
def 					__lowerCAmelCase    (    a__						) ->						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.'''						)

				__a    							=					{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.'''						)

				__a    							=					len(a__						)

				__a    							=					0
				__a    							=					[]
				# put the x and x to the power values in a matrix
				while count_of_line < x:
								__a    							=					0
								__a    							=					coordinates[count_of_line][0]
								__a    							=					[]
								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

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

				__a    							=					0

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

				__a    							=					0
				__a    							=					'''f(x)='''

				while count < x:
								__a    							=					solution[count].split('''E'''						)
								if len(a__						) > 1:
												__a    							=					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]]))

import importlib
import importlib.metadata
import os
import warnings
from distutils.util import strtobool
from functools import lru_cache

import torch
from packaging import version
from packaging.version import parse

from .environment import parse_flag_from_env
from .versions import compare_versions, is_torch_version


try:
					import torch_xla.core.xla_model as xm  # noqa: F401

					A  :			List[str]     				=				True
except ImportError:
					A  :			List[Any]     				=				False


# Cache this result has it's a C FFI call which can be pretty time-consuming
A  :			str     				=				torch.distributed.is_available()


def 					__lowerCAmelCase    (    a__						) ->						List[str]:
				# Check we're not importing a "pkg_name" directory somewhere but the actual library by trying to grab the version
				__a    							=					importlib.util.find_spec(a__						) is not None
				if package_exists:
								try:
												__a    							=					importlib.metadata.metadata(a__						)
												return True
								except importlib.metadata.PackageNotFoundError:
												return False


def 					__lowerCAmelCase    (    ) ->						bool:
				return _torch_distributed_available


def 					__lowerCAmelCase    (    ) ->						Tuple:
				try:
								pass
				except ImportError:
								print(
								    '''Intel(R) oneCCL Bindings for PyTorch* is required to run DDP on Intel(R) GPUs, but it is not'''
								    ''' detected. If you see \"ValueError: Invalid backend: \'ccl\'\" error, please install Intel(R) oneCCL'''
								    ''' Bindings for PyTorch*.'''						)
				return (
				    importlib.util.find_spec('''torch_ccl'''						) is not None
				    or importlib.util.find_spec('''oneccl_bindings_for_pytorch'''						) is not None
				)


def 					__lowerCAmelCase    (    ) ->						Union[str, Any]:
				return importlib.metadata.version('''oneccl_bind_pt'''						)


def 					__lowerCAmelCase    (    ) ->						Dict:
				return _is_package_available('''transformer_engine'''						)


@lru_cache
def 					__lowerCAmelCase    (    a__=True						) ->						Union[str, Any]:
				# Due to bugs on the amp series GPUs, we disable torch-xla on them
				if torch.cuda.is_available():
								return False
				if _tpu_available and check_device:
								try:
												# Will raise a RuntimeError if no XLA configuration is found
												__a    							=					xm.xla_device()
												return True
								except RuntimeError:
												return False
				return _tpu_available


def 					__lowerCAmelCase    (    ) ->						Any:
				return _is_package_available('''deepspeed'''						)


def 					__lowerCAmelCase    (    a__=False						) ->						Optional[Any]:
				if is_tpu_available():
								return not ignore_tpu
				if torch.cuda.is_available():
								return torch.cuda.is_bfaa_supported()
				if is_npu_available():
								return False
				return True


def 					__lowerCAmelCase    (    ) ->						List[str]:
				__a    							=					_is_package_available('''bitsandbytes'''						)
				if package_exists:
								__a    							=					version.parse(importlib.metadata.version('''bitsandbytes'''						)						)
								return compare_versions(a__		,					'''>='''		,					'''0.39.0'''						)
				return False


def 					__lowerCAmelCase    (    ) ->						Any:
				__a    							=					_is_package_available('''bitsandbytes'''						)
				if package_exists:
								__a    							=					version.parse(importlib.metadata.version('''bitsandbytes'''						)						)
								return compare_versions(a__		,					'''>='''		,					'''0.37.2'''						)
				return False


def 					__lowerCAmelCase    (    ) ->						int:
				return _is_package_available('''bitsandbytes'''						)


def 					__lowerCAmelCase    (    ) ->						str:
				if strtobool(os.environ.get('''ACCELERATE_USE_MEGATRON_LM'''		,					'''False'''						)						) == 1:
								__a    							=					importlib.util.find_spec('''megatron'''						) is not None
								if package_exists:
												try:
																__a    							=					parse(importlib.metadata.version('''megatron-lm'''						)						)
																return compare_versions(a__		,					'''>='''		,					'''2.2.0'''						)
												except Exception as e:
																warnings.warn(F"""Parse Megatron version failed. Exception:{e}"""						)
																return False


def 					__lowerCAmelCase    (    ) ->						List[str]:
				return _is_package_available('''safetensors'''						)


def 					__lowerCAmelCase    (    ) ->						Optional[Any]:
				return _is_package_available('''transformers'''						)


def 					__lowerCAmelCase    (    ) ->						int:
				return _is_package_available('''datasets'''						)


def 					__lowerCAmelCase    (    ) ->						Optional[int]:
				return _is_package_available('''aim'''						)


def 					__lowerCAmelCase    (    ) ->						List[Any]:
				return _is_package_available('''tensorboard'''						) or _is_package_available('''tensorboardX'''						)


def 					__lowerCAmelCase    (    ) ->						List[Any]:
				return _is_package_available('''wandb'''						)


def 					__lowerCAmelCase    (    ) ->						Optional[int]:
				return _is_package_available('''comet_ml'''						)


def 					__lowerCAmelCase    (    ) ->						str:
				return _is_package_available('''boto3'''						)


def 					__lowerCAmelCase    (    ) ->						Optional[Any]:
				if _is_package_available('''rich'''						):
								if "ACCELERATE_DISABLE_RICH" in os.environ:
												warnings.warn(
												    '''`ACCELERATE_DISABLE_RICH` is deprecated and will be removed in v0.22.0 and deactivated by default. Please use `ACCELERATE_ENABLE_RICH` if you wish to use `rich`.'''						)
												return not parse_flag_from_env('''ACCELERATE_DISABLE_RICH'''		,					a__						)
								return parse_flag_from_env('''ACCELERATE_ENABLE_RICH'''		,					a__						)
				return False


def 					__lowerCAmelCase    (    ) ->						Any:
				return _is_package_available('''sagemaker'''						)


def 					__lowerCAmelCase    (    ) ->						int:
				return _is_package_available('''tqdm'''						)


def 					__lowerCAmelCase    (    ) ->						Union[str, Any]:
				return _is_package_available('''mlflow'''						)


def 					__lowerCAmelCase    (    ) ->						Optional[Any]:
				return is_torch_version('''>='''		,					'''1.12'''						) and torch.backends.mps.is_available() and torch.backends.mps.is_built()


def 					__lowerCAmelCase    (    ) ->						str:
				def get_major_and_minor_from_version(a__						):
								return str(version.parse(a__						).major						) + "." + str(version.parse(a__						).minor						)

				__a    							=					importlib.metadata.version('''torch'''						)
				if importlib.util.find_spec('''intel_extension_for_pytorch'''						) is None:
								return False
				__a    							=					'''N/A'''
				try:
								__a    							=					importlib.metadata.version('''intel_extension_for_pytorch'''						)
				except importlib.metadata.PackageNotFoundError:
								return False
				__a    							=					get_major_and_minor_from_version(_torch_version						)
				__a    							=					get_major_and_minor_from_version(_ipex_version						)
				if torch_major_and_minor != ipex_major_and_minor:
								warnings.warn(
								    F"""Intel Extension for PyTorch {ipex_major_and_minor} needs to work with PyTorch {ipex_major_and_minor}.*,"""
								    F""" but PyTorch {_torch_version} is found. Please switch to the matching version and run again."""						)
								return False
				return True


@lru_cache
def 					__lowerCAmelCase    (    a__=False						) ->						List[Any]:
				if importlib.util.find_spec('''torch'''						) is None or importlib.util.find_spec('''torch_npu'''						) is None:
								return False

				import torch
				import torch_npu  # noqa: F401

				if check_device:
								try:
												# Will raise a RuntimeError if no NPU is found
												__a    							=					torch.npu.device_count()
												return torch.npu.is_available()
								except RuntimeError:
												return False
				return hasattr(a__		,					'''npu'''						) and torch.npu.is_available()






@lru_cache
def 					__lowerCAmelCase    (    a__=False						) ->						Union[str, Any]:
				if not parse_flag_from_env('''ACCELERATE_USE_XPU'''		,					default=a__						):
								return False
				if is_ipex_available():
								import torch

								if is_torch_version('''<='''		,					'''1.12'''						):
												return False
				else:
								return False

				import intel_extension_for_pytorch  # noqa: F401

				if check_device:
								try:
												# Will raise a RuntimeError if no XPU  is found
												__a    							=					torch.xpu.device_count()
												return torch.xpu.is_available()
								except RuntimeError:
												return False
				return hasattr(a__		,					'''xpu'''						) and torch.xpu.is_available()