from math import isclose, sqrt


def        __lowerCAmelCase      (	_A			,_A			,_A       ):



							"""simple docstring"""


							_lowercase					=						point_y / 4 / point_x
							_lowercase					=						2 * normal_gradient / (1 + normal_gradient * normal_gradient)
							_lowercase					=						(1 - normal_gradient * normal_gradient) / (
							    1 + normal_gradient * normal_gradient
							)
							_lowercase					=						(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

							# to find the next point, solve the simultaeneous equations:
							# y^2 + 4x^2 = 100
							# y - b = m * (x - a)
							# ==> A x^2 + B x + C = 0
							_lowercase					=						outgoing_gradient**2 + 4
							_lowercase					=						2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
							_lowercase					=						(point_y - outgoing_gradient * point_x) ** 2 - 100

							_lowercase					=						(
							    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
							) / (2 * quadratic_term)
							_lowercase					=						(
							    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
							) / (2 * quadratic_term)

							# two solutions, one of which is our input point
							_lowercase					=						x_minus if isclose(__UpperCAmelCase			,__UpperCAmelCase       ) else x_plus
							_lowercase					=						point_y + outgoing_gradient * (next_x - point_x)

							return next_x, next_y, outgoing_gradient


def        __lowerCAmelCase      (	_A = 1.4			,_A = -9.6       ):



							"""simple docstring"""


							_lowercase					=						0
							_lowercase					=						first_x_coord
							_lowercase					=						first_y_coord
							_lowercase					=						(10.1 - point_y) / (0.0 - point_x)

							while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
														_lowercase					,							_lowercase					,							_lowercase					=						next_point(__UpperCAmelCase			,__UpperCAmelCase			,__UpperCAmelCase       )
														num_reflections += 1

							return num_reflections


if __name__ == "__main__":
				print(F'''{solution() = }''')








'''simple docstring'''

from __future__ import annotations

import unittest

from transformers import is_tf_available
from transformers.testing_utils import require_tf, slow

from ...test_configuration_common import ConfigTester
from ...test_modeling_tf_common import TFModelTesterMixin, ids_tensor, random_attention_mask
from ...test_pipeline_mixin import PipelineTesterMixin


if is_tf_available():
							import numpy
							import tensorflow as tf

							from transformers import (
							    TF_DPR_CONTEXT_ENCODER_PRETRAINED_MODEL_ARCHIVE_LIST,
							    TF_DPR_QUESTION_ENCODER_PRETRAINED_MODEL_ARCHIVE_LIST,
							    TF_DPR_READER_PRETRAINED_MODEL_ARCHIVE_LIST,
							    BertConfig,
							    DPRConfig,
							    TFDPRContextEncoder,
							    TFDPRQuestionEncoder,
							    TFDPRReader,
							)






class 		a							:


				def __init__(						self       :   Any			,							lowercase_       :   int			,							lowercase_       :   str=13			,							lowercase_       :   Union[str, Any]=7			,							lowercase_       :   Any=True			,							lowercase_       :   List[str]=True			,							lowercase_       :   Tuple=True			,							lowercase_       :   List[Any]=True			,							lowercase_       :   str=99			,							lowercase_       :   int=32			,							lowercase_       :   int=2			,							lowercase_       :   Any=4			,							lowercase_       :   Union[str, Any]=37			,							lowercase_       :   Optional[int]="gelu"			,							lowercase_       :   Dict=0.1			,							lowercase_       :   Optional[Any]=0.1			,							lowercase_       :   Union[str, Any]=512			,							lowercase_       :   Dict=16			,							lowercase_       :   int=2			,							lowercase_       :   Any=0.02			,							lowercase_       :   int=3			,							lowercase_       :   Any=4			,							lowercase_       :   List[Any]=None			,							lowercase_       :   List[str]=0			,							):
									snake_case_		     =							parent
									snake_case_		     =							batch_size
									snake_case_		     =							seq_length
									snake_case_		     =							is_training
									snake_case_		     =							use_input_mask
									snake_case_		     =							use_token_type_ids
									snake_case_		     =							use_labels
									snake_case_		     =							vocab_size
									snake_case_		     =							hidden_size
									snake_case_		     =							num_hidden_layers
									snake_case_		     =							num_attention_heads
									snake_case_		     =							intermediate_size
									snake_case_		     =							hidden_act
									snake_case_		     =							hidden_dropout_prob
									snake_case_		     =							attention_probs_dropout_prob
									snake_case_		     =							max_position_embeddings
									snake_case_		     =							type_vocab_size
									snake_case_		     =							type_sequence_label_size
									snake_case_		     =							initializer_range
									snake_case_		     =							num_labels
									snake_case_		     =							num_choices
									snake_case_		     =							scope
									snake_case_		     =							projection_dim


				def     A_					(						self       :   int   ):
									snake_case_		     =							ids_tensor([self.batch_size, self.seq_length]			,							self.vocab_size   )

									snake_case_		     =							None
									if self.use_input_mask:
														# follow test_modeling_tf_ctrl.py
														snake_case_		     =							random_attention_mask([self.batch_size, self.seq_length]   )

									snake_case_		     =							None
									if self.use_token_type_ids:
														snake_case_		     =							ids_tensor([self.batch_size, self.seq_length]			,							self.type_vocab_size   )

									snake_case_		     =							None
									snake_case_		     =							None
									snake_case_		     =							None
									if self.use_labels:
														snake_case_		     =							ids_tensor([self.batch_size]			,							self.type_sequence_label_size   )
														snake_case_		     =							ids_tensor([self.batch_size, self.seq_length]			,							self.num_labels   )
														snake_case_		     =							ids_tensor([self.batch_size]			,							self.num_choices   )

									snake_case_		     =							BertConfig(
									    vocab_size=self.vocab_size			,							hidden_size=self.hidden_size			,							num_hidden_layers=self.num_hidden_layers			,							num_attention_heads=self.num_attention_heads			,							intermediate_size=self.intermediate_size			,							hidden_act=self.hidden_act			,							hidden_dropout_prob=self.hidden_dropout_prob			,							attention_probs_dropout_prob=self.attention_probs_dropout_prob			,							max_position_embeddings=self.max_position_embeddings			,							type_vocab_size=self.type_vocab_size			,							is_decoder=lowercase_			,							initializer_range=self.initializer_range			,							)
									snake_case_		     =							DPRConfig(projection_dim=self.projection_dim			,							**config.to_dict()   )

									return config, input_ids, token_type_ids, input_mask, sequence_labels, token_labels, choice_labels


				def     A_					(						self       :   Dict			,							lowercase_       :   Optional[Any]			,							lowercase_       :   int			,							lowercase_       :   Optional[int]			,							lowercase_       :   int			,							lowercase_       :   int			,							lowercase_       :   Union[str, Any]			,							lowercase_       :   str   ):
									snake_case_		     =							TFDPRContextEncoder(config=lowercase_   )
									snake_case_		     =							model(lowercase_			,							attention_mask=lowercase_			,							token_type_ids=lowercase_   )
									snake_case_		     =							model(lowercase_			,							token_type_ids=lowercase_   )
									snake_case_		     =							model(lowercase_   )
									self.parent.assertEqual(result.pooler_output.shape			,							(self.batch_size, self.projection_dim or self.hidden_size)   )


				def     A_					(						self       :   Any			,							lowercase_       :   Optional[Any]			,							lowercase_       :   List[str]			,							lowercase_       :   Union[str, Any]			,							lowercase_       :   Dict			,							lowercase_       :   Dict			,							lowercase_       :   Dict			,							lowercase_       :   Any   ):
									snake_case_		     =							TFDPRQuestionEncoder(config=lowercase_   )
									snake_case_		     =							model(lowercase_			,							attention_mask=lowercase_			,							token_type_ids=lowercase_   )
									snake_case_		     =							model(lowercase_			,							token_type_ids=lowercase_   )
									snake_case_		     =							model(lowercase_   )
									self.parent.assertEqual(result.pooler_output.shape			,							(self.batch_size, self.projection_dim or self.hidden_size)   )


				def     A_					(						self       :   str			,							lowercase_       :   Tuple			,							lowercase_       :   Tuple			,							lowercase_       :   Tuple			,							lowercase_       :   Tuple			,							lowercase_       :   Tuple			,							lowercase_       :   Dict			,							lowercase_       :   Dict   ):
									snake_case_		     =							TFDPRReader(config=lowercase_   )
									snake_case_		     =							model(lowercase_			,							attention_mask=lowercase_   )

									self.parent.assertEqual(result.start_logits.shape			,							(self.batch_size, self.seq_length)   )
									self.parent.assertEqual(result.end_logits.shape			,							(self.batch_size, self.seq_length)   )
									self.parent.assertEqual(result.relevance_logits.shape			,							(self.batch_size,)   )


				def     A_					(						self       :   Union[str, Any]   ):
									snake_case_		     =							self.prepare_config_and_inputs()
									(
									    (
									    snake_case_
									)     ,(
									    snake_case_
									)     ,(
									    snake_case_
									)     ,(
									    snake_case_
									)     ,(
									    snake_case_
									)     ,(
									    snake_case_
									)     ,(
									    snake_case_
									)     ,
									)		     =							config_and_inputs
									snake_case_		     =							{'''input_ids''': input_ids}
									return config, inputs_dict







@require_tf
class 		a							(  _lowerCamelCase  ,	_lowerCamelCase  ,	unittest.TestCase	):
				snake_case_           =						(
				    (
				        TFDPRContextEncoder,
				        TFDPRQuestionEncoder,
				        TFDPRReader,
				    )
				    if is_tf_available()
				    else ()
				)
				snake_case_           =						{"feature-extraction": TFDPRQuestionEncoder} if is_tf_available() else {}

				snake_case_           =						False
				snake_case_           =						False
				snake_case_           =						False
				snake_case_           =						False
				snake_case_           =						False


				def     A_					(						self       :   List[Any]   ):
									snake_case_		     =							TFDPRModelTester(self   )
									snake_case_		     =							ConfigTester(self			,							config_class=lowercase_			,							hidden_size=37   )


				def     A_					(						self       :   Any   ):
									self.config_tester.run_common_tests()


				def     A_					(						self       :   int   ):
									snake_case_		     =							self.model_tester.prepare_config_and_inputs()
									self.model_tester.create_and_check_dpr_context_encoder(*lowercase_   )


				def     A_					(						self       :   List[Any]   ):
									snake_case_		     =							self.model_tester.prepare_config_and_inputs()
									self.model_tester.create_and_check_dpr_question_encoder(*lowercase_   )


				def     A_					(						self       :   Any   ):
									snake_case_		     =							self.model_tester.prepare_config_and_inputs()
									self.model_tester.create_and_check_dpr_reader(*lowercase_   )


				@slow
				def     A_					(						self       :   int   ):
									for model_name in TF_DPR_CONTEXT_ENCODER_PRETRAINED_MODEL_ARCHIVE_LIST[:1]:
														snake_case_		     =							TFDPRContextEncoder.from_pretrained(lowercase_   )
														self.assertIsNotNone(lowercase_   )

									for model_name in TF_DPR_CONTEXT_ENCODER_PRETRAINED_MODEL_ARCHIVE_LIST[:1]:
														snake_case_		     =							TFDPRContextEncoder.from_pretrained(lowercase_   )
														self.assertIsNotNone(lowercase_   )

									for model_name in TF_DPR_QUESTION_ENCODER_PRETRAINED_MODEL_ARCHIVE_LIST[:1]:
														snake_case_		     =							TFDPRQuestionEncoder.from_pretrained(lowercase_   )
														self.assertIsNotNone(lowercase_   )

									for model_name in TF_DPR_READER_PRETRAINED_MODEL_ARCHIVE_LIST[:1]:
														snake_case_		     =							TFDPRReader.from_pretrained(lowercase_   )
														self.assertIsNotNone(lowercase_   )







@require_tf
class 		a							(  unittest.TestCase	):


				@slow
				def     A_					(						self       :   Dict   ):
									snake_case_		     =							TFDPRQuestionEncoder.from_pretrained('''facebook/dpr-question_encoder-single-nq-base'''   )

									snake_case_		     =							tf.constant(
									    [[101, 7592, 1010, 2003, 2026, 3899, 1_0140, 1029, 102]]   )  # [CLS] hello, is my dog cute? [SEP]
									snake_case_		     =							model(lowercase_   )[0]  # embedding shape = (1, 768)
									# compare the actual values for a slice.
									snake_case_		     =							tf.constant(
									    [
									        [
									            0.0323_6253,
									            0.1275_3335,
									            0.1681_8509,
									            0.0027_9786,
									            0.389_6933,
									            0.2426_4945,
									            0.217_8971,
									            -0.0233_5227,
									            -0.0848_1959,
									            -0.1432_4117,
									        ]
									    ]   )
									self.assertTrue(numpy.allclose(output[:, :10].numpy()			,							expected_slice.numpy()			,							atol=1e-4   )   )

