
from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__a							  =						2_0

# Time between time steps in seconds
__a							  =						INTERVAL / 1_0_0_0

class        UpperCamelCase__:

	"""simple docstring"""

	def __init__(      self				:			Tuple							,			snake_case__				:			float							,			snake_case__				:			float							,			snake_case__				:			float							,			snake_case__				:			float							,			snake_case__				:			float = 1.0							,			snake_case__				:			float = 1.0							,			snake_case__				:			str = "blue"							,			):
			"""simple docstring"""





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







	@property
	def        _a					(      self				:			int ):
			"""simple docstring"""





			return self.position_x, self.position_y







	@property
	def        _a					(      self				:			str ):
			"""simple docstring"""





			return self.velocity_x, self.velocity_y







	def        _a					(      self				:			Optional[int]							,			snake_case__				:			float							,			snake_case__				:			float							,			snake_case__				:			float ):
			"""simple docstring"""





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







	def        _a					(      self				:			Tuple							,			snake_case__				:			float ):
			"""simple docstring"""





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


class        UpperCamelCase__:

	"""simple docstring"""

	def __init__(      self				:			str							,			snake_case__				:			list[Body]							,			snake_case__				:			float = 1.0							,			snake_case__				:			float = 1.0							,			snake_case__				:			float = 0.0							,			):
			"""simple docstring"""





			A				=bodies
			A				=gravitation_constant
			A				=time_factor
			A				=softening_factor







	def __len__(      self				:			str ):
			"""simple docstring"""





			return len(self.bodies )







	def        _a					(      self				:			List[Any]							,			snake_case__				:			float ):
			"""simple docstring"""





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

									# Calculation of the distance using Pythagoras's theorem
									# Extra factor due to the softening technique
									A				=(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(snake_case__							,			snake_case__							,			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    UpperCamelCase_ (	a_  ,				a_  ,				a_		)     ->None:
		body_system.update_system(a_		)

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

def    UpperCamelCase_ (	a_  ,				a_  ,				a_ = -1  ,				a_ = 1  ,				a_ = -1  ,				a_ = 1  ,				)     ->None:
		A				=plt.figure()
		fig.canvas.set_window_title(a_		)
		A				=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
		A				=[
		    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(a_		)

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

		A				=animation.FuncAnimation(  # noqa: F841
		    a_  ,				a_  ,				interval=a_  ,				blit=a_		)

		plt.show()

def    UpperCamelCase_ (	)     ->BodySystem:
		A				=0.9700436
		A				=-0.24308753
		A				=0.466203685
		A				=0.43236573

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

def    UpperCamelCase_ (	)     ->BodySystem:
		A				=7.3476E22
		A				=5.972E24
		A				=1022
		A				=3_8439_9000
		A				=6.674E-11

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

		A				=Body(-earth_moon_distance  ,				0  ,				0  ,				a_  ,				a_  ,				1000_0000  ,				"grey"		)
		A				=Body(0  ,				0  ,				0  ,				a_  ,				a_  ,				5000_0000  ,				"blue"		)
		return BodySystem([earth, moon]  ,				a_  ,				time_factor=100_0000		)

def    UpperCamelCase_ (	)     ->BodySystem:
		A				=[]
		for _ in range(10		):
				A				=random.uniform(-0.5  ,				0.5		)
				A				=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		)  ,				a_  ,				a_  ,				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(a_  ,				0.01  ,				10  ,				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_3_0_0_0_0_0_0_0,
	    4_3_0_0_0_0_0_0_0,
	    -4_3_0_0_0_0_0_0_0,
	    4_3_0_0_0_0_0_0_0,
	)
	plot("""Random system with many bodies""", example_a(), -1.5, 1.5, -1.5, 1.5)






import functools
from math import sqrt
from time import time

def    UpperCamelCase_ (	a_  ,				*a_  ,				**a_		)     ->List[Any]:
		A				=time()
		A				=func(*a_  ,				**a_		)
		A				=time()
		if int(end - start		) > 0:
				print(f'''{func.__name__} runtime: {(end - start):0.4f} s'''		)
		else:
				print(f'''{func.__name__} runtime: {(end - start) * 1000:0.4f} ms'''		)
		return output

def    UpperCamelCase_ (	a_		)     ->list[int]:
		if n < 0:
				raise Exception("n is negative"		)
		if n == 0:
				return [0]
		A				=[0, 1]
		for _ in range(n - 1		):
				fib.append(fib[-1] + fib[-2]		)
		return fib

def    UpperCamelCase_ (	a_		)     ->list[int]:

		def fib_recursive_term(a_		) -> int:
				if i < 0:
						raise Exception("n is negative"		)
				if i < 2:
						return i
				return fib_recursive_term(i - 1		) + fib_recursive_term(i - 2		)

		if n < 0:
				raise Exception("n is negative"		)
		return [fib_recursive_term(a_		) for i in range(n + 1		)]

def    UpperCamelCase_ (	a_		)     ->list[int]:

		@functools.cache
		def fib_recursive_term(a_		) -> int:
				if i < 0:
						raise Exception("n is negative"		)
				if i < 2:
						return i
				return fib_recursive_term(i - 1		) + fib_recursive_term(i - 2		)

		if n < 0:
				raise Exception("n is negative"		)
		return [fib_recursive_term(a_		) for i in range(n + 1		)]

def    UpperCamelCase_ (	a_		)     ->list[int]:
		if n < 0:
				raise Exception("n is negative"		)
		# Cache must be outside recursuive function
		# other it will reset every time it calls itself.
		A				={0: 0, 1: 1, 2: 1}  # Prefilled cache

		def rec_fn_memoized(a_		) -> int:
				if num in cache:
						return cache[num]

				A				=rec_fn_memoized(num - 1		) + rec_fn_memoized(num - 2		)
				A				=value
				return value

		return [rec_fn_memoized(a_		) for i in range(n + 1		)]

def    UpperCamelCase_ (	a_		)     ->list[int]:
		if n < 0:
				raise Exception("n is negative"		)
		if n >= 1475:
				raise Exception("n is too large"		)
		A				=sqrt(5		)
		A				=(1 + sqrt_a) / 2
		return [round(phi**i / sqrt_a		) for i in range(n + 1		)]


if __name__ == "__main__":
	__a							  =						3_0
	time_func(fib_iterative, num)
	time_func(fib_recursive, num)  # Around 3s runtime
	time_func(fib_recursive_cached, num)  # Around 0ms runtime
	time_func(fib_memoization, num)
	time_func(fib_binet, num)




