


"""simple docstring"""




from math import isclose, sqrt


def     lowercase			(SCREAMING_SNAKE_CASE_		:				float ,			SCREAMING_SNAKE_CASE_		:				float ,			SCREAMING_SNAKE_CASE_		:				float  )   ->						tuple[float, float, float]:
						SCREAMING_SNAKE_CASE													=					point_y / 4 / point_x
						SCREAMING_SNAKE_CASE													=					2 * normal_gradient / (1 + normal_gradient * normal_gradient)
						SCREAMING_SNAKE_CASE													=					(1 - normal_gradient * normal_gradient) / (
						    1 + normal_gradient * normal_gradient
						)
						SCREAMING_SNAKE_CASE													=					(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
						SCREAMING_SNAKE_CASE													=					outgoing_gradient**2 + 4
						SCREAMING_SNAKE_CASE													=					2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
						SCREAMING_SNAKE_CASE													=					(point_y - outgoing_gradient * point_x) ** 2 - 1_00

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

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

						return next_x, next_y, outgoing_gradient


def     lowercase			(SCREAMING_SNAKE_CASE_		:				float = 1.4 ,			SCREAMING_SNAKE_CASE_		:				float = -9.6  )   ->						int:
						SCREAMING_SNAKE_CASE													=					0
						SCREAMING_SNAKE_CASE													=					first_x_coord
						SCREAMING_SNAKE_CASE													=					first_y_coord
						SCREAMING_SNAKE_CASE													=					(10.1 - point_y) / (0.0 - point_x)

						while not (-0.01 <= point_x <= 0.01 and point_y > 0):
												SCREAMING_SNAKE_CASE							,				SCREAMING_SNAKE_CASE							,				SCREAMING_SNAKE_CASE													=					next_point(SCREAMING_SNAKE_CASE_ ,			SCREAMING_SNAKE_CASE_ ,			SCREAMING_SNAKE_CASE_  )
												num_reflections += 1

						return num_reflections


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






"""simple docstring"""




__UpperCamelCase						=      frozenset(
    [
        '''prompt''',
        '''height''',
        '''width''',
        '''guidance_scale''',
        '''negative_prompt''',
        '''prompt_embeds''',
        '''negative_prompt_embeds''',
        '''cross_attention_kwargs''',
    ]
)

__UpperCamelCase						=      frozenset(['''prompt''', '''negative_prompt'''])

__UpperCamelCase						=      frozenset([])

__UpperCamelCase						=      frozenset(['''image'''])

__UpperCamelCase						=      frozenset(
    [
        '''image''',
        '''height''',
        '''width''',
        '''guidance_scale''',
    ]
)

__UpperCamelCase						=      frozenset(['''image'''])

__UpperCamelCase						=      frozenset(
    [
        '''prompt''',
        '''image''',
        '''height''',
        '''width''',
        '''guidance_scale''',
        '''negative_prompt''',
        '''prompt_embeds''',
        '''negative_prompt_embeds''',
    ]
)

__UpperCamelCase						=      frozenset(['''prompt''', '''image''', '''negative_prompt'''])

__UpperCamelCase						=      frozenset(
    [
        # Text guided image variation with an image mask
        '''prompt''',
        '''image''',
        '''mask_image''',
        '''height''',
        '''width''',
        '''guidance_scale''',
        '''negative_prompt''',
        '''prompt_embeds''',
        '''negative_prompt_embeds''',
    ]
)

__UpperCamelCase						=      frozenset(['''prompt''', '''image''', '''mask_image''', '''negative_prompt'''])

__UpperCamelCase						=      frozenset(
    [
        # image variation with an image mask
        '''image''',
        '''mask_image''',
        '''height''',
        '''width''',
        '''guidance_scale''',
    ]
)

__UpperCamelCase						=      frozenset(['''image''', '''mask_image'''])

__UpperCamelCase						=      frozenset(
    [
        '''example_image''',
        '''image''',
        '''mask_image''',
        '''height''',
        '''width''',
        '''guidance_scale''',
    ]
)

__UpperCamelCase						=      frozenset(['''example_image''', '''image''', '''mask_image'''])

__UpperCamelCase						=      frozenset(['''class_labels'''])

__UpperCamelCase						=      frozenset(['''class_labels'''])

__UpperCamelCase						=      frozenset(['''batch_size'''])

__UpperCamelCase						=      frozenset([])

__UpperCamelCase						=      frozenset(['''batch_size'''])

__UpperCamelCase						=      frozenset([])

__UpperCamelCase						=      frozenset(
    [
        '''prompt''',
        '''audio_length_in_s''',
        '''guidance_scale''',
        '''negative_prompt''',
        '''prompt_embeds''',
        '''negative_prompt_embeds''',
        '''cross_attention_kwargs''',
    ]
)

__UpperCamelCase						=      frozenset(['''prompt''', '''negative_prompt'''])
__UpperCamelCase						=      frozenset(['''input_tokens'''])

__UpperCamelCase						=      frozenset(['''input_tokens'''])


