from typing import Dict, Optional, Tuple

import torch





def 	lowerCAmelCase_			(					__lowerCAmelCase    )->	torch.Tensor:






      '''simple docstring'''





      UpperCAmelCase     :			List[str]     	=boundaries[1] - boundaries[0]
      UpperCAmelCase     :			str     	=boundaries + step / 2
      UpperCAmelCase     :			int     	=torch.cat([bin_centers, (bin_centers[-1] + step).unsqueeze(-1    )]		,       dim=0    )
      return bin_centers





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase		,       )->	Tuple[torch.Tensor, torch.Tensor]:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[Any]     	=_calculate_bin_centers(__lowerCAmelCase    )
      return (
          torch.sum(aligned_distance_error_probs * bin_centers		,       dim=-1    ),
          bin_centers[-1],
      )





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase = 31		,       __lowerCAmelCase = 64		,       **__lowerCAmelCase		,       )->	Dict[str, torch.Tensor]:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[int]     	=torch.linspace(0		,       __lowerCAmelCase		,       steps=(no_bins - 1)		,       device=logits.device    )

      UpperCAmelCase     :			Dict     	=torch.nn.functional.softmax(__lowerCAmelCase		,       dim=-1    )
      UpperCAmelCase	,    UpperCAmelCase     :			Optional[int]     	=_calculate_expected_aligned_error(
          alignment_confidence_breaks=__lowerCAmelCase		,       aligned_distance_error_probs=__lowerCAmelCase		,       )

      return {
          "aligned_confidence_probs": aligned_confidence_probs,
          "predicted_aligned_error": predicted_aligned_error,
          "max_predicted_aligned_error": max_predicted_aligned_error,
      }





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase = None		,       __lowerCAmelCase = 31		,       __lowerCAmelCase = 64		,       __lowerCAmelCase = 1e-8		,       **__lowerCAmelCase		,       )->	torch.Tensor:






      '''simple docstring'''





      if residue_weights is None:
            UpperCAmelCase     :			List[str]     	=logits.new_ones(logits.shape[-2]    )

      UpperCAmelCase     :			List[str]     	=torch.linspace(0		,       __lowerCAmelCase		,       steps=(no_bins - 1)		,       device=logits.device    )

      UpperCAmelCase     :			Any     	=_calculate_bin_centers(__lowerCAmelCase    )
      torch.sum(__lowerCAmelCase    )
      UpperCAmelCase     :			str     	=logits.shape[-2]
      UpperCAmelCase     :			Union[str, Any]     	=max(__lowerCAmelCase		,       19    )

      UpperCAmelCase     :			Optional[Any]     	=1.24 * (clipped_n - 15) ** (1.0 / 3) - 1.8

      UpperCAmelCase     :			List[str]     	=torch.nn.functional.softmax(__lowerCAmelCase		,       dim=-1    )

      UpperCAmelCase     :			List[str]     	=1.0 / (1 + (bin_centers**2) / (da**2))
      UpperCAmelCase     :			Optional[Any]     	=torch.sum(probs * tm_per_bin		,       dim=-1    )

      UpperCAmelCase     :			Any     	=residue_weights / (eps + residue_weights.sum())
      UpperCAmelCase     :			Optional[int]     	=torch.sum(predicted_tm_term * normed_residue_mask		,       dim=-1    )

      UpperCAmelCase     :			Optional[int]     	=per_alignment * residue_weights

      UpperCAmelCase     :			int     	=(weighted == torch.max(__lowerCAmelCase    )).nonzero()[0]
      return per_alignment[tuple(__lowerCAmelCase    )]

from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__snake_case         =     20

# Time between time steps in seconds
__snake_case         =     INTERVAL / 10_00







class       __snake_case :
 def __init__(	self		, snake_case__		, snake_case__		, snake_case__		, snake_case__		, snake_case__ = 1.0		, snake_case__ = 1.0		, snake_case__ = "blue"		, )						->				None:
       '''simple docstring'''





       UpperCAmelCase     :			Any     	=position_x
       UpperCAmelCase     :			Dict     	=position_y
       UpperCAmelCase     :			List[Any]     	=velocity_x
       UpperCAmelCase     :			Optional[int]     	=velocity_y
       UpperCAmelCase     :			List[str]     	=mass
       UpperCAmelCase     :			Any     	=size
       UpperCAmelCase     :			List[str]     	=color
 @property
 def 						UpperCAmelCase__							(	self    )						->				tuple[float, float]:
       '''simple docstring'''





       return self.position_x, self.position_y
 @property
 def 						UpperCAmelCase__							(	self    )						->				tuple[float, float]:
       '''simple docstring'''





       return self.velocity_x, self.velocity_y
 def 						UpperCAmelCase__							(	self		, snake_case__		, snake_case__		, snake_case__    )						->				None:
       '''simple docstring'''





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






 def 						UpperCAmelCase__							(	self		, snake_case__    )						->				None:
       '''simple docstring'''





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







class       __snake_case :
 def __init__(	self		, snake_case__		, snake_case__ = 1.0		, snake_case__ = 1.0		, snake_case__ = 0.0		, )						->				None:
       '''simple docstring'''





       UpperCAmelCase     :			Tuple     	=bodies
       UpperCAmelCase     :			List[str]     	=gravitation_constant
       UpperCAmelCase     :			int     	=time_factor
       UpperCAmelCase     :			List[Any]     	=softening_factor
 def __len__(	self    )						->				int:
       '''simple docstring'''





       return len(self.bodies    )






 def 						UpperCAmelCase__							(	self		, snake_case__    )						->				None:
       '''simple docstring'''





       for bodya in self.bodies:
             UpperCAmelCase     :			Tuple     	=0.0
             UpperCAmelCase     :			Union[str, Any]     	=0.0
             for bodya in self.bodies:
                   if bodya != bodya:
                         UpperCAmelCase     :			Optional[Any]     	=bodya.position_x - bodya.position_x
                         UpperCAmelCase     :			Any     	=bodya.position_y - bodya.position_y

                         # Calculation of the distance using Pythagoras's theorem
                         # Extra factor due to the softening technique
                         UpperCAmelCase     :			Optional[Any]     	=(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 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase    )->	None:






      '''simple docstring'''





      body_system.update_system(__lowerCAmelCase    )

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





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase = -1		,       __lowerCAmelCase = 1		,       __lowerCAmelCase = -1		,       __lowerCAmelCase = 1		,       )->	None:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[Any]     	=plt.figure()
      fig.canvas.set_window_title(__lowerCAmelCase    )
      UpperCAmelCase     :			Dict     	=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     :			Tuple     	=[
          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(__lowerCAmelCase    )

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

      UpperCAmelCase     :			List[Any]     	=animation.FuncAnimation(  # noqa: F841
          __lowerCAmelCase		,       __lowerCAmelCase		,       interval=__lowerCAmelCase		,       blit=__lowerCAmelCase    )

      plt.show()





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			int     	=0.9700436
      UpperCAmelCase     :			Dict     	=-0.24308753
      UpperCAmelCase     :			List[Any]     	=0.466203685
      UpperCAmelCase     :			Dict     	=0.43236573

      UpperCAmelCase     :			Optional[Any]     	=[
          Body(__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase		,       size=0.2		,       color='''red'''    ),
          Body(-position_x		,       -position_y		,       __lowerCAmelCase		,       __lowerCAmelCase		,       size=0.2		,       color='''green'''    ),
          Body(0		,       0		,       -2 * velocity_x		,       -2 * velocity_y		,       size=0.2		,       color='''blue'''    ),
      ]
      return BodySystem(__lowerCAmelCase		,       time_factor=3    )





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			List[str]     	=7.3_4_7_6e2_2
      UpperCAmelCase     :			Tuple     	=5.9_7_2e2_4
      UpperCAmelCase     :			str     	=10_22
      UpperCAmelCase     :			Any     	=3_84_39_90_00
      UpperCAmelCase     :			List[Any]     	=6.6_7_4e-1_1

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

      UpperCAmelCase     :			Optional[int]     	=Body(-earth_moon_distance		,       0		,       0		,       __lowerCAmelCase		,       __lowerCAmelCase		,       10_00_00_00		,       '''grey'''    )
      UpperCAmelCase     :			Dict     	=Body(0		,       0		,       0		,       __lowerCAmelCase		,       __lowerCAmelCase		,       50_00_00_00		,       '''blue'''    )
      return BodySystem([earth, moon]		,       __lowerCAmelCase		,       time_factor=1_00_00_00    )





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[int]     	=[]
      for _ in range(10    ):
            UpperCAmelCase     :			Union[str, Any]     	=random.uniform(-0.5		,       0.5    )
            UpperCAmelCase     :			List[Any]     	=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    )		,       __lowerCAmelCase		,       __lowerCAmelCase		,       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(__lowerCAmelCase		,       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_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)
