


'''simple docstring'''







import re

from flax.core.frozen_dict import freeze
from flax.traverse_util import flatten_dict, unflatten_dict
from jax.experimental import PartitionSpec as P


# Sentinels
A__:    int					    =       object()

# For specifying empty leaf dict `{}`
A__:    str					    =       object()


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	List[Any]    ,_UpperCAmelCase	:	int  )  ->			Any:
       _a    : List[str]     					=tuple((re.compile(x + """$"""  ) for x in qs)  )
       for i in range(len(_lowercase  ) - len(_lowercase  ) + 1  ):
              _a    : Union[str, Any]     					=[x.match(_lowercase  ) for x, y in zip(_lowercase    ,ks[i:]  )]
              if matches and all(_lowercase  ):
                     return True
       return False


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	Optional[Any]  )  ->			List[Any]:


       def replace(_UpperCAmelCase	:	Tuple    ,_UpperCAmelCase	:	Any  ):
              for rule, replacement in rules:
                     if _match(_lowercase    ,_lowercase  ):
                            return replacement
              return val

       return replace


def 	SCREAMING_SNAKE_CASE_					(  )  ->			Dict:
       return [
           # embeddings
           (("transformer", "wpe", "embedding"), P("""mp"""    ,_lowercase  )),
           (("transformer", "wte", "embedding"), P("""mp"""    ,_lowercase  )),
           # atention
           (("attention", "(q_proj|k_proj|v_proj)", "kernel"), P(_lowercase    ,"""mp"""  )),
           (("attention", "out_proj", "kernel"), P("""mp"""    ,_lowercase  )),
           (("attention", "out_proj", "bias"), None),
           # mlp
           (("mlp", "c_fc", "kernel"), P(_lowercase    ,"""mp"""  )),
           (("mlp", "c_fc", "bias"), P("""mp"""  )),
           (("mlp", "c_proj", "kernel"), P("""mp"""    ,_lowercase  )),
           (("mlp", "c_proj", "bias"), None),
           # layer norms
           ((r"ln_\d+", "bias"), None),
           ((r"\d+", r"ln_\d+", "scale"), None),
           (("ln_f", "bias"), None),
           (("ln_f", "scale"), None),
       ]


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	List[str]  )  ->			List[Any]:
       _a    : Optional[Any]     					=_get_partition_rules()
       _a    : Dict     					=_replacement_rules(_lowercase  )
       _a    : Tuple     					={k: _unmatched for k in flatten_dict(_lowercase  )}
       _a    : List[str]     					={k: replace(_lowercase    ,_lowercase  ) for k, v in initd.items()}
       assert _unmatched not in result.values(), "Incomplete partition spec."
       return freeze(unflatten_dict(_lowercase  )  )







"""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))
