



'''simple docstring'''
from typing import List

import numpy as np






def A__    (     UpperCAmelCase_							):
						_UpperCamelCase					:		Optional[Any] 				=   {key: len(UpperCAmelCase_							) for key, value in gen_kwargs.items() if isinstance(UpperCAmelCase_  ,  UpperCAmelCase_							)}
						if len(set(lists_lengths.values()							)							) > 1:
												raise RuntimeError(
												    (
												        'Sharding is ambiguous for this dataset: '
												        + 'we found several data sources lists of different lengths, and we don\'t know over which list we should parallelize:\n'
												        + '\n'.join(f'\t- key {key} has length {length}' for key, length in lists_lengths.items()							)
												        + '\nTo fix this, check the \'gen_kwargs\' and make sure to use lists only for data sources, '
												        + 'and use tuples otherwise. In the end there should only be one single list, or several lists with the same length.'
												    )							)
						_UpperCamelCase					:		Union[str, Any] 				=   max(lists_lengths.values()  ,  default=0							)
						return max(1  ,  UpperCAmelCase_							)






def A__    (     UpperCAmelCase_  ,  UpperCAmelCase_							):
						_UpperCamelCase					:		Dict 				=   []
						for group_idx in range(UpperCAmelCase_							):
												_UpperCamelCase					:		Tuple 				=   num_shards // max_num_jobs + (group_idx < (num_shards % max_num_jobs))
												if num_shards_to_add == 0:
																		break
												_UpperCamelCase					:		Union[str, Any] 				=   shards_indices_per_group[-1].stop if shards_indices_per_group else 0
												_UpperCamelCase					:		Union[str, Any] 				=   range(UpperCAmelCase_  ,  start + num_shards_to_add							)
												shards_indices_per_group.append(UpperCAmelCase_							)
						return shards_indices_per_group






def A__    (     UpperCAmelCase_  ,  UpperCAmelCase_							):
						_UpperCamelCase					:		List[str] 				=   _number_of_shards_in_gen_kwargs(UpperCAmelCase_							)
						if num_shards == 1:
												return [dict(UpperCAmelCase_							)]
						else:
												_UpperCamelCase					:		str 				=   _distribute_shards(num_shards=UpperCAmelCase_  ,  max_num_jobs=UpperCAmelCase_							)
												return [
												    {
												        key: [value[shard_idx] for shard_idx in shard_indices_per_group[group_idx]]
												        if isinstance(UpperCAmelCase_  ,  UpperCAmelCase_							)
												        else value
												        for key, value in gen_kwargs.items()
												    }
												    for group_idx in range(len(UpperCAmelCase_							)							)
												]






def A__    (     UpperCAmelCase_							):
						return {
						    key: [value for gen_kwargs in gen_kwargs_list for value in gen_kwargs[key]]
						    if isinstance(gen_kwargs_list[0][key]  ,  UpperCAmelCase_							)
						    else gen_kwargs_list[0][key]
						    for key in gen_kwargs_list[0]
						}






def A__    (     UpperCAmelCase_  ,  UpperCAmelCase_							):
						_UpperCamelCase					:		Optional[Any] 				=   {len(UpperCAmelCase_							) for value in gen_kwargs.values() if isinstance(UpperCAmelCase_  ,  UpperCAmelCase_							)}
						_UpperCamelCase					:		Tuple 				=   {}
						for size in list_sizes:
												_UpperCamelCase					:		str 				=   list(range(UpperCAmelCase_							)							)
												rng.shuffle(indices_per_size[size]							)
						# Now let's copy the gen_kwargs and shuffle the lists based on their sizes
						_UpperCamelCase					:		Tuple 				=   dict(UpperCAmelCase_							)
						for key, value in shuffled_kwargs.items():
												if isinstance(UpperCAmelCase_  ,  UpperCAmelCase_							):
																		_UpperCamelCase					:		Tuple 				=   [value[i] for i in indices_per_size[len(UpperCAmelCase_							)]]
						return shuffled_kwargs







"""simple docstring"""
from math import factorial
class 				lowerCamelCase  :






					'''simple docstring'''





					def __init__(       self:							Optional[int]						,     snake_case:							Dict						,     snake_case:							int						)   ->	Tuple:
											snake_case_ :List[Any]						=			real
											if isinstance(snake_case						,     snake_case						):
																	snake_case_ :Tuple						=			[1] * rank
											else:
																	snake_case_ :Optional[Any]						=			rank






					def __repr__(       self:							List[str]						)   ->	Tuple:
											return (
											    f"""{self.real}+"""
											    f"""{'+'.join(str(snake_case						)+'E'+str(n+1						)for n,dual in enumerate(self.duals						)						)}"""
											)






					def    lowerCAmelCase_				(       self:							Optional[int]						)   ->	Optional[int]:
											snake_case_ :Any						=			self.duals.copy()
											while cur[-1] == 0:
																	cur.pop(-1						)
											return Dual(self.real						,     snake_case						)






					def __add__(       self:							Optional[int]						,     snake_case:							Dict						)   ->	List[str]:
											if not isinstance(snake_case						,     snake_case						):
																	return Dual(self.real + other						,     self.duals						)
											snake_case_ :List[Any]						=			self.duals.copy()
											snake_case_ :Tuple						=			other.duals.copy()
											if len(snake_case						) > len(snake_case						):
																	o_dual.extend([1] * (len(snake_case						) - len(snake_case						))						)
											elif len(snake_case						) < len(snake_case						):
																	s_dual.extend([1] * (len(snake_case						) - len(snake_case						))						)
											snake_case_ :Dict						=			[]
											for i in range(len(snake_case						)						):
																	new_duals.append(s_dual[i] + o_dual[i]						)
											return Dual(self.real + other.real						,     snake_case						)

					_A		:			str						=     __add__






					def __sub__(       self:							Tuple						,     snake_case:							Union[str, Any]						)   ->	Tuple:
											return self + other * -1






					def __mul__(       self:							str						,     snake_case:							Tuple						)   ->	Optional[Any]:
											if not isinstance(snake_case						,     snake_case						):
																	snake_case_ :Dict						=			[]
																	for i in self.duals:
																							new_duals.append(i * other						)
																	return Dual(self.real * other						,     snake_case						)
											snake_case_ :int						=			[0] * (len(self.duals						) + len(other.duals						) + 1)
											for i, item in enumerate(self.duals						):
																	for j, jtem in enumerate(other.duals						):
																							new_duals[i + j + 1] += item * jtem
											for k in range(len(self.duals						)						):
																	new_duals[k] += self.duals[k] * other.real
											for index in range(len(other.duals						)						):
																	new_duals[index] += other.duals[index] * self.real
											return Dual(self.real * other.real						,     snake_case						)

					_A		:			int						=     __mul__






					def __truediv__(       self:							List[str]						,     snake_case:							List[str]						)   ->	List[str]:
											if not isinstance(snake_case						,     snake_case						):
																	snake_case_ :Optional[Any]						=			[]
																	for i in self.duals:
																							new_duals.append(i / other						)
																	return Dual(self.real / other						,     snake_case						)
											raise ValueError






					def __floordiv__(       self:							int						,     snake_case:							List[Any]						)   ->	Any:
											if not isinstance(snake_case						,     snake_case						):
																	snake_case_ :Optional[int]						=			[]
																	for i in self.duals:
																							new_duals.append(i // other						)
																	return Dual(self.real // other						,     snake_case						)
											raise ValueError






					def __pow__(       self:							Optional[Any]						,     snake_case:							Optional[int]						)   ->	List[Any]:
											if n < 0 or isinstance(snake_case						,     snake_case						):
																	raise ValueError("""power must be a positive integer"""						)
											if n == 0:
																	return 1
											if n == 1:
																	return self
											snake_case_ :str						=			self
											for _ in range(n - 1						):
																	x *= self
											return x

def    A_  (			_lowercase,					_lowercase,					_lowercase	):







						'''simple docstring'''


						if not callable(_lowercase	):
												raise ValueError("""differentiate() requires a function as input for func"""	)
						if not isinstance(_lowercase,					(float, int)	):
												raise ValueError("""differentiate() requires a float as input for position"""	)
						if not isinstance(_lowercase,					_lowercase	):
												raise ValueError("""differentiate() requires an int as input for order"""	)
						snake_case_ :Optional[Any]						=			Dual(_lowercase,					1	)
						snake_case_ :List[Any]						=			func(_lowercase	)
						if order == 0:
												return result.real
						return result.duals[order - 1] * factorial(_lowercase	)


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						def    A_  (			_lowercase	):







												'''simple docstring'''


												return y**2 * y**4

						print(differentiate(f, 9, 2))
