'''simple docstring'''

import torch
import torch.nn as nn
from transformers.modeling_utils import ModuleUtilsMixin
from transformers.models.ta.modeling_ta import (
    TaBlock,
    TaConfig,
    TaLayerNorm,
)

from ...configuration_utils import ConfigMixin, register_to_config
from ...models import ModelMixin




class    lowerCamelCase__  (   __snake_case       ,				__snake_case       ,				__snake_case     ):



   @register_to_config
   def __init__(    self					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__					,      lowerCAmelCase__ = False					,      )				->     Dict:
        """simple docstring"""





        super().__init__()

        _UpperCamelCase  :List[Any]       =nn.Linear(lowerCAmelCase__					,      lowerCAmelCase__					,      bias=lowerCAmelCase__     )

        _UpperCamelCase  :List[str]       =nn.Embedding(lowerCAmelCase__					,      lowerCAmelCase__     )
        _UpperCamelCase  :Any       =False

        _UpperCamelCase  :Optional[Any]       =nn.Dropout(p=lowerCAmelCase__     )

        _UpperCamelCase  :List[str]       =TaConfig(
            d_model=lowerCAmelCase__					,      num_heads=lowerCAmelCase__					,      d_kv=lowerCAmelCase__					,      d_ff=lowerCAmelCase__					,      feed_forward_proj=lowerCAmelCase__					,      dropout_rate=lowerCAmelCase__					,      is_decoder=lowerCAmelCase__					,      is_encoder_decoder=lowerCAmelCase__					,      )
        _UpperCamelCase  :Dict       =nn.ModuleList()
        for lyr_num in range(lowerCAmelCase__     ):
             _UpperCamelCase  :List[str]       =TaBlock(lowerCAmelCase__     )
             self.encoders.append(lowerCAmelCase__     )

        _UpperCamelCase  :str       =TaLayerNorm(lowerCAmelCase__     )
        _UpperCamelCase  :List[str]       =nn.Dropout(p=lowerCAmelCase__     )




   def  _UpperCamelCase		(    self					,      lowerCAmelCase__					,      lowerCAmelCase__     )				->     Any:
        """simple docstring"""





        _UpperCamelCase  :str       =self.input_proj(lowerCAmelCase__     )

        # terminal relative positional encodings
        _UpperCamelCase  :List[str]       =encoder_inputs.shape[1]
        _UpperCamelCase  :Union[str, Any]       =torch.arange(lowerCAmelCase__					,      device=encoder_inputs.device     )

        _UpperCamelCase  :Dict       =encoder_inputs_mask.sum(-1     )
        _UpperCamelCase  :Optional[Any]       =torch.roll(input_positions.unsqueeze(0     )					,      tuple(seq_lens.tolist()     )					,      dims=0     )
        x += self.position_encoding(lowerCAmelCase__     )

        _UpperCamelCase  :List[str]       =self.dropout_pre(lowerCAmelCase__     )

        # inverted the attention mask
        _UpperCamelCase  :Dict       =encoder_inputs.size()
        _UpperCamelCase  :Any       =self.get_extended_attention_mask(lowerCAmelCase__					,      lowerCAmelCase__     )

        for lyr in self.encoders:
             _UpperCamelCase  :Union[str, Any]       =lyr(lowerCAmelCase__					,      lowerCAmelCase__     )[0]
        _UpperCamelCase  :str       =self.layer_norm(lowerCAmelCase__     )

        return self.dropout_post(lowerCAmelCase__     ), encoder_inputs_mask





def 		lowercase__ (		__A:						list[list[int]]					):





					'''simple docstring'''






					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.'''					)

					__magic_name__				:						Union[str, 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.'''					)

					__magic_name__				:						List[Any]     				=				len(__A					)

					__magic_name__				:						int     				=				0
					__magic_name__				:						list[list[float]]     				=				[]
					# put the x and x to the power values in a matrix
					while count_of_line < x:
										__magic_name__				:						List[Any]     				=				0
										__magic_name__				:						List[str]     				=				coordinates[count_of_line][0]
										__magic_name__				:						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

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

					__magic_name__				:						int     				=				0

					while count < x:
										__magic_name__				:						int     				=				0
										while zahlen < x:
															if count == zahlen:
																				zahlen += 1
															if zahlen == x:
																				break
															__magic_name__				:						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

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

					__magic_name__				:						List[Any]     				=				0
					__magic_name__				:						Union[str, Any]     				=				'''f(x)='''

					while count < x:
										__magic_name__				:						list[str]     				=				solution[count].split('''E'''					)
										if len(__A					) > 1:
															__magic_name__				:						Union[str, 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]]))



