

from math import isclose, sqrt






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__     ,			lowerCamelCase__     ,			lowerCamelCase__      )  ->					tuple[float, float, float]:
	__lowerCamelCase    :    str 	=			point_y / 4 / point_x
	__lowerCamelCase    :    Optional[int] 	=			2 * normal_gradient / (1 + normal_gradient * normal_gradient)
	__lowerCamelCase    :    str 	=			(1 - normal_gradient * normal_gradient) / (
	    1 + normal_gradient * normal_gradient
	)
	__lowerCamelCase    :    int 	=			(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
	__lowerCamelCase    :    Dict 	=			outgoing_gradient**2 + 4
	__lowerCamelCase    :    Optional[Any] 	=			2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
	__lowerCamelCase    :    List[str] 	=			(point_y - outgoing_gradient * point_x) ** 2 - 1_0_0

	__lowerCamelCase    :    Optional[int] 	=			(
	    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term      )
	) / (2 * quadratic_term)
	__lowerCamelCase    :    Dict 	=			(
	    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term      )
	) / (2 * quadratic_term)

	# two solutions, one of which is our input point
	__lowerCamelCase    :    Optional[int] 	=			x_minus if isclose(lowerCamelCase__     ,			lowerCamelCase__      ) else x_plus
	__lowerCamelCase    :    Optional[Any] 	=			point_y + outgoing_gradient * (next_x - point_x)

	return next_x, next_y, outgoing_gradient






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__ = 1.4     ,			lowerCamelCase__ = -9.6      )  ->					int:
	__lowerCamelCase    :    int 	=			0
	__lowerCamelCase    :    float 	=			first_x_coord
	__lowerCamelCase    :    float 	=			first_y_coord
	__lowerCamelCase    :    float 	=			(10.1 - point_y) / (0.0 - point_x)

	while not (-0.01 <= point_x <= 0.01 and point_y > 0):
		__lowerCamelCase	,			__lowerCamelCase	,			__lowerCamelCase    :    Optional[int] 	=			next_point(lowerCamelCase__     ,			lowerCamelCase__     ,			lowerCamelCase__      )
		num_reflections += 1

	return num_reflections


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







from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


a		={
    """configuration_bigbird_pegasus""": [
        """BIGBIRD_PEGASUS_PRETRAINED_CONFIG_ARCHIVE_MAP""",
        """BigBirdPegasusConfig""",
        """BigBirdPegasusOnnxConfig""",
    ],
}

try:
					if not is_torch_available():
										raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
					pass
else:
					a		=[
					    """BIGBIRD_PEGASUS_PRETRAINED_MODEL_ARCHIVE_LIST""",
					    """BigBirdPegasusForCausalLM""",
					    """BigBirdPegasusForConditionalGeneration""",
					    """BigBirdPegasusForQuestionAnswering""",
					    """BigBirdPegasusForSequenceClassification""",
					    """BigBirdPegasusModel""",
					    """BigBirdPegasusPreTrainedModel""",
					]


if TYPE_CHECKING:
					from .configuration_bigbird_pegasus import (
					    BIGBIRD_PEGASUS_PRETRAINED_CONFIG_ARCHIVE_MAP,
					    BigBirdPegasusConfig,
					    BigBirdPegasusOnnxConfig,
					)

					try:
										if not is_torch_available():
															raise OptionalDependencyNotAvailable()
					except OptionalDependencyNotAvailable:
										pass
					else:
										from .modeling_bigbird_pegasus import (
										    BIGBIRD_PEGASUS_PRETRAINED_MODEL_ARCHIVE_LIST,
										    BigBirdPegasusForCausalLM,
										    BigBirdPegasusForConditionalGeneration,
										    BigBirdPegasusForQuestionAnswering,
										    BigBirdPegasusForSequenceClassification,
										    BigBirdPegasusModel,
										    BigBirdPegasusPreTrainedModel,
										)


else:
					import sys

					a		=_LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)




