



'''simple docstring'''





from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt





def   __snake_case			(    _UpperCAmelCase       ):
    __a		 =       int(number**0.5       )
    return number == sq * sq





def   __snake_case			(    _UpperCAmelCase     ,				_UpperCAmelCase     ,				_UpperCAmelCase     ,				_UpperCAmelCase     ,				_UpperCAmelCase     ,				_UpperCAmelCase       ):
    __a		 =       x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
    __a		 =       x_den * y_den * z_den
    __a		 =       gcd(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
    top //= hcf
    bottom //= hcf
    return top, bottom





def   __snake_case			(    _UpperCAmelCase = 35       ):
    __a		 =       set()
    __a		 =       42
    __a		 =       Fraction(0       )
    __a		 =       42

    for x_num in range(1     ,				order + 1       ):
        for x_den in range(x_num + 1     ,				order + 1       ):
            for y_num in range(1     ,				order + 1       ):
                for y_den in range(y_num + 1     ,				order + 1       ):
                    # n=1
                    __a		 =       x_num * y_den + x_den * y_num
                    __a		 =       x_den * y_den
                    __a		 =       gcd(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                    z_num //= hcf
                    z_den //= hcf
                    if 0 < z_num < z_den <= order:
                        __a		 =       add_three(
                            _SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                        unique_s.add(_SCREAMING_SNAKE_CASE       )

                    # n=2
                    __a		 =       (
                        x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
                    )
                    __a		 =       x_den * x_den * y_den * y_den
                    if is_sq(_SCREAMING_SNAKE_CASE       ) and is_sq(_SCREAMING_SNAKE_CASE       ):
                        __a		 =       int(sqrt(_SCREAMING_SNAKE_CASE       )       )
                        __a		 =       int(sqrt(_SCREAMING_SNAKE_CASE       )       )
                        __a		 =       gcd(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                        z_num //= hcf
                        z_den //= hcf
                        if 0 < z_num < z_den <= order:
                            __a		 =       add_three(
                                _SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                            unique_s.add(_SCREAMING_SNAKE_CASE       )

                    # n=-1
                    __a		 =       x_num * y_num
                    __a		 =       x_den * y_num + x_num * y_den
                    __a		 =       gcd(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                    z_num //= hcf
                    z_den //= hcf
                    if 0 < z_num < z_den <= order:
                        __a		 =       add_three(
                            _SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                        unique_s.add(_SCREAMING_SNAKE_CASE       )

                    # n=2
                    __a		 =       x_num * x_num * y_num * y_num
                    __a		 =       (
                        x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
                    )
                    if is_sq(_SCREAMING_SNAKE_CASE       ) and is_sq(_SCREAMING_SNAKE_CASE       ):
                        __a		 =       int(sqrt(_SCREAMING_SNAKE_CASE       )       )
                        __a		 =       int(sqrt(_SCREAMING_SNAKE_CASE       )       )
                        __a		 =       gcd(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                        z_num //= hcf
                        z_den //= hcf
                        if 0 < z_num < z_den <= order:
                            __a		 =       add_three(
                                _SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )
                            unique_s.add(_SCREAMING_SNAKE_CASE       )

    for num, den in unique_s:
        total += Fraction(_SCREAMING_SNAKE_CASE     ,				_SCREAMING_SNAKE_CASE       )

    return total.denominator + total.numerator


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










import inspect
from typing import List, Optional, Tuple, Union

import numpy as np
import PIL
import torch
import torch.utils.checkpoint

from ...models import UNetaDModel, VQModel
from ...schedulers import (
    DDIMScheduler,
    DPMSolverMultistepScheduler,
    EulerAncestralDiscreteScheduler,
    EulerDiscreteScheduler,
    LMSDiscreteScheduler,
    PNDMScheduler,
)
from ...utils import PIL_INTERPOLATION, randn_tensor
from ..pipeline_utils import DiffusionPipeline, ImagePipelineOutput





def   __snake_case			(    _UpperCAmelCase       ):
    __a ,     __a		 =       image.size
    __a ,     __a		 =       (x - x % 32 for x in (w, h))  # resize to integer multiple of 32
    __a		 =       image.resize((w, h)     ,				resample=PIL_INTERPOLATION['''lanczos''']       )
    __a		 =       np.array(_UpperCAmelCase       ).astype(np.floataa       ) / 2_55.0
    __a		 =       image[None].transpose(0     ,				3     ,				1     ,				2       )
    __a		 =       torch.from_numpy(_UpperCAmelCase       )
    return 2.0 * image - 1.0

class 					_A (			__UpperCAmelCase					):



       def __init__(	self							:  Any  ,	__SCREAMING_SNAKE_CASE							:  VQModel  ,	__SCREAMING_SNAKE_CASE							:  UNetaDModel  ,	__SCREAMING_SNAKE_CASE							:  Union[
               DDIMScheduler,
               PNDMScheduler,
               LMSDiscreteScheduler,
               EulerDiscreteScheduler,
               EulerAncestralDiscreteScheduler,
               DPMSolverMultistepScheduler,
           ]  ,	):


           '''simple docstring'''

           super().__init__()
           self.register_modules(vqvae=__SCREAMING_SNAKE_CASE  ,	unet=__SCREAMING_SNAKE_CASE  ,	scheduler=__SCREAMING_SNAKE_CASE)





       @torch.no_grad()
       def __call__(	self							:  List[Any]  ,	__SCREAMING_SNAKE_CASE							:  Union[torch.Tensor, PIL.Image.Image] = None  ,	__SCREAMING_SNAKE_CASE							:  Optional[int] = 1  ,	__SCREAMING_SNAKE_CASE							:  Optional[int] = 100  ,	__SCREAMING_SNAKE_CASE							:  Optional[float] = 0.0  ,	__SCREAMING_SNAKE_CASE							:  Optional[Union[torch.Generator, List[torch.Generator]]] = None  ,	__SCREAMING_SNAKE_CASE							:  Optional[str] = "pil"  ,	__SCREAMING_SNAKE_CASE							:  bool = True  ,	):


           '''simple docstring'''

           if isinstance(__SCREAMING_SNAKE_CASE  ,	PIL.Image.Image):
               __a		 =       1
           elif isinstance(__SCREAMING_SNAKE_CASE  ,	torch.Tensor):
               __a		 =       image.shape[0]
           else:
               raise ValueError(F'`image` has to be of type `PIL.Image.Image` or `torch.Tensor` but is {type(__SCREAMING_SNAKE_CASE)}')

           if isinstance(__SCREAMING_SNAKE_CASE  ,	PIL.Image.Image):
               __a		 =       preprocess(__SCREAMING_SNAKE_CASE)

           __a ,     __a		 =       image.shape[-2:]

           # in_channels should be 6: 3 for latents, 3 for low resolution image
           __a		 =       (batch_size, self.unet.config.in_channels // 2, height, width)
           __a		 =       next(self.unet.parameters()).dtype

           __a		 =       randn_tensor(__SCREAMING_SNAKE_CASE  ,	generator=__SCREAMING_SNAKE_CASE  ,	device=self.device  ,	dtype=__SCREAMING_SNAKE_CASE)

           __a		 =       image.to(device=self.device  ,	dtype=__SCREAMING_SNAKE_CASE)

           # set timesteps and move to the correct device
           self.scheduler.set_timesteps(__SCREAMING_SNAKE_CASE  ,	device=self.device)
           __a		 =       self.scheduler.timesteps

           # scale the initial noise by the standard deviation required by the scheduler
           __a		 =       latents * self.scheduler.init_noise_sigma

           # prepare extra kwargs for the scheduler step, since not all schedulers have the same signature.
           # eta (η) is only used with the DDIMScheduler, it will be ignored for other schedulers.
           # eta corresponds to η in DDIM paper: https://arxiv.org/abs/2010.02502
           # and should be between [0, 1]
           __a		 =       '''eta''' in set(inspect.signature(self.scheduler.step).parameters.keys())
           __a		 =       {}
           if accepts_eta:
               __a		 =       eta

           for t in self.progress_bar(__SCREAMING_SNAKE_CASE):
               # concat latents and low resolution image in the channel dimension.
               __a		 =       torch.cat([latents, image]  ,	dim=1)
               __a		 =       self.scheduler.scale_model_input(__SCREAMING_SNAKE_CASE  ,	__SCREAMING_SNAKE_CASE)
               # predict the noise residual
               __a		 =       self.unet(__SCREAMING_SNAKE_CASE  ,	__SCREAMING_SNAKE_CASE).sample
               # compute the previous noisy sample x_t -> x_t-1
               __a		 =       self.scheduler.step(__SCREAMING_SNAKE_CASE  ,	__SCREAMING_SNAKE_CASE  ,	__SCREAMING_SNAKE_CASE  ,	**__SCREAMING_SNAKE_CASE).prev_sample

           # decode the image latents with the VQVAE
           __a		 =       self.vqvae.decode(__SCREAMING_SNAKE_CASE).sample
           __a		 =       torch.clamp(__SCREAMING_SNAKE_CASE  ,	-1.0  ,	1.0)
           __a		 =       image / 2 + 0.5
           __a		 =       image.cpu().permute(0  ,	2  ,	3  ,	1).numpy()

           if output_type == "pil":
               __a		 =       self.numpy_to_pil(__SCREAMING_SNAKE_CASE)

           if not return_dict:
               return (image,)

           return ImagePipelineOutput(images=__SCREAMING_SNAKE_CASE)





