


'''simple docstring'''


import unittest

import numpy as np

from transformers import is_flax_available
from transformers.testing_utils import require_flax

from ..test_modeling_flax_common import ids_tensor


if is_flax_available():
						import jax
						import jax.numpy as jnp

						from transformers.generation import (
						    FlaxForcedBOSTokenLogitsProcessor,
						    FlaxForcedEOSTokenLogitsProcessor,
						    FlaxLogitsProcessorList,
						    FlaxMinLengthLogitsProcessor,
						    FlaxTemperatureLogitsWarper,
						    FlaxTopKLogitsWarper,
						    FlaxTopPLogitsWarper,
						)






@require_flax
class 	lowerCAmelCase__			(			unittest.TestCase	):




			def 							lowerCAmelCase__    (					self  :     List[str]	,  lowerCamelCase__  :     int	,  lowerCamelCase__  :     int )   ->Dict:



					'''simple docstring'''







					_UpperCAmelCase		:       Union[str, Any]								=      jnp.ones((batch_size, length) ) / length
					return scores




			def 							lowerCAmelCase__    (					self  :     Optional[int] )   ->Any:



					'''simple docstring'''







					_UpperCAmelCase		:       List[Any]								=      None
					_UpperCAmelCase		:       Dict								=      20

					_UpperCAmelCase		:       List[str]								=      self._get_uniform_logits(batch_size=2	,  length=lowerCamelCase__ )

					# tweak scores to not be uniform anymore
					_UpperCAmelCase		:       Optional[int]								=      scores.at[1, 5].set((1 / length) + 0.1 )  # peak, 1st batch
					_UpperCAmelCase		:       Optional[Any]								=      scores.at[1, 10].set((1 / length) - 0.4 )  # valley, 1st batch

					# compute softmax
					_UpperCAmelCase		:       List[Any]								=      jax.nn.softmax(lowerCamelCase__	,  axis=-1 )

					_UpperCAmelCase		:       int								=      FlaxTemperatureLogitsWarper(temperature=0.5 )
					_UpperCAmelCase		:       List[Any]								=      FlaxTemperatureLogitsWarper(temperature=1.3 )

					_UpperCAmelCase		:       List[Any]								=      jax.nn.softmax(temp_dist_warper_sharper(lowerCamelCase__	,  scores.copy()	,  cur_len=lowerCamelCase__ )	,  axis=-1 )
					_UpperCAmelCase		:       Any								=      jax.nn.softmax(temp_dist_warper_smoother(lowerCamelCase__	,  scores.copy()	,  cur_len=lowerCamelCase__ )	,  axis=-1 )

					# uniform distribution stays uniform
					self.assertTrue(jnp.allclose(probs[0, :]	,  warped_prob_sharp[0, :]	,  atol=1E-3 ) )
					self.assertTrue(jnp.allclose(probs[0, :]	,  warped_prob_smooth[0, :]	,  atol=1E-3 ) )

					# sharp peaks get higher, valleys get lower
					self.assertLess(probs[1, :].max()	,  warped_prob_sharp[1, :].max() )
					self.assertGreater(probs[1, :].min()	,  warped_prob_sharp[1, :].min() )

					# smooth peaks get lower, valleys get higher
					self.assertGreater(probs[1, :].max()	,  warped_prob_smooth[1, :].max() )
					self.assertLess(probs[1, :].min()	,  warped_prob_smooth[1, :].min() )




			def 							lowerCAmelCase__    (					self  :     Optional[int] )   ->Union[str, Any]:



					'''simple docstring'''







					_UpperCAmelCase		:       Any								=      None
					_UpperCAmelCase		:       int								=      10
					_UpperCAmelCase		:       Tuple								=      2

					# create ramp distribution
					_UpperCAmelCase		:       Optional[int]								=      np.broadcast_to(np.arange(lowerCamelCase__ )[None, :]	,  (batch_size, vocab_size) ).copy()
					_UpperCAmelCase		:       Tuple								=      ramp_logits[1:, : vocab_size // 2] + vocab_size

					_UpperCAmelCase		:       int								=      FlaxTopKLogitsWarper(3 )

					_UpperCAmelCase		:       List[str]								=      top_k_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )

					# check that correct tokens are filtered
					self.assertListEqual(jnp.isinf(scores[0] ).tolist()	,  7 * [True] + 3 * [False] )
					self.assertListEqual(jnp.isinf(scores[1] ).tolist()	,  2 * [True] + 3 * [False] + 5 * [True] )

					# check special case
					_UpperCAmelCase		:       str								=      5
					_UpperCAmelCase		:       Any								=      FlaxTopKLogitsWarper(top_k=1	,  filter_value=0.0	,  min_tokens_to_keep=3 )

					_UpperCAmelCase		:       int								=      np.broadcast_to(np.arange(lowerCamelCase__ )[None, :]	,  (batch_size, length) ).copy()
					_UpperCAmelCase		:       List[Any]								=      top_k_warp_safety_check(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )

					# min_tokens overwrites k: 3 tokens are kept => 2 tokens are nullified
					self.assertListEqual((scores == 0.0).sum(axis=-1 ).tolist()	,  [2, 2] )




			def 							lowerCAmelCase__    (					self  :     Dict )   ->List[str]:



					'''simple docstring'''







					_UpperCAmelCase		:       Optional[int]								=      None
					_UpperCAmelCase		:       Optional[int]								=      10
					_UpperCAmelCase		:       int								=      2

					# create distribution and take log (inverse to Softmax as taken in TopPLogitsWarper)
					_UpperCAmelCase		:       Tuple								=      np.log(np.array([[0.3, 0.1, 0.1, 0.5], [0.1_5, 0.3, 0.3, 0.2_5]] ) )

					_UpperCAmelCase		:       Tuple								=      FlaxTopPLogitsWarper(0.8 )
					_UpperCAmelCase		:       List[Any]								=      np.exp(top_p_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ ) )

					# dist should be filtered to keep min num values so that sum is >= top_p
					# exp (-inf) => 0
					_UpperCAmelCase		:       Tuple								=      np.array([[0.3, 0.0, 0.0, 0.5], [0.0, 0.3, 0.3, 0.2_5]] )
					self.assertTrue(np.allclose(lowerCamelCase__	,  lowerCamelCase__	,  atol=1E-3 ) )

					# check edge cases with negative and extreme logits
					_UpperCAmelCase		:       Optional[Any]								=      np.broadcast_to(np.arange(lowerCamelCase__ )[None, :]	,  (batch_size, vocab_size) ).copy() - (
					    vocab_size // 2
					)

					# make ramp_logits more extreme
					_UpperCAmelCase		:       str								=      ramp_logits[1] * 1_00.0

					# make sure at least 2 tokens are kept
					_UpperCAmelCase		:       int								=      FlaxTopPLogitsWarper(0.9	,  min_tokens_to_keep=2	,  filter_value=0.0 )
					_UpperCAmelCase		:       Tuple								=      top_p_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )

					# first batch should keep three tokens, second batch would keep only 1, but due to `min_tokens_to_keep=2` keeps 2.
					self.assertListEqual((filtered_dist != 0.0).sum(axis=-1 ).tolist()	,  [3, 2] )




			def 							lowerCAmelCase__    (					self  :     Dict )   ->List[Any]:



					'''simple docstring'''







					_UpperCAmelCase		:       Union[str, Any]								=      20
					_UpperCAmelCase		:       str								=      4
					_UpperCAmelCase		:       Any								=      0

					_UpperCAmelCase		:       Optional[Any]								=      FlaxMinLengthLogitsProcessor(min_length=10	,  eos_token_id=lowerCamelCase__ )

					# check that min length is applied at length 5
					_UpperCAmelCase		:       Union[str, Any]								=      ids_tensor((batch_size, 20)	,  vocab_size=20 )
					_UpperCAmelCase		:       int								=      5
					_UpperCAmelCase		:       List[Any]								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Any								=      min_dist_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertListEqual(scores_before_min_length[:, eos_token_id].tolist()	,  4 * [-float("inf" )] )

					# check that min length is not applied anymore at length 15
					_UpperCAmelCase		:       Dict								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Dict								=      15
					_UpperCAmelCase		:       str								=      min_dist_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertFalse(jnp.isinf(lowerCamelCase__ ).any() )




			def 							lowerCAmelCase__    (					self  :     List[str] )   ->Any:



					'''simple docstring'''







					_UpperCAmelCase		:       int								=      20
					_UpperCAmelCase		:       int								=      4
					_UpperCAmelCase		:       List[str]								=      0

					_UpperCAmelCase		:       Dict								=      FlaxForcedBOSTokenLogitsProcessor(bos_token_id=lowerCamelCase__ )

					# check that all scores are -inf except the bos_token_id score
					_UpperCAmelCase		:       int								=      ids_tensor((batch_size, 1)	,  vocab_size=20 )
					_UpperCAmelCase		:       Any								=      1
					_UpperCAmelCase		:       str								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       int								=      logits_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertTrue(jnp.isneginf(scores[:, bos_token_id + 1 :] ).all() )
					self.assertListEqual(scores[:, bos_token_id].tolist()	,  4 * [0] )  # score for bos_token_id shold be zero

					# check that bos_token_id is not forced if current length is greater than 1
					_UpperCAmelCase		:       int								=      3
					_UpperCAmelCase		:       str								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Optional[int]								=      logits_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertFalse(jnp.isinf(lowerCamelCase__ ).any() )




			def 							lowerCAmelCase__    (					self  :     List[Any] )   ->Union[str, Any]:



					'''simple docstring'''







					_UpperCAmelCase		:       Any								=      20
					_UpperCAmelCase		:       Union[str, Any]								=      4
					_UpperCAmelCase		:       List[Any]								=      0
					_UpperCAmelCase		:       str								=      5

					_UpperCAmelCase		:       Tuple								=      FlaxForcedEOSTokenLogitsProcessor(max_length=lowerCamelCase__	,  eos_token_id=lowerCamelCase__ )

					# check that all scores are -inf except the eos_token_id when max_length is reached
					_UpperCAmelCase		:       Tuple								=      ids_tensor((batch_size, 4)	,  vocab_size=20 )
					_UpperCAmelCase		:       List[str]								=      4
					_UpperCAmelCase		:       Any								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       int								=      logits_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertTrue(jnp.isneginf(scores[:, eos_token_id + 1 :] ).all() )
					self.assertListEqual(scores[:, eos_token_id].tolist()	,  4 * [0] )  # score for eos_token_id should be zero

					# check that eos_token_id is not forced if max_length is not reached
					_UpperCAmelCase		:       List[Any]								=      3
					_UpperCAmelCase		:       Dict								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       List[str]								=      logits_processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					self.assertFalse(jnp.isinf(lowerCamelCase__ ).any() )




			def 							lowerCAmelCase__    (					self  :     Optional[int] )   ->Dict:



					'''simple docstring'''







					_UpperCAmelCase		:       Dict								=      4
					_UpperCAmelCase		:       int								=      10
					_UpperCAmelCase		:       str								=      15
					_UpperCAmelCase		:       int								=      2
					_UpperCAmelCase		:       str								=      1
					_UpperCAmelCase		:       str								=      15

					# dummy input_ids and scores
					_UpperCAmelCase		:       Optional[int]								=      ids_tensor((batch_size, sequence_length)	,  lowerCamelCase__ )
					_UpperCAmelCase		:       List[str]								=      input_ids.copy()

					_UpperCAmelCase		:       List[Any]								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       List[Any]								=      scores.copy()

					# instantiate all dist processors
					_UpperCAmelCase		:       int								=      FlaxTemperatureLogitsWarper(temperature=0.5 )
					_UpperCAmelCase		:       Optional[Any]								=      FlaxTopKLogitsWarper(3 )
					_UpperCAmelCase		:       Dict								=      FlaxTopPLogitsWarper(0.8 )

					# instantiate all logits processors
					_UpperCAmelCase		:       Any								=      FlaxMinLengthLogitsProcessor(min_length=10	,  eos_token_id=lowerCamelCase__ )
					_UpperCAmelCase		:       Optional[Any]								=      FlaxForcedBOSTokenLogitsProcessor(bos_token_id=lowerCamelCase__ )
					_UpperCAmelCase		:       Dict								=      FlaxForcedEOSTokenLogitsProcessor(max_length=lowerCamelCase__	,  eos_token_id=lowerCamelCase__ )

					_UpperCAmelCase		:       List[Any]								=      10

					# no processor list
					_UpperCAmelCase		:       Tuple								=      temp_dist_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					_UpperCAmelCase		:       Dict								=      top_k_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					_UpperCAmelCase		:       str								=      top_p_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					_UpperCAmelCase		:       Union[str, Any]								=      min_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					_UpperCAmelCase		:       List[str]								=      bos_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
					_UpperCAmelCase		:       int								=      eos_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )

					# with processor list
					_UpperCAmelCase		:       Any								=      FlaxLogitsProcessorList(
					    [temp_dist_warp, top_k_warp, top_p_warp, min_dist_proc, bos_dist_proc, eos_dist_proc] )
					_UpperCAmelCase		:       Dict								=      processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )

					# scores should be equal
					self.assertTrue(jnp.allclose(lowerCamelCase__	,  lowerCamelCase__	,  atol=1E-3 ) )

					# input_ids should never be changed
					self.assertListEqual(input_ids.tolist()	,  input_ids_comp.tolist() )






			def 							lowerCAmelCase__    (					self  :     Any )   ->List[str]:



					'''simple docstring'''







					_UpperCAmelCase		:       int								=      4
					_UpperCAmelCase		:       Tuple								=      10
					_UpperCAmelCase		:       Optional[int]								=      15
					_UpperCAmelCase		:       Tuple								=      2
					_UpperCAmelCase		:       List[Any]								=      1
					_UpperCAmelCase		:       Dict								=      15

					# dummy input_ids and scores
					_UpperCAmelCase		:       List[str]								=      ids_tensor((batch_size, sequence_length)	,  lowerCamelCase__ )
					_UpperCAmelCase		:       List[Any]								=      input_ids.copy()

					_UpperCAmelCase		:       Optional[Any]								=      self._get_uniform_logits(lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       int								=      scores.copy()

					# instantiate all dist processors
					_UpperCAmelCase		:       Union[str, Any]								=      FlaxTemperatureLogitsWarper(temperature=0.5 )
					_UpperCAmelCase		:       Any								=      FlaxTopKLogitsWarper(3 )
					_UpperCAmelCase		:       List[Any]								=      FlaxTopPLogitsWarper(0.8 )

					# instantiate all logits processors
					_UpperCAmelCase		:       Dict								=      FlaxMinLengthLogitsProcessor(min_length=10	,  eos_token_id=lowerCamelCase__ )
					_UpperCAmelCase		:       Dict								=      FlaxForcedBOSTokenLogitsProcessor(bos_token_id=lowerCamelCase__ )
					_UpperCAmelCase		:       Optional[int]								=      FlaxForcedEOSTokenLogitsProcessor(max_length=lowerCamelCase__	,  eos_token_id=lowerCamelCase__ )

					_UpperCAmelCase		:       Union[str, Any]								=      10

					# no processor list
					def run_no_processor_list(lowerCamelCase__  :     Any	,  lowerCamelCase__  :     int	,  lowerCamelCase__  :     int ):
							_UpperCAmelCase		:       Optional[Any]								=      temp_dist_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							_UpperCAmelCase		:       str								=      top_k_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							_UpperCAmelCase		:       Optional[int]								=      top_p_warp(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							_UpperCAmelCase		:       List[Any]								=      min_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							_UpperCAmelCase		:       Any								=      bos_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							_UpperCAmelCase		:       Tuple								=      eos_dist_proc(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							return scores

					# with processor list
					def run_processor_list(lowerCamelCase__  :     Dict	,  lowerCamelCase__  :     Any	,  lowerCamelCase__  :     Dict ):
							_UpperCAmelCase		:       int								=      FlaxLogitsProcessorList(
							    [temp_dist_warp, top_k_warp, top_p_warp, min_dist_proc, bos_dist_proc, eos_dist_proc] )
							_UpperCAmelCase		:       Optional[int]								=      processor(lowerCamelCase__	,  lowerCamelCase__	,  cur_len=lowerCamelCase__ )
							return scores

					_UpperCAmelCase		:       Tuple								=      jax.jit(lowerCamelCase__ )
					_UpperCAmelCase		:       str								=      jax.jit(lowerCamelCase__ )

					_UpperCAmelCase		:       List[str]								=      jitted_run_no_processor_list(lowerCamelCase__	,  lowerCamelCase__	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Dict								=      jitted_run_processor_list(lowerCamelCase__	,  lowerCamelCase__	,  lowerCamelCase__ )

					# scores should be equal
					self.assertTrue(jnp.allclose(lowerCamelCase__	,  lowerCamelCase__	,  atol=1E-3 ) )

					# input_ids should never be changed
					self.assertListEqual(input_ids.tolist()	,  input_ids_comp.tolist() )







'''simple docstring'''


from math import factorial






class 	lowerCAmelCase__			:




			def __init__(		self  :     int	,  lowerCamelCase__  :     Union[str, Any]	,  lowerCamelCase__  :     str )   ->Dict:



					'''simple docstring'''







					_UpperCAmelCase		:       Optional[Any]								=      real
					if isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Any								=      [1] * rank
					else:
							_UpperCAmelCase		:       Dict								=      rank




			def __repr__(		self  :     str )   ->List[str]:



					'''simple docstring'''







					return (
					    F"""{self.real}+"""
					    F"""{'+'.join(str(lowerCamelCase__ )+'E'+str(n+1 )for n,dual in enumerate(self.duals ) )}"""
					)




			def 							lowerCAmelCase__    (		self  :     Dict )   ->Tuple:



					'''simple docstring'''







					_UpperCAmelCase		:       Union[str, Any]								=      self.duals.copy()
					while cur[-1] == 0:
							cur.pop(-1 )
					return Dual(self.real	,  lowerCamelCase__ )




			def __add__(		self  :     Dict	,  lowerCamelCase__  :     List[Any] )   ->Any:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							return Dual(self.real + other	,  self.duals )
					_UpperCAmelCase		:       Optional[int]								=      self.duals.copy()
					_UpperCAmelCase		:       Optional[int]								=      other.duals.copy()
					if len(lowerCamelCase__ ) > len(lowerCamelCase__ ):
							o_dual.extend([1] * (len(lowerCamelCase__ ) - len(lowerCamelCase__ )) )
					elif len(lowerCamelCase__ ) < len(lowerCamelCase__ ):
							s_dual.extend([1] * (len(lowerCamelCase__ ) - len(lowerCamelCase__ )) )
					_UpperCAmelCase		:       Union[str, Any]								=      []
					for i in range(len(lowerCamelCase__ ) ):
							new_duals.append(s_dual[i] + o_dual[i] )
					return Dual(self.real + other.real	,  lowerCamelCase__ )

			lowerCAmelCase :	Tuple          =     __add__




			def __sub__(		self  :     List[Any]	,  lowerCamelCase__  :     Union[str, Any] )   ->Dict:



					'''simple docstring'''







					return self + other * -1




			def __mul__(		self  :     List[str]	,  lowerCamelCase__  :     Optional[Any] )   ->Union[str, Any]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Optional[int]								=      []
							for i in self.duals:
									new_duals.append(i * other )
							return Dual(self.real * other	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Union[str, Any]								=      [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	,  lowerCamelCase__ )

			lowerCAmelCase :	Union[str, Any]          =     __mul__




			def __truediv__(		self  :     Optional[Any]	,  lowerCamelCase__  :     List[Any] )   ->Union[str, Any]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Union[str, Any]								=      []
							for i in self.duals:
									new_duals.append(i / other )
							return Dual(self.real / other	,  lowerCamelCase__ )
					raise ValueError




			def __floordiv__(		self  :     str	,  lowerCamelCase__  :     str )   ->List[str]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Tuple								=      []
							for i in self.duals:
									new_duals.append(i // other )
							return Dual(self.real // other	,  lowerCamelCase__ )
					raise ValueError






			def __pow__(		self  :     Tuple	,  lowerCamelCase__  :     Optional[Any] )   ->Optional[int]:



					'''simple docstring'''







					if n < 0 or isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							raise ValueError("power must be a positive integer" )
					if n == 0:
							return 1
					if n == 1:
							return self
					_UpperCAmelCase		:       str								=      self
					for _ in range(n - 1 ):
							x *= self
					return x






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


if __name__ == "__main__":
						import doctest

						doctest.testmod()






						def  __lowerCAmelCase (__lowerCAmelCase   ):
								return y**2 * y**4





						print(differentiate(f, 9, 2))



