




from typing import Optional, Tuple, Union

import torch
from einops import rearrange, reduce

from diffusers import DDIMScheduler, DDPMScheduler, DiffusionPipeline, ImagePipelineOutput, UNetaDConditionModel
from diffusers.schedulers.scheduling_ddim import DDIMSchedulerOutput
from diffusers.schedulers.scheduling_ddpm import DDPMSchedulerOutput


lowerCamelCase__			:       List[str]         =							8

def  UpperCAmelCase_							(				__UpperCAmelCase						:     Tuple      ,      __UpperCAmelCase						:     Any=BITS							) ->			str:
      SCREAMING_SNAKE_CASE_     						=      x.device

      SCREAMING_SNAKE_CASE_     						=      (x * 2_55).int().clamp(0      ,      2_55							)

      SCREAMING_SNAKE_CASE_     						=      2 ** torch.arange(bits - 1      ,      -1      ,      -1      ,      device=__UpperCAmelCase							)
      SCREAMING_SNAKE_CASE_     						=      rearrange(__UpperCAmelCase      ,      'd -> d 1 1'							)
      SCREAMING_SNAKE_CASE_     						=      rearrange(__UpperCAmelCase      ,      'b c h w -> b c 1 h w'							)

      SCREAMING_SNAKE_CASE_     						=      ((x & mask) != 0).float()
      SCREAMING_SNAKE_CASE_     						=      rearrange(__UpperCAmelCase      ,      'b c d h w -> b (c d) h w'							)
      SCREAMING_SNAKE_CASE_     						=      bits * 2 - 1
      return bits

def  UpperCAmelCase_							(				__UpperCAmelCase						:     List[Any]      ,      __UpperCAmelCase						:     Tuple=BITS							) ->			str:
      SCREAMING_SNAKE_CASE_     						=      x.device

      SCREAMING_SNAKE_CASE_     						=      (x > 0).int()
      SCREAMING_SNAKE_CASE_     						=      2 ** torch.arange(bits - 1      ,      -1      ,      -1      ,      device=__UpperCAmelCase      ,      dtype=torch.intaa							)

      SCREAMING_SNAKE_CASE_     						=      rearrange(__UpperCAmelCase      ,      'd -> d 1 1'							)
      SCREAMING_SNAKE_CASE_     						=      rearrange(__UpperCAmelCase      ,      'b (c d) h w -> b c d h w'      ,      d=8							)
      SCREAMING_SNAKE_CASE_     						=      reduce(x * mask      ,      'b c d h w -> b c h w'      ,      'sum'							)
      return (dec / 2_55).clamp(0.0      ,      1.0							)

def  UpperCAmelCase_							(				self						:     str      ,      __UpperCAmelCase						:     torch.FloatTensor      ,      __UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     torch.FloatTensor      ,      __UpperCAmelCase						:     float = 0.0      ,      __UpperCAmelCase						:     bool = True      ,      __UpperCAmelCase						:     List[str]=None      ,      __UpperCAmelCase						:     bool = True      ,      ) ->			Union[DDIMSchedulerOutput, Tuple]:
      if self.num_inference_steps is None:
            raise ValueError(
                'Number of inference steps is \'None\', you need to run \'set_timesteps\' after creating the scheduler'							)

      # See formulas (12) and (16) of DDIM paper https://arxiv.org/pdf/2010.02502.pdf
      # Ideally, read DDIM paper in-detail understanding

      # Notation (<variable name> -> <name in paper>
      # - pred_noise_t -> e_theta(x_t, t)
      # - pred_original_sample -> f_theta(x_t, t) or x_0
      # - std_dev_t -> sigma_t
      # - eta -> η
      # - pred_sample_direction -> "direction pointing to x_t"
      # - pred_prev_sample -> "x_t-1"

      # 1. get previous step value (=t-1)
      SCREAMING_SNAKE_CASE_     						=      timestep - self.config.num_train_timesteps // self.num_inference_steps

      # 2. compute alphas, betas
      SCREAMING_SNAKE_CASE_     						=      self.alphas_cumprod[timestep]
      SCREAMING_SNAKE_CASE_     						=      self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod

      SCREAMING_SNAKE_CASE_     						=      1 - alpha_prod_t

      # 3. compute predicted original sample from predicted noise also called
      # "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
      SCREAMING_SNAKE_CASE_     						=      (sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5

      # 4. Clip "predicted x_0"
      SCREAMING_SNAKE_CASE_     						=      self.bit_scale
      if self.config.clip_sample:
            SCREAMING_SNAKE_CASE_     						=      torch.clamp(__UpperCAmelCase      ,      -scale      ,      __UpperCAmelCase							)

      # 5. compute variance: "sigma_t(η)" -> see formula (16)
      # σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
      SCREAMING_SNAKE_CASE_     						=      self._get_variance(__UpperCAmelCase      ,      __UpperCAmelCase							)
      SCREAMING_SNAKE_CASE_     						=      eta * variance ** 0.5

      if use_clipped_model_output:
            # the model_output is always re-derived from the clipped x_0 in Glide
            SCREAMING_SNAKE_CASE_     						=      (sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5

      # 6. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
      SCREAMING_SNAKE_CASE_     						=      (1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * model_output

      # 7. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
      SCREAMING_SNAKE_CASE_     						=      alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction

      if eta > 0:
            # randn_like does not support generator https://github.com/pytorch/pytorch/issues/27072
            SCREAMING_SNAKE_CASE_     						=      model_output.device if torch.is_tensor(__UpperCAmelCase							) else 'cpu'
            SCREAMING_SNAKE_CASE_     						=      torch.randn(model_output.shape      ,      dtype=model_output.dtype      ,      generator=__UpperCAmelCase							).to(__UpperCAmelCase							)
            SCREAMING_SNAKE_CASE_     						=      self._get_variance(__UpperCAmelCase      ,      __UpperCAmelCase							) ** 0.5 * eta * noise

            SCREAMING_SNAKE_CASE_     						=      prev_sample + variance

      if not return_dict:
            return (prev_sample,)

      return DDIMSchedulerOutput(prev_sample=__UpperCAmelCase      ,      pred_original_sample=__UpperCAmelCase							)

def  UpperCAmelCase_							(				self						:     Union[str, Any]      ,      __UpperCAmelCase						:     torch.FloatTensor      ,      __UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     torch.FloatTensor      ,      __UpperCAmelCase						:     Optional[int]="epsilon"      ,      __UpperCAmelCase						:     Tuple=None      ,      __UpperCAmelCase						:     bool = True      ,      ) ->			Union[DDPMSchedulerOutput, Tuple]:
      SCREAMING_SNAKE_CASE_     						=      timestep

      if model_output.shape[1] == sample.shape[1] * 2 and self.variance_type in ["learned", "learned_range"]:
            SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      torch.split(__UpperCAmelCase      ,      sample.shape[1]      ,      dim=1							)
      else:
            SCREAMING_SNAKE_CASE_     						=      None

      # 1. compute alphas, betas
      SCREAMING_SNAKE_CASE_     						=      self.alphas_cumprod[t]
      SCREAMING_SNAKE_CASE_     						=      self.alphas_cumprod[t - 1] if t > 0 else self.one
      SCREAMING_SNAKE_CASE_     						=      1 - alpha_prod_t
      SCREAMING_SNAKE_CASE_     						=      1 - alpha_prod_t_prev

      # 2. compute predicted original sample from predicted noise also called
      # "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
      if prediction_type == "epsilon":
            SCREAMING_SNAKE_CASE_     						=      (sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
      elif prediction_type == "sample":
            SCREAMING_SNAKE_CASE_     						=      model_output
      else:
            raise ValueError(f"Unsupported prediction_type {prediction_type}."							)

      # 3. Clip "predicted x_0"
      SCREAMING_SNAKE_CASE_     						=      self.bit_scale
      if self.config.clip_sample:
            SCREAMING_SNAKE_CASE_     						=      torch.clamp(__UpperCAmelCase      ,      -scale      ,      __UpperCAmelCase							)

      # 4. Compute coefficients for pred_original_sample x_0 and current sample x_t
      # See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
      SCREAMING_SNAKE_CASE_     						=      (alpha_prod_t_prev ** 0.5 * self.betas[t]) / beta_prod_t
      SCREAMING_SNAKE_CASE_     						=      self.alphas[t] ** 0.5 * beta_prod_t_prev / beta_prod_t

      # 5. Compute predicted previous sample µ_t
      # See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
      SCREAMING_SNAKE_CASE_     						=      pred_original_sample_coeff * pred_original_sample + current_sample_coeff * sample

      # 6. Add noise
      SCREAMING_SNAKE_CASE_     						=      0
      if t > 0:
            SCREAMING_SNAKE_CASE_     						=      torch.randn(
                model_output.size()      ,      dtype=model_output.dtype      ,      layout=model_output.layout      ,      generator=__UpperCAmelCase							).to(model_output.device							)
            SCREAMING_SNAKE_CASE_     						=      (self._get_variance(__UpperCAmelCase      ,      predicted_variance=__UpperCAmelCase							) ** 0.5) * noise

      SCREAMING_SNAKE_CASE_     						=      pred_prev_sample + variance

      if not return_dict:
            return (pred_prev_sample,)

      return DDPMSchedulerOutput(prev_sample=__UpperCAmelCase      ,      pred_original_sample=__UpperCAmelCase							)





class 	lowerCamelCase_							(				_SCREAMING_SNAKE_CASE ):


 '''simple docstring'''







 def __init__(							self  :		Optional[Any]    ,    _lowerCAmelCase  :		UNetaDConditionModel    ,    _lowerCAmelCase  :		Union[DDIMScheduler, DDPMScheduler]    ,    _lowerCAmelCase  :		Optional[float] = 1.0    ,    ):
       super().__init__()
       SCREAMING_SNAKE_CASE_     						=      bit_scale
       SCREAMING_SNAKE_CASE_     						=      (
           ddim_bit_scheduler_step if isinstance(_lowerCAmelCase    ,    _lowerCAmelCase						) else ddpm_bit_scheduler_step
       )

       self.register_modules(unet=_lowerCAmelCase    ,    scheduler=_lowerCAmelCase						)




 @torch.no_grad()
 def __call__(							self  :		Optional[int]    ,    _lowerCAmelCase  :		Optional[int] = 256    ,    _lowerCAmelCase  :		Optional[int] = 256    ,    _lowerCAmelCase  :		Optional[int] = 50    ,    _lowerCAmelCase  :		Optional[torch.Generator] = None    ,    _lowerCAmelCase  :		Optional[int] = 1    ,    _lowerCAmelCase  :		Optional[str] = "pil"    ,    _lowerCAmelCase  :		bool = True    ,    **_lowerCAmelCase  :		Tuple    ,    ):
       SCREAMING_SNAKE_CASE_     						=      torch.randn(
           (batch_size, self.unet.config.in_channels, height, width)    ,    generator=_lowerCAmelCase    ,    )
       SCREAMING_SNAKE_CASE_     						=      decimal_to_bits(_lowerCAmelCase						) * self.bit_scale
       SCREAMING_SNAKE_CASE_     						=      latents.to(self.device						)

       self.scheduler.set_timesteps(_lowerCAmelCase						)

       for t in self.progress_bar(self.scheduler.timesteps						):
             # predict the noise residual
             SCREAMING_SNAKE_CASE_     						=      self.unet(_lowerCAmelCase    ,    _lowerCAmelCase						).sample

             # compute the previous noisy sample x_t -> x_t-1
             SCREAMING_SNAKE_CASE_     						=      self.scheduler.step(_lowerCAmelCase    ,    _lowerCAmelCase    ,    _lowerCAmelCase						).prev_sample

       SCREAMING_SNAKE_CASE_     						=      bits_to_decimal(_lowerCAmelCase						)

       if output_type == "pil":
             SCREAMING_SNAKE_CASE_     						=      self.numpy_to_pil(_lowerCAmelCase						)

       if not return_dict:
             return (image,)

       return ImagePipelineOutput(images=_lowerCAmelCase						)





def  UpperCAmelCase_							(				) ->			int:
      return 1

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else two_pence(x - 2							) + one_pence()

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else five_pence(x - 5							) + two_pence(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else ten_pence(x - 10							) + five_pence(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else twenty_pence(x - 20							) + ten_pence(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else fifty_pence(x - 50							) + twenty_pence(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else one_pound(x - 1_00							) + fifty_pence(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return 0 if x < 0 else two_pound(x - 2_00							) + one_pound(__UpperCAmelCase							)

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int = 2_00							) ->			int:
      return two_pound(__UpperCAmelCase							)


if __name__ == "__main__":
   print(solution(int(input().strip())))