

'''simple docstring'''





def 				_lowerCAmelCase			(   __magic_name__						:     list[list]						)				->  list[list]:
						lowercase :							Dict    		=current_set.copy()
						for row_index, row in enumerate(__magic_name__						):
												lowercase :							Union[str, Any]    		=row[0]
												for column_index, column in enumerate(__magic_name__						):
																		if magnitude == 0:
																								lowercase :							str    		=column
																								continue
																		lowercase :							Any    		=column / magnitude
    # Subtract to cancel term
						lowercase :							str    		=current_set[0]
						lowercase :							int    		=[first_row]
						lowercase :							List[str]    		=current_set[1::]
						for row in current_set:
												lowercase :							Optional[Any]    		=[]
												# If first term is 0, it is already in form we want, so we preserve it
												if row[0] == 0:
																		final_set.append(__magic_name__						)
																		continue
												for column_index in range(len(__magic_name__						)						):
																		temp_row.append(first_row[column_index] - row[column_index]						)
												final_set.append(__magic_name__						)
						# Create next recursion iteration set
						if len(final_set[0]						) != 3:
												lowercase :							Optional[int]    		=final_set[0]
												lowercase :							List[Any]    		=[]
												lowercase :							Tuple    		=[]
												for row in final_set[1::]:
																		current_first_column.append(row[0]						)
																		next_iteration.append(row[1::]						)
												lowercase :							Optional[int]    		=simplify(__magic_name__						)
												for i in range(len(__magic_name__						)						):
																		resultant[i].insert(0   , current_first_column[i]						)
												resultant.insert(0   , __magic_name__						)
												lowercase :							List[Any]    		=resultant
						return final_set






def 				_lowerCAmelCase			(   __magic_name__						:     list[list]						)				->  list:
						if len(__magic_name__						) == 0:
												raise IndexError('''solve_simultaneous() requires n lists of length n+1'''						)
						lowercase :							Optional[int]    		=len(__magic_name__						) + 1
						if any(len(__magic_name__						) != _length for item in equations						):
												raise IndexError('''solve_simultaneous() requires n lists of length n+1'''						)
						for row in equations:
												if any(not isinstance(__magic_name__   , (int, float)						) for column in row						):
																		raise ValueError('''solve_simultaneous() requires lists of integers'''						)
						if len(__magic_name__						) == 1:
												return [equations[0][-1] / equations[0][0]]
						lowercase :							List[str]    		=equations.copy()
						if any(0 in row for row in data_set						):
												lowercase :							List[str]    		=data_set.copy()
												lowercase :							str    		=[]
												for row_index, row in enumerate(__magic_name__						):
																		if 0 not in row:
																								lowercase :							Any    		=data_set.pop(__magic_name__						)
																								break
												if not full_row:
																		raise ValueError('''solve_simultaneous() requires at least 1 full equation'''						)
												data_set.insert(0   , __magic_name__						)
						lowercase :							Dict    		=data_set.copy()
						lowercase :							List[Any]    		=simplify(__magic_name__						)
						lowercase :							Dict    		=simplified[::-1]
						lowercase :							list    		=[]
						for row in simplified:
												lowercase :							int    		=row[-1]
												if not solutions:
																		if row[-2] == 0:
																								solutions.append(0						)
																								continue
																		solutions.append(current_solution / row[-2]						)
																		continue
												lowercase :							Union[str, Any]    		=row.copy()[: len(__magic_name__						) - 1 :]
												while temp_row[0] == 0:
																		temp_row.pop(0						)
												if len(__magic_name__						) == 0:
																		solutions.append(0						)
																		continue
												lowercase :							str    		=temp_row[1::]
												lowercase :							Union[str, Any]    		=temp_row[::-1]
												for column_index, column in enumerate(__magic_name__						):
																		current_solution -= column * solutions[column_index]
												solutions.append(__magic_name__						)
						lowercase :							int    		=[]
						for item in solutions:
												final.append(float(round(__magic_name__   , 5						)						)						)
						return final[::-1]


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							UpperCamelCase_    						=       [
							    [2, 1, 1, 1, 1, 4],
							    [1, 2, 1, 1, 1, 5],
							    [1, 1, 2, 1, 1, 6],
							    [1, 1, 1, 2, 1, 7],
							    [1, 1, 1, 1, 2, 8],
							]
							print(solve_simultaneous(eq))
							print(solve_simultaneous([[4, 2]]))








'''simple docstring'''





import math

import flax.linen as nn
import jax.numpy as jnp






def 				_lowerCAmelCase			(   __magic_name__						:     jnp.ndarray   , __magic_name__						:     int   , __magic_name__						:     float = 1   , __magic_name__						:     float = 1   , __magic_name__						:     float = 1.0E4   , __magic_name__						:     bool = False   , __magic_name__						:     float = 1.0   , )				->  jnp.ndarray:
						assert timesteps.ndim == 1, "Timesteps should be a 1d-array"
						assert embedding_dim % 2 == 0, f'''Embedding dimension {embedding_dim} should be even'''
						lowercase :							int    		=float(embedding_dim // 2						)
						lowercase :							Optional[int]    		=math.log(max_timescale / min_timescale						) / (num_timescales - freq_shift)
						lowercase :							Any    		=min_timescale * jnp.exp(jnp.arange(__magic_name__   , dtype=jnp.floataa						) * -log_timescale_increment						)
						lowercase :							List[Any]    		=jnp.expand_dims(__magic_name__   , 1						) * jnp.expand_dims(__magic_name__   , 0						)

						# scale embeddings
						lowercase :							Tuple    		=scale * emb

						if flip_sin_to_cos:
												lowercase :							Dict    		=jnp.concatenate([jnp.cos(__magic_name__						), jnp.sin(__magic_name__						)]   , axis=1						)
						else:
												lowercase :							Any    		=jnp.concatenate([jnp.sin(__magic_name__						), jnp.cos(__magic_name__						)]   , axis=1						)
						lowercase :							List[str]    		=jnp.reshape(__magic_name__   , [jnp.shape(__magic_name__						)[0], embedding_dim]						)
						return signal





class 		__SCREAMING_SNAKE_CASE							(			nn.Module      ):
				lowerCamelCase_		       =    32
				lowerCamelCase_		       =    jnp.floataa







				@nn.compact
				def __call__(       self     :		Tuple		,					UpperCAmelCase__     :		int   ):
										'''simple docstring'''
										lowercase :							List[Any]    		=nn.Dense(self.time_embed_dim		,					dtype=self.dtype		,					name='''linear_1'''   )(UpperCAmelCase__   )
										lowercase :							Any    		=nn.silu(UpperCAmelCase__   )
										lowercase :							int    		=nn.Dense(self.time_embed_dim		,					dtype=self.dtype		,					name='''linear_2'''   )(UpperCAmelCase__   )
										return temb





class 		__SCREAMING_SNAKE_CASE							(			nn.Module      ):
				lowerCamelCase_		       =    32
				lowerCamelCase_		       =    False
				lowerCamelCase_		       =    1







				@nn.compact
				def __call__(       self     :		int		,					UpperCAmelCase__     :		str   ):
										'''simple docstring'''
										return get_sinusoidal_embeddings(
										    UpperCAmelCase__		,					embedding_dim=self.dim		,					flip_sin_to_cos=self.flip_sin_to_cos		,					freq_shift=self.freq_shift   )





