no_grad():
                    UpperCAmelCase__	:   Dict												=	self.vae.encode(tfms.ToTensor()(_UpperCAmelCase						).unsqueeze(0						).to(self.device						) * 2 - 1						)
                    UpperCAmelCase__	:   int												=	0.1_82_15 * latent.latent_dist.sample()
             return latent




      def 							lowerCamelCase     (					self  ,		_UpperCAmelCase						):
             UpperCAmelCase__	:   int												=	(1 / 0.1_82_15) * latent
             with torch.no_grad():
                    UpperCAmelCase__	:   Optional[Any]												=	self.vae.decode(_UpperCAmelCase						).sample
             UpperCAmelCase__	:   List[str]												=	(img / 2 + 0.5).clamp(0  ,		1						)
             UpperCAmelCase__	:   List[Any]												=	img.detach().cpu().permute(0  ,		2  ,		3  ,		1						).numpy()
             UpperCAmelCase__	:   Union[str, Any]												=	(img * 255).round().astype('''uint8'''						)
             return Image.fromarray(img[0]						)




      def 							lowerCamelCase     (					self  ,		_UpperCAmelCase						):
             UpperCAmelCase__	:   Dict												=	self.tokenizer(
                 _UpperCAmelCase  ,		padding='''max_length'''  ,		max_length=self.tokenizer.model_max_length  ,		truncation=_UpperCAmelCase  ,		return_tensors='''pt'''  ,		)

             UpperCAmelCase__	:   Tuple												=	self.text_encoder(text_input.input_ids.to(self.device						)						)[0]

             UpperCAmelCase__	:   Dict												=	self.tokenizer(
                 ''''''  ,		padding='''max_length'''  ,		max_length=self.tokenizer.model_max_length  ,		truncation=_UpperCAmelCase  ,		return_tensors='''pt'''  ,		)

             UpperCAmelCase__	:   List[Any]												=	self.text_encoder(uncond_input.input_ids.to(self.device						)						)[0]

             return torch.cat([uncond_embedding, text_embedding]						)




      def __call__(					self  ,		_UpperCAmelCase  ,		_UpperCAmelCase  ,		_UpperCAmelCase = 0.3  ,		_UpperCAmelCase = 0.6  ,		_UpperCAmelCase = 0.5  ,		_UpperCAmelCase = 42  ,		_UpperCAmelCase = 50  ,		_UpperCAmelCase = 7.5  ,		):
             UpperCAmelCase__	:   Any												=	steps - int(kmin * steps						)
             UpperCAmelCase__	:   Any												=	steps - int(kmax * steps						)

             UpperCAmelCase__	:   List[str]												=	self.prep_text(_UpperCAmelCase						)

             self.scheduler.set_timesteps(_UpperCAmelCase						)

             UpperCAmelCase__							,							UpperCAmelCase__	:   List[str]												=	img.size
             UpperCAmelCase__	:   str												=	self.encode(_UpperCAmelCase						)

             torch.manual_seed(_UpperCAmelCase						)
             UpperCAmelCase__	:   Union[str, Any]												=	torch.randn(
                 (1, self.unet.config.in_channels, height // 8, width // 8)  ,		).to(self.device						)

             UpperCAmelCase__	:   Optional[int]												=	self.scheduler.add_noise(
                 _UpperCAmelCase  ,		_UpperCAmelCase  ,		timesteps=self.scheduler.timesteps[tmax]  ,		)

             UpperCAmelCase__	:   Any												=	torch.cat([latents] * 2						)

             UpperCAmelCase__	:   Dict												=	self.scheduler.scale_model_input(_UpperCAmelCase  ,		self.scheduler.timesteps[tmax]						)

             with torch.no_grad():
                    UpperCAmelCase__	:   Tuple												=	self.unet(
                        _UpperCAmelCase  ,		self.scheduler.timesteps[tmax]  ,		encoder_hidden_states=_UpperCAmelCase  ,		).sample

             UpperCAmelCase__							,							UpperCAmelCase__	:   List[Any]												=	pred.chunk(2						)
             UpperCAmelCase__	:   List[str]												=	pred_uncond + guidance_scale * (pred_text - pred_uncond)

             UpperCAmelCase__	:   Union[str, Any]												=	self.scheduler.step(_UpperCAmelCase  ,		self.scheduler.timesteps[tmax]  ,		_UpperCAmelCase						).prev_sample

             for i, t in enumerate(tqdm(self.scheduler.timesteps						)						):
                    if i > tmax:
                           if i < tmin:  # layout generation phase
                                  UpperCAmelCase__	:   Dict												=	self.scheduler.add_noise(
                                      _UpperCAmelCase  ,		_UpperCAmelCase  ,		timesteps=_UpperCAmelCase  ,		)

                                  UpperCAmelCase__	:   Union[str, Any]												=	(mix_factor * latents) + (
                                      1 - mix_factor
                                  ) * orig_latents  # interpolating between layout noise and conditionally generated noise to preserve layout sematics
                                  UpperCAmelCase__	:   Tuple												=	torch.cat([input] * 2						)

                           else:  # content generation phase
                                  UpperCAmelCase__	:   Any												=	torch.cat([latents] * 2						)

                           UpperCAmelCase__	:   str												=	self.scheduler.scale_model_input(_UpperCAmelCase  ,		_UpperCAmelCase						)

                           with torch.no_grad():
                                  UpperCAmelCase__	:   Any												=	self.unet(
                                      _UpperCAmelCase  ,		_UpperCAmelCase  ,		encoder_hidden_states=_UpperCAmelCase  ,		).sample

                           UpperCAmelCase__							,							UpperCAmelCase__	:   Tuple												=	pred.chunk(2						)
                           UpperCAmelCase__	:   str												=	pred_uncond + guidance_scale * (pred_text - pred_uncond)

                           UpperCAmelCase__	:   Union[str, Any]												=	self.scheduler.step(_UpperCAmelCase  ,		_UpperCAmelCase  ,		_UpperCAmelCase						).prev_sample

             return self.decode(_UpperCAmelCase						)