





def  SCREAMING_SNAKE_CASE__							(				UpperCamelCase__:							list[list[int]]			):




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

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

		SCREAMING_SNAKE_CASE__								=					{x for x, _ in coordinates}
		if len(UpperCamelCase__			) == 1:
				return f'''x={coordinates[0][0]}'''

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

		SCREAMING_SNAKE_CASE__								=					len(UpperCamelCase__			)

		SCREAMING_SNAKE_CASE__								=					0
		SCREAMING_SNAKE_CASE__								=					[]
		# put the x and x to the power values in a matrix
		while count_of_line < x:
				SCREAMING_SNAKE_CASE__								=					0
				SCREAMING_SNAKE_CASE__								=					coordinates[count_of_line][0]
				SCREAMING_SNAKE_CASE__								=					[]
				while count_in_line < x:
						count_line.append(a ** (x - (count_in_line + 1))			)
						count_in_line += 1
				matrix.append(UpperCamelCase__			)
				count_of_line += 1

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

		SCREAMING_SNAKE_CASE__								=					0

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

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

		SCREAMING_SNAKE_CASE__								=					0
		SCREAMING_SNAKE_CASE__								=					"""f(x)="""

		while count < x:
				SCREAMING_SNAKE_CASE__								=					solution[count].split("""E"""			)
				if len(UpperCamelCase__			) > 1:
						SCREAMING_SNAKE_CASE__								=					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]]))





import os
import random
from pathlib import Path
from typing import List

import numpy as np
import torch
from torch.cuda.amp import GradScaler

from .utils import (
    MODEL_NAME,
    OPTIMIZER_NAME,
    RNG_STATE_NAME,
    SCALER_NAME,
    SCHEDULER_NAME,
    get_pretty_name,
    is_tpu_available,
    is_xpu_available,
    save,
)


if is_tpu_available(check_device=False):
      import torch_xla.core.xla_model as xm

from .logging import get_logger
from .state import PartialState


__SCREAMING_SNAKE_CASE			:      Union[str, Any]      			=get_logger(__name__)

def 	lowerCAmelCase_	(     lowerCamelCase:				str   , lowerCamelCase:				List[dict]   , lowerCamelCase:				list   , lowerCamelCase:				list   , lowerCamelCase:				int   , lowerCamelCase:				GradScaler = None   , )    ->	Any:


      '''simple docstring'''




      for i, state in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = f"""{MODEL_NAME}.bin""" if i == 0 else f"""{MODEL_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            save(lowerCamelCase   , lowerCamelCase  )
            logger.info(f"""Model weights saved in {output_model_file}"""  )
      # Optimizer states
      for i, opt in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = opt.state_dict()
            lowerCamelCase__					     = f"""{OPTIMIZER_NAME}.bin""" if i == 0 else f"""{OPTIMIZER_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            save(lowerCamelCase   , lowerCamelCase  )
            logger.info(f"""Optimizer state saved in {output_optimizer_file}"""  )
      # Scheduler states
      for i, scheduler in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = scheduler.state_dict()
            lowerCamelCase__					     = f"""{SCHEDULER_NAME}.bin""" if i == 0 else f"""{SCHEDULER_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            save(lowerCamelCase   , lowerCamelCase  )
            logger.info(f"""Scheduler state saved in {output_scheduler_file}"""  )
      # GradScaler state
      if scaler is not None:
            lowerCamelCase__					     = scaler.state_dict()
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            torch.save(lowerCamelCase   , lowerCamelCase  )
            logger.info(f"""Gradient scaler state saved in {output_scaler_file}"""  )
      # Random number generator states
      lowerCamelCase__					     = {}
      lowerCamelCase__					     = f"""{RNG_STATE_NAME}_{process_index}.pkl"""
      lowerCamelCase__					     = random.getstate()
      lowerCamelCase__					     = np.random.get_state()
      lowerCamelCase__					     = torch.get_rng_state()
      if is_xpu_available():
            lowerCamelCase__					     = torch.xpu.get_rng_state_all()
      else:
            lowerCamelCase__					     = torch.cuda.get_rng_state_all()
      if is_tpu_available():
            lowerCamelCase__					     = xm.get_rng_state()
      lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
      torch.save(lowerCamelCase   , lowerCamelCase  )
      logger.info(f"""Random states saved in {output_states_file}"""  )
      return output_dir

def 	lowerCAmelCase_	(     lowerCamelCase:				int   , lowerCamelCase:				Dict   , lowerCamelCase:				Union[str, Any]   , lowerCamelCase:				Any   , lowerCamelCase:				Any   , lowerCamelCase:				Any=None   , lowerCamelCase:				Dict=None   , **lowerCamelCase:				Optional[Any]   , )    ->	Any:


      '''simple docstring'''




      if map_location not in [None, "cpu", "on_device"]:
            raise TypeError(
                'Unsupported optimizer map location passed, please choose one of `None`, `\'cpu\'`, or `\'on_device\'`'  )
      if map_location is None:
            lowerCamelCase__					     = 'cpu'
      elif map_location == "on_device":
            lowerCamelCase__					     = PartialState().device
      # Model states
      for i, model in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = f"""{MODEL_NAME}.bin""" if i == 0 else f"""{MODEL_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            models[i].load_state_dict(torch.load(lowerCamelCase   , map_location=lowerCamelCase  )   , **lowerCamelCase  )
      logger.info('All model weights loaded successfully'  )

      # Optimizer states
      for i, opt in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = f"""{OPTIMIZER_NAME}.bin""" if i == 0 else f"""{OPTIMIZER_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            lowerCamelCase__					     = torch.load(lowerCamelCase   , map_location=lowerCamelCase  )
            optimizers[i].load_state_dict(lowerCamelCase  )
      logger.info('All optimizer states loaded successfully'  )

      # Scheduler states
      for i, scheduler in enumerate(lowerCamelCase  ):
            lowerCamelCase__					     = f"""{SCHEDULER_NAME}.bin""" if i == 0 else f"""{SCHEDULER_NAME}_{i}.bin"""
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            scheduler.load_state_dict(torch.load(lowerCamelCase  )  )
      logger.info('All scheduler states loaded successfully'  )

      # GradScaler state
      if scaler is not None:
            lowerCamelCase__					     = os.path.join(lowerCamelCase   , lowerCamelCase  )
            scaler.load_state_dict(torch.load(lowerCamelCase  )  )
            logger.info('GradScaler state loaded successfully'  )

      # Random states
      try:
            lowerCamelCase__					     = torch.load(os.path.join(lowerCamelCase   , f"""{RNG_STATE_NAME}_{process_index}.pkl"""  )  )
            random.setstate(states['random_state']  )
            np.random.set_state(states['numpy_random_seed']  )
            torch.set_rng_state(states['torch_manual_seed']  )
            if is_xpu_available():
                  torch.xpu.set_rng_state_all(states['torch_xpu_manual_seed']  )
            else:
                  torch.cuda.set_rng_state_all(states['torch_cuda_manual_seed']  )
            if is_tpu_available():
                  xm.set_rng_state(states['xm_seed']  )
            logger.info('All random states loaded successfully'  )
      except Exception:
            logger.info('Could not load random states'  )

def 	lowerCAmelCase_	(     lowerCamelCase:				Dict   , lowerCamelCase:				Any   , lowerCamelCase:				int = 0  )    ->	Union[str, Any]:


      '''simple docstring'''




      lowerCamelCase__					     = Path(lowerCamelCase  ) / f"""custom_checkpoint_{index}.pkl"""
      logger.info(f"""Saving the state of {get_pretty_name(lowerCamelCase  )} to {save_location}"""  )
      torch.save(obj.state_dict()   , lowerCamelCase  )


def 	lowerCAmelCase_	(     lowerCamelCase:				int   , lowerCamelCase:				Optional[int]   , lowerCamelCase:				int = 0  )    ->	Any:


      '''simple docstring'''




      lowerCamelCase__					     = f"""{path}/custom_checkpoint_{index}.pkl"""
      logger.info(f"""Loading the state of {get_pretty_name(lowerCamelCase  )} from {load_location}"""  )
      obj.load_state_dict(torch.load(lowerCamelCase   , map_location='cpu'  )  )