




from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__a			:	Union[str, Any]							     =		20

# Time between time steps in seconds
__a			:	Optional[Any]							     =		INTERVAL / 10_00




class     A					:







							def __init__(    self   :	str   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float = 1.0   ,     __UpperCAmelCase   :	float = 1.0   ,     __UpperCAmelCase   :	str = "blue"   ,     )		->					None:

										"""simple docstring"""



										UpperCamelCase_     				= position_x
										UpperCamelCase_     				= position_y
										UpperCamelCase_     				= velocity_x
										UpperCamelCase_     				= velocity_y
										UpperCamelCase_     				= mass
										UpperCamelCase_     				= size
										UpperCamelCase_     				= color







							@property
							def        lowercase__							(    self   :	Union[str, Any]				)		->					tuple[float, float]:

										"""simple docstring"""



										return self.position_x, self.position_y







							@property
							def        lowercase__							(    self   :	Tuple				)		->					tuple[float, float]:

										"""simple docstring"""



										return self.velocity_x, self.velocity_y







							def        lowercase__							(    self   :	Optional[int]   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float   ,     __UpperCAmelCase   :	float				)		->					None:

										"""simple docstring"""



										self.velocity_x += force_x * delta_time
										self.velocity_y += force_y * delta_time







							def        lowercase__							(    self   :	Tuple   ,     __UpperCAmelCase   :	float				)		->					None:

										"""simple docstring"""



										self.position_x += self.velocity_x * delta_time
										self.position_y += self.velocity_y * delta_time







class     A					:







							def __init__(    self   :	List[str]   ,     __UpperCAmelCase   :	list[Body]   ,     __UpperCAmelCase   :	float = 1.0   ,     __UpperCAmelCase   :	float = 1.0   ,     __UpperCAmelCase   :	float = 0.0   ,     )		->					None:

										"""simple docstring"""



										UpperCamelCase_     				= bodies
										UpperCamelCase_     				= gravitation_constant
										UpperCamelCase_     				= time_factor
										UpperCamelCase_     				= softening_factor







							def __len__(    self   :	List[Any]				)		->					int:

										"""simple docstring"""



										return len(self.bodies				)







							def        lowercase__							(    self   :	Union[str, Any]   ,     __UpperCAmelCase   :	float				)		->					None:

										"""simple docstring"""



										for bodya in self.bodies:
													UpperCamelCase_     				= 0.0
													UpperCamelCase_     				= 0.0
													for bodya in self.bodies:
																if bodya != bodya:
																			UpperCamelCase_     				= bodya.position_x - bodya.position_x
																			UpperCamelCase_     				= bodya.position_y - bodya.position_y

																			# Calculation of the distance using Pythagoras's theorem
																			# Extra factor due to the softening technique
																			UpperCamelCase_     				= (dif_x**2 + dif_y**2 + self.softening_factor) ** (
																			    1 / 2
																			)

																			# Newton's law of universal gravitation.
																			force_x += (
																			    self.gravitation_constant * bodya.mass * dif_x / distance**3
																			)
																			force_y += (
																			    self.gravitation_constant * bodya.mass * dif_y / distance**3
																			)

            # Update the body's velocity once all the force components have been added
													bodya.update_velocity(__UpperCAmelCase   ,     __UpperCAmelCase   ,     delta_time * self.time_factor				)

										# Update the positions only after all the velocities have been updated
										for body in self.bodies:
													body.update_position(delta_time * self.time_factor				)







def  a_		(							__snake_case     ,		__snake_case     ,		__snake_case					)	->			None:

			'''simple docstring'''


			body_system.update_system(__snake_case					)

			# Update the positions of the patches
			for patch, body in zip(__snake_case     ,		body_system.bodies					):
						UpperCamelCase_     				= (body.position_x, body.position_y)




def  a_		(							__snake_case     ,		__snake_case     ,		__snake_case = -1     ,		__snake_case = 1     ,		__snake_case = -1     ,		__snake_case = 1     ,		)	->			None:

			'''simple docstring'''


			UpperCamelCase_     				= plt.figure()
			fig.canvas.set_window_title(__snake_case					)
			UpperCamelCase_     				= plt.axes(
			    xlim=(x_start, x_end)     ,		ylim=(y_start, y_end)					)  # Set section to be plotted
			plt.gca().set_aspect('equal'					)  # Fix aspect ratio

			# Each body is drawn as a patch by the plt-function
			UpperCamelCase_     				= [
			    plt.Circle((body.position_x, body.position_y)     ,		body.size     ,		fc=body.color					)
			    for body in body_system.bodies
			]

			for patch in patches:
						ax.add_patch(__snake_case					)

			# Function called at each step of the animation
			def update(__snake_case					) -> list[plt.Circle]:
						update_step(__snake_case     ,		__snake_case     ,		__snake_case					)
						return patches

			UpperCamelCase_     				= animation.FuncAnimation(  # noqa: F841
			    __snake_case     ,		__snake_case     ,		interval=__snake_case     ,		blit=__snake_case					)

			plt.show()




def  a_		(							)	->			BodySystem:

			'''simple docstring'''


			UpperCamelCase_     				= 0.9_70_04_36
			UpperCamelCase_     				= -0.24_30_87_53
			UpperCamelCase_     				= 0.4_66_20_36_85
			UpperCamelCase_     				= 0.43_23_65_73

			UpperCamelCase_     				= [
			    Body(__snake_case     ,		__snake_case     ,		__snake_case     ,		__snake_case     ,		size=0.2     ,		color='red'					),
			    Body(-position_x     ,		-position_y     ,		__snake_case     ,		__snake_case     ,		size=0.2     ,		color='green'					),
			    Body(0     ,		0     ,		-2 * velocity_x     ,		-2 * velocity_y     ,		size=0.2     ,		color='blue'					),
			]
			return BodySystem(__snake_case     ,		time_factor=3					)




def  a_		(							)	->			BodySystem:

			'''simple docstring'''


			UpperCamelCase_     				= 7.3_476E22
			UpperCamelCase_     				= 5.972E24
			UpperCamelCase_     				= 1_0_2_2
			UpperCamelCase_     				= 3_8_4_3_9_9_0_0_0
			UpperCamelCase_     				= 6.674E-11

			# Calculation of the respective velocities so that total impulse is zero,
			# i.e. the two bodies together don't move
			UpperCamelCase_     				= earth_mass * velocity_dif / (earth_mass + moon_mass)
			UpperCamelCase_     				= moon_velocity - velocity_dif

			UpperCamelCase_     				= Body(-earth_moon_distance     ,		0     ,		0     ,		__snake_case     ,		__snake_case     ,		1_0_0_0_0_0_0_0     ,		'grey'					)
			UpperCamelCase_     				= Body(0     ,		0     ,		0     ,		__snake_case     ,		__snake_case     ,		5_0_0_0_0_0_0_0     ,		'blue'					)
			return BodySystem([earth, moon]     ,		__snake_case     ,		time_factor=1_0_0_0_0_0_0					)




def  a_		(							)	->			BodySystem:

			'''simple docstring'''


			UpperCamelCase_     				= []
			for _ in range(1_0					):
						UpperCamelCase_     				= random.uniform(-0.5     ,		0.5					)
						UpperCamelCase_     				= random.uniform(-0.5     ,		0.5					)

						# Bodies are created pairwise with opposite velocities so that the
						# total impulse remains zero
						bodies.append(
						    Body(
						        random.uniform(-0.5     ,		0.5					)     ,		random.uniform(-0.5     ,		0.5					)     ,		__snake_case     ,		__snake_case     ,		size=0.05     ,		)					)
						bodies.append(
						    Body(
						        random.uniform(-0.5     ,		0.5					)     ,		random.uniform(-0.5     ,		0.5					)     ,		-velocity_x     ,		-velocity_y     ,		size=0.05     ,		)					)
			return BodySystem(__snake_case     ,		0.01     ,		1_0     ,		0.1					)


if __name__ == "__main__":
				plot("""Figure-8 solution to the 3-body-problem""", example_a(), -2, 2, -2, 2)
				plot(
				    """Moon's orbit around the earth""",
				    example_a(),
				    -4_30_00_00_00,
				    4_30_00_00_00,
				    -4_30_00_00_00,
				    4_30_00_00_00,
				)
				plot("""Random system with many bodies""", example_a(), -1.5, 1.5, -1.5, 1.5)






import numpy as np




def  a_		(							__snake_case     ,		__snake_case					)	->			List[str]:

			'''simple docstring'''


			UpperCamelCase_     				= np.array(__snake_case					)
			UpperCamelCase_     				= np.array(__snake_case					)

			UpperCamelCase_     				= abs(predict - actual					)
			UpperCamelCase_     				= difference.mean()

			return score




def  a_		(							__snake_case     ,		__snake_case					)	->			Optional[Any]:

			'''simple docstring'''


			UpperCamelCase_     				= np.array(__snake_case					)
			UpperCamelCase_     				= np.array(__snake_case					)

			UpperCamelCase_     				= predict - actual
			UpperCamelCase_     				= np.square(__snake_case					)

			UpperCamelCase_     				= square_diff.mean()
			return score




def  a_		(							__snake_case     ,		__snake_case					)	->			Optional[Any]:

			'''simple docstring'''


			UpperCamelCase_     				= np.array(__snake_case					)
			UpperCamelCase_     				= np.array(__snake_case					)

			UpperCamelCase_     				= predict - actual
			UpperCamelCase_     				= np.square(__snake_case					)
			UpperCamelCase_     				= square_diff.mean()
			UpperCamelCase_     				= np.sqrt(__snake_case					)
			return score




def  a_		(							__snake_case     ,		__snake_case					)	->			Optional[int]:

			'''simple docstring'''


			UpperCamelCase_     				= np.array(__snake_case					)
			UpperCamelCase_     				= np.array(__snake_case					)

			UpperCamelCase_     				= np.log(predict + 1					)
			UpperCamelCase_     				= np.log(actual + 1					)

			UpperCamelCase_     				= log_predict - log_actual
			UpperCamelCase_     				= np.square(__snake_case					)
			UpperCamelCase_     				= square_diff.mean()

			UpperCamelCase_     				= np.sqrt(__snake_case					)

			return score




def  a_		(							__snake_case     ,		__snake_case					)	->			int:

			'''simple docstring'''


			UpperCamelCase_     				= np.array(__snake_case					)
			UpperCamelCase_     				= np.array(__snake_case					)

			UpperCamelCase_     				= predict - actual
			UpperCamelCase_     				= np.sum(__snake_case					) / len(__snake_case					)
			UpperCamelCase_     				= np.sum(__snake_case					) / len(__snake_case					)
			# print(numerator, denumerator)
			UpperCamelCase_     				= float(__snake_case					) / denumerator * 1_0_0

			return score





def  a_		(							__snake_case     ,		__snake_case					)	->			Optional[Any]:

			'''simple docstring'''


			return np.mean(np.array(__snake_case					) == np.array(__snake_case					)					)
