code: from itertools import count
  from math import asin, pi, sqrt
  
  
  def SCREAMING_SNAKE_CASE				(     a:	Optional[int]				):
         return (
             (1 - 2 * point) * sqrt(point - point**2				) + 2 * point + asin(sqrt(1 - point				)				)
         ) / 4
  
  
  def SCREAMING_SNAKE_CASE				(     a:	Dict				):
         _UpperCamelCase     :       Optional[int]  						=  (circles_number + 1 - sqrt(2 * circles_number				)) / (
             2 * (circles_number**2 + 1)
         )
         _UpperCamelCase     :       Union[str, Any]  						=  circles_number * intersection_y
  
         _UpperCamelCase     :       int  						=  intersection_x * intersection_y / 2
         _UpperCamelCase     :       Dict  						=  circle_bottom_arc_integral(
             1 / 2				) - circle_bottom_arc_integral(snake_case_				)
  
         return triangle_area + concave_region_area
  
  
  def SCREAMING_SNAKE_CASE				(     a:	int = 1 / 1000				):
         _UpperCamelCase     :       Optional[Any]  						=  (1 - pi / 4) / 4
  
         for n in count(1				):
                if concave_triangle_area(snake_case_				) / l_section_area < fraction:
                       return n
  
         return -1
  
  
  if __name__ == "__main__":
   print(f'''{solution() = }''')
  
  
  
  
style_context: import tempfile
  import unittest
  
  import torch
  
  from diffusers import UNetaDConditionModel
  from diffusers.training_utils import EMAModel
  from diffusers.utils.testing_utils import enable_full_determinism, skip_mps, torch_device
  
  
  enable_full_determinism()
  
  
  
  
  class    __lowerCAmelCase  (       unittest.TestCase						):
  
  
  
   '''simple docstring'''
   _lowerCamelCase						:	str				    = 'hf-internal-testing/tiny-stable-diffusion-pipe'
   _lowerCamelCase						:	int				    = 1
   _lowerCamelCase						:	Any				    = 77
   _lowerCamelCase						:	List[Any]				    = 32
   _lowerCamelCase						:	List[str]				    = 4
   _lowerCamelCase						:	Dict				    = 64
   _lowerCamelCase						:	List[str]				    = torch.manual_seed(0						)
  
   def 					__snake_case					(			self       :		Any       ,			lowerCamelCase       :		Union[str, Any]=0.9_9_9_9  ) ->							Dict:
          _UpperCamelCase     :       int  						=  UNetaDConditionModel.from_pretrained(self.model_id       ,			subfolder='unet'  )
          _UpperCamelCase     :       Any  						=  unet.to(lowerCamelCase  )
          _UpperCamelCase     :       Union[str, Any]  						=  EMAModel(unet.parameters()       ,			decay=lowerCamelCase       ,			model_cls=lowerCamelCase       ,			model_config=unet.config  )
          return unet, ema_unet
  
   def 					__snake_case					(			self       :		Tuple  ) ->							Union[str, Any]:
          _UpperCamelCase     :       Tuple  						=  torch.randn(
              self.batch_size       ,			self.num_in_channels       ,			self.latent_height       ,			self.latent_width       ,			generator=self.generator  ).to(lowerCamelCase  )
          _UpperCamelCase     :       Union[str, Any]  						=  torch.randint(0       ,			1_0_0_0       ,			size=(self.batch_size,)       ,			generator=self.generator  ).to(lowerCamelCase  )
          _UpperCamelCase     :       int  						=  torch.randn(
              self.batch_size       ,			self.prompt_length       ,			self.text_encoder_hidden_dim       ,			generator=self.generator  ).to(lowerCamelCase  )
          return noisy_latents, timesteps, encoder_hidden_states
  
   def 					__snake_case					(			self       :		str       ,			lowerCamelCase       :		List[Any]  ) ->							List[Any]:
          _UpperCamelCase     :       int  						=  {}
          for k, param in unet.state_dict().items():
                 _UpperCamelCase     :       Optional[Any]  						=  torch.randn_like(lowerCamelCase  ) + (param * torch.randn_like(lowerCamelCase  ))
                 updated_state_dict.update({k: updated_param}  )
          unet.load_state_dict(lowerCamelCase  )
          return unet
  
   def 					__snake_case					(			self       :		List[Any]  ) ->							Tuple:
          _UpperCamelCase						,      _UpperCamelCase     :       Union[str, Any]  						=  self.get_models()
          # Take the first (hypothetical) EMA step.
          ema_unet.step(unet.parameters()  )
          assert ema_unet.optimization_step == 1
  
          # Take two more.
          for _ in range(2  ):
                 ema_unet.step(unet.parameters()  )
          assert ema_unet.optimization_step == 3
  
   def 					__snake_case					(			self       :		int  ) ->							Any:
          _UpperCamelCase						,      _UpperCamelCase     :       Any  						=  self.get_models()
          # Since the `unet` is not being updated (i.e., backprop'd)
          # there won't be any difference between the `params` of `unet`
          # and `ema_unet` even if we call `ema_unet.step(unet.parameters())`.
          ema_unet.step(unet.parameters()  )
          _UpperCamelCase     :       List[Any]  						=  list(unet.parameters()  )
          for s_param, param in zip(ema_unet.shadow_params       ,			lowerCamelCase  ):
                 assert torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
          # The above holds true even if we call `ema.step()` multiple times since
          # `unet` params are still not being updated.
          for _ in range(4  ):
                 ema_unet.step(unet.parameters()  )
          for s_param, param in zip(ema_unet.shadow_params       ,			lowerCamelCase  ):
                 assert torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
   def 					__snake_case					(			self       :		Dict  ) ->							Dict:
          _UpperCamelCase						,      _UpperCamelCase     :       Dict  						=  self.get_models()
          # Here we simulate the parameter updates for `unet`. Since there might
          # be some parameters which are initialized to zero we take extra care to
          # initialize their values to something non-zero before the multiplication.
          _UpperCamelCase     :       str  						=  self.simulate_backprop(lowerCamelCase  )
  
          # Take the EMA step.
          ema_unet.step(unet_pseudo_updated_step_one.parameters()  )
  
          # Now the EMA'd parameters won't be equal to the original model parameters.
          _UpperCamelCase     :       Optional[Any]  						=  list(unet_pseudo_updated_step_one.parameters()  )
          for s_param, param in zip(ema_unet.shadow_params       ,			lowerCamelCase  ):
                 assert ~torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
          # Ensure this is the case when we take multiple EMA steps.
          for _ in range(4  ):
                 ema_unet.step(unet.parameters()  )
          for s_param, param in zip(ema_unet.shadow_params       ,			lowerCamelCase  ):
                 assert ~torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
   def 					__snake_case					(			self       :		Union[str, Any]  ) ->							Tuple:
          # If we call EMA step after a backpropagation consecutively for two times,
          # the shadow params from those two steps should be different.
          _UpperCamelCase						,      _UpperCamelCase     :       Dict  						=  self.get_models()
  
          # First backprop + EMA
          _UpperCamelCase     :       Optional[Any]  						=  self.simulate_backprop(lowerCamelCase  )
          ema_unet.step(unet_step_one.parameters()  )
          _UpperCamelCase     :       Optional[Any]  						=  ema_unet.shadow_params
  
          # Second backprop + EMA
          _UpperCamelCase     :       Optional[int]  						=  self.simulate_backprop(lowerCamelCase  )
          ema_unet.step(unet_step_two.parameters()  )
          _UpperCamelCase     :       List[str]  						=  ema_unet.shadow_params
  
          for step_one, step_two in zip(lowerCamelCase       ,			lowerCamelCase  ):
                 assert ~torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
   def 					__snake_case					(			self       :		str  ) ->							int:
          # If there's no decay even if there are backprops, EMA steps
          # won't take any effect i.e., the shadow params would remain the
          # same.
          _UpperCamelCase						,      _UpperCamelCase     :       List[str]  						=  self.get_models(decay=0.0  )
          _UpperCamelCase     :       Optional[Any]  						=  self.simulate_backprop(lowerCamelCase  )
          ema_unet.step(unet_step_one.parameters()  )
          _UpperCamelCase     :       Dict  						=  ema_unet.shadow_params
  
          _UpperCamelCase     :       List[str]  						=  self.simulate_backprop(lowerCamelCase  )
          ema_unet.step(unet_step_two.parameters()  )
          _UpperCamelCase     :       Union[str, Any]  						=  ema_unet.shadow_params
  
          for step_one, step_two in zip(lowerCamelCase       ,			lowerCamelCase  ):
                 assert torch.allclose(lowerCamelCase       ,			lowerCamelCase  )
  
   @skip_mps
   def 					__snake_case					(			self       :		Dict  ) ->							str:
          _UpperCamelCase						,      _UpperCamelCase     :       Optional[Any]  						=  self.get_models()
          _UpperCamelCase						,      _UpperCamelCase						,      _UpperCamelCase     :       Any  						=  self.get_dummy_inputs()
  
          with tempfile.TemporaryDirectory() as tmpdir:
                 ema_unet.save_pretrained(lowerCamelCase  )
                 _UpperCamelCase     :       List[str]  						=  UNetaDConditionModel.from_pretrained(lowerCamelCase       ,			model_cls=lowerCamelCase  )
                 _UpperCamelCase     :       Dict  						=  loaded_unet.to(unet.device  )
  
          # Since no EMA step has been performed the outputs should match.
          _UpperCamelCase     :       Optional[Any]  						=  unet(lowerCamelCase       ,			lowerCamelCase       ,			lowerCamelCase  ).sample
          _UpperCamelCase     :       List[str]  						=  loaded_unet(lowerCamelCase       ,			lowerCamelCase       ,			lowerCamelCase  ).sample
  
          assert torch.allclose(lowerCamelCase       ,			lowerCamelCase       ,			atol=1e-4  )
  
  
  
  