




'''simple docstring'''





from dataclasses import dataclass
from typing import Tuple

import numpy as np
import torch




@dataclass
class 			_snake_case		:
			lowerCAmelCase_					:				torch.Tensor  # [batch_size x 3]
			lowerCAmelCase_					:				torch.Tensor  # [batch_size x 3]
			lowerCAmelCase_					:				torch.Tensor  # [batch_size x 3]
			lowerCAmelCase_					:				torch.Tensor  # [batch_size x 3]
			lowerCAmelCase_					:				int
			lowerCAmelCase_					:				int
			lowerCAmelCase_					:				float
			lowerCAmelCase_					:				float
			lowerCAmelCase_					:				Tuple[int]




			def        lowerCAmelCase__    ( self							)    ->    Union[str, Any]:

										'''simple docstring'''




										assert self.x.shape[0] == self.y.shape[0] == self.z.shape[0] == self.origin.shape[0]
										assert self.x.shape[1] == self.y.shape[1] == self.z.shape[1] == self.origin.shape[1] == 3
										assert len(self.x.shape							) == len(self.y.shape							) == len(self.z.shape							) == len(self.origin.shape							) == 2




			def        lowerCAmelCase__    ( self							)    ->    Any:

										'''simple docstring'''




										return torch.from_numpy(np.array([self.width, self.height]    ,							dtype=np.floataa							)							)




			def        lowerCAmelCase__    ( self							)    ->    Tuple:

										'''simple docstring'''




										return torch.from_numpy(np.array([self.x_fov, self.y_fov]    ,							dtype=np.floataa							)							)




			def        lowerCAmelCase__    ( self							)    ->    str:

										'''simple docstring'''




										snake_case_      =				torch.arange(self.height * self.width							)
										snake_case_      =				torch.stack(
										    [
										        pixel_indices % self.width,
										        torch.div(__lowercase    ,							self.width    ,							rounding_mode="trunc"							),
										    ]    ,							axis=1    ,							)
										return coords




			@property
			def        lowerCAmelCase__    ( self							)    ->    str:

										'''simple docstring'''




										snake_case_      =				self.shape
										snake_case_      =				int(np.prod(__lowercase							)							)

										snake_case_      =				self.get_image_coords()
										snake_case_      =				torch.broadcast_to(coords.unsqueeze(0							)    ,							[batch_size * inner_batch_size, *coords.shape]							)
										snake_case_      =				self.get_camera_rays(__lowercase							)

										snake_case_      =				rays.view(__lowercase    ,							inner_batch_size * self.height * self.width    ,							2    ,							3							)

										return rays




			def        lowerCAmelCase__    ( self    ,							a__							)    ->    int:

										'''simple docstring'''




										snake_case_      =				coords.shape
										assert n_coords == 2
										assert batch_size == self.origin.shape[0]

										snake_case_      =				coords.view(__lowercase    ,							-1    ,							2							)

										snake_case_      =				self.resolution()
										snake_case_      =				self.fov()

										snake_case_      =				(flat.float() / (res - 1)) * 2 - 1
										snake_case_      =				fracs * torch.tan(fov / 2							)

										snake_case_      =				fracs.view(__lowercase    ,							-1    ,							2							)
										snake_case_      =				(
										    self.z.view(__lowercase    ,							1    ,							3							)
										    + self.x.view(__lowercase    ,							1    ,							3							) * fracs[:, :, :1]
										    + self.y.view(__lowercase    ,							1    ,							3							) * fracs[:, :, 1:]
										)
										snake_case_      =				directions / directions.norm(dim=-1    ,							keepdim=__lowercase							)
										snake_case_      =				torch.stack(
										    [
										        torch.broadcast_to(self.origin.view(__lowercase    ,							1    ,							3							)    ,							[batch_size, directions.shape[1], 3]							),
										        directions,
										    ]    ,							dim=2    ,							)
										return rays.view(__lowercase    ,							*__lowercase    ,							2    ,							3							)




			def        lowerCAmelCase__    ( self    ,							a__    ,							a__							)    ->    int:

										'''simple docstring'''




										assert width * self.height == height * self.width, "The aspect ratio should not change."
										return DifferentiableProjectiveCamera(
										    origin=self.origin    ,							x=self.x    ,							y=self.y    ,							z=self.z    ,							width=__lowercase    ,							height=__lowercase    ,							x_fov=self.x_fov    ,							y_fov=self.y_fov    ,							)


def 					UpperCamelCase_(     snake_case							:	Optional[Any]					):

							'''simple docstring'''






							snake_case_      =				[]
							snake_case_      =				[]
							snake_case_      =				[]
							snake_case_      =				[]
							for theta in np.linspace(0							,    2 * np.pi							,    num=2_0					):
														snake_case_      =				np.array([np.sin(__lowerCAmelCase					), np.cos(__lowerCAmelCase					), -0.5]					)
														z /= np.sqrt(np.sum(z**2					)					)
														snake_case_      =				-z * 4
														snake_case_      =				np.array([np.cos(__lowerCAmelCase					), -np.sin(__lowerCAmelCase					), 0.0]					)
														snake_case_      =				np.cross(__lowerCAmelCase							,    __lowerCAmelCase					)
														origins.append(__lowerCAmelCase					)
														xs.append(__lowerCAmelCase					)
														ys.append(__lowerCAmelCase					)
														zs.append(__lowerCAmelCase					)
							return DifferentiableProjectiveCamera(
							    origin=torch.from_numpy(np.stack(__lowerCAmelCase							,    axis=0					)					).float()							,    x=torch.from_numpy(np.stack(__lowerCAmelCase							,    axis=0					)					).float()							,    y=torch.from_numpy(np.stack(__lowerCAmelCase							,    axis=0					)					).float()							,    z=torch.from_numpy(np.stack(__lowerCAmelCase							,    axis=0					)					).float()							,    width=__lowerCAmelCase							,    height=__lowerCAmelCase							,    x_fov=0.7							,    y_fov=0.7							,    shape=(1, len(__lowerCAmelCase					))							,    )










'''simple docstring'''





def 					UpperCamelCase_(     snake_case							:	int = 1_0_0_0_0_0_0					):

							'''simple docstring'''






							snake_case_      =				set(range(3							,    snake_case							,    2					)					)
							primes.add(2					)
							for p in range(3							,    snake_case							,    2					):
														if p not in primes:
																					continue
														primes.difference_update(set(range(p * p							,    snake_case							,    snake_case					)					)					)

							snake_case_      =				[float(snake_case					) for n in range(limit + 1					)]

							for p in primes:
														for n in range(snake_case							,    limit + 1							,    snake_case					):
																					phi[n] *= 1 - 1 / p

							return int(sum(phi[2:]					)					)


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




