


'''simple docstring'''







from itertools import count
from math import asin, pi, sqrt







def _UpperCamelCase	(					__A   )							->		float:


      '''simple docstring'''



      return (
          (1 - 2 * point) * sqrt(point - point**2   ) + 2 * point + asin(sqrt(1 - point   )   )
      ) / 4







def _UpperCamelCase	(					__A   )							->		float:


      '''simple docstring'''



      UpperCamelCase__  =    (circles_number + 1 - sqrt(2 * circles_number   )) / (
          2 * (circles_number**2 + 1)
      )
      UpperCamelCase__  =    circles_number * intersection_y

      UpperCamelCase__  =    intersection_x * intersection_y / 2
      UpperCamelCase__  =    circle_bottom_arc_integral(
          1 / 2   ) - circle_bottom_arc_integral(__A   )

      return triangle_area + concave_region_area







def _UpperCamelCase	(					__A = 1 / 1000   )							->		int:


      '''simple docstring'''



      UpperCamelCase__  =    (1 - pi / 4) / 4

      for n in count(1   ):
            if concave_triangle_area(__A   ) / l_section_area < fraction:
                  return n

      return -1


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









'''simple docstring'''







import tempfile

import torch

from diffusers import (
    DEISMultistepScheduler,
    DPMSolverMultistepScheduler,
    DPMSolverSinglestepScheduler,
    UniPCMultistepScheduler,
)

from .test_schedulers import SchedulerCommonTest
class  lowercase_					(   a__       ):
    __UpperCAmelCase             =     (DPMSolverMultistepScheduler,)
    __UpperCAmelCase             =     (('num_inference_steps', 25),)


    def        __a							(					self  , **a   ):
          UpperCamelCase__  =    {
              "num_train_timesteps": 10_00,
              "beta_start": 0.0001,
              "beta_end": 0.02,
              "beta_schedule": "linear",
              "solver_order": 2,
              "prediction_type": "epsilon",
              "thresholding": False,
              "sample_max_value": 1.0,
              "algorithm_type": "dpmsolver++",
              "solver_type": "midpoint",
              "lower_order_final": False,
              "lambda_min_clipped": -float("inf"   ),
              "variance_type": None,
          }

          config.update(**a   )
          return config


    def        __a							(					self  , a=0  , **a   ):
          UpperCamelCase__  =    dict(self.forward_default_kwargs   )
          UpperCamelCase__  =    kwargs.pop("num_inference_steps"  , a   )
          UpperCamelCase__  =    self.dummy_sample
          UpperCamelCase__  =    0.1 * sample
          UpperCamelCase__  =    [residual + 0.2, residual + 0.15, residual + 0.10]

          for scheduler_class in self.scheduler_classes:
                UpperCamelCase__  =    self.get_scheduler_config(**a   )
                UpperCamelCase__  =    scheduler_class(**a   )
                scheduler.set_timesteps(a   )
                # copy over dummy past residuals
                UpperCamelCase__  =    dummy_past_residuals[: scheduler.config.solver_order]

                with tempfile.TemporaryDirectory() as tmpdirname:
                      scheduler.save_config(a   )
                      UpperCamelCase__  =    scheduler_class.from_pretrained(a   )
                      new_scheduler.set_timesteps(a   )
                      # copy over dummy past residuals
                      UpperCamelCase__  =    dummy_past_residuals[: new_scheduler.config.solver_order]

                UpperCamelCase__ ,     UpperCamelCase__  =    sample, sample
                for t in range(a  , time_step + scheduler.config.solver_order + 1   ):
                      UpperCamelCase__  =    scheduler.step(a  , a  , a  , **a   ).prev_sample
                      UpperCamelCase__  =    new_scheduler.step(a  , a  , a  , **a   ).prev_sample

                      assert torch.sum(torch.abs(output - new_output   )   ) < 1e-5, "Scheduler outputs are not identical"


    def        __a							(					self   ):
          pass


    def        __a							(					self  , a=0  , **a   ):
          UpperCamelCase__  =    dict(self.forward_default_kwargs   )
          UpperCamelCase__  =    kwargs.pop("num_inference_steps"  , a   )
          UpperCamelCase__  =    self.dummy_sample
          UpperCamelCase__  =    0.1 * sample
          UpperCamelCase__  =    [residual + 0.2, residual + 0.15, residual + 0.10]

          for scheduler_class in self.scheduler_classes:
                UpperCamelCase__  =    self.get_scheduler_config()
                UpperCamelCase__  =    scheduler_class(**a   )
                scheduler.set_timesteps(a   )

                # copy over dummy past residuals (must be after setting timesteps)
                UpperCamelCase__  =    dummy_past_residuals[: scheduler.config.solver_order]

                with tempfile.TemporaryDirectory() as tmpdirname:
                      scheduler.save_config(a   )
                      UpperCamelCase__  =    scheduler_class.from_pretrained(a   )
                      # copy over dummy past residuals
                      new_scheduler.set_timesteps(a   )

                      # copy over dummy past residual (must be after setting timesteps)
                      UpperCamelCase__  =    dummy_past_residuals[: new_scheduler.config.solver_order]

                UpperCamelCase__  =    scheduler.step(a  , a  , a  , **a   ).prev_sample
                UpperCamelCase__  =    new_scheduler.step(a  , a  , a  , **a   ).prev_sample

                assert torch.sum(torch.abs(output - new_output   )   ) < 1e-5, "Scheduler outputs are not identical"


    def        __a							(					self  , a=None  , **a   ):
          if scheduler is None:
                UpperCamelCase__  =    self.scheduler_classes[0]
                UpperCamelCase__  =    self.get_scheduler_config(**a   )
                UpperCamelCase__  =    scheduler_class(**a   )

          UpperCamelCase__  =    10
          UpperCamelCase__  =    self.dummy_model()
          UpperCamelCase__  =    self.dummy_sample_deter
          scheduler.set_timesteps(a   )

          for i, t in enumerate(scheduler.timesteps   ):
                UpperCamelCase__  =    model(a  , a   )
                UpperCamelCase__  =    scheduler.step(a  , a  , a   ).prev_sample

          return sample


    def        __a							(					self   ):
          UpperCamelCase__  =    dict(self.forward_default_kwargs   )

          UpperCamelCase__  =    kwargs.pop("num_inference_steps"  , a   )

          for scheduler_class in self.scheduler_classes:
                UpperCamelCase__  =    self.get_scheduler_config()
                UpperCamelCase__  =    scheduler_class(**a   )

                UpperCamelCase__  =    self.dummy_sample
                UpperCamelCase__  =    0.1 * sample

                if num_inference_steps is not None and hasattr(a  , "set_timesteps"   ):
                      scheduler.set_timesteps(a   )
                elif num_inference_steps is not None and not hasattr(a  , "set_timesteps"   ):
                      UpperCamelCase__  =    num_inference_steps

                # copy over dummy past residuals (must be done after set_timesteps)
                UpperCamelCase__  =    [residual + 0.2, residual + 0.15, residual + 0.10]
                UpperCamelCase__  =    dummy_past_residuals[: scheduler.config.solver_order]

                UpperCamelCase__  =    scheduler.timesteps[5]
                UpperCamelCase__  =    scheduler.timesteps[6]

                UpperCamelCase__  =    scheduler.step(a  , a  , a  , **a   ).prev_sample
                UpperCamelCase__  =    scheduler.step(a  , a  , a  , **a   ).prev_sample

                self.assertEqual(output_a.shape  , sample.shape   )
                self.assertEqual(output_a.shape  , output_a.shape   )


    def        __a							(					self   ):
          for timesteps in [25, 50, 1_00, 9_99, 10_00]:
                self.check_over_configs(num_train_timesteps=a   )


    def        __a							(					self   ):
          self.check_over_configs(thresholding=a   )
          for order in [1, 2, 3]:
                for solver_type in ["midpoint", "heun"]:
                      for threshold in [0.5, 1.0, 2.0]:
                            for prediction_type in ["epsilon", "sample"]:
                                  self.check_over_configs(
                                      thresholding=a  , prediction_type=a  , sample_max_value=a  , algorithm_type="dpmsolver++"  , solver_order=a  , solver_type=a  , )


    def        __a							(					self   ):
          for prediction_type in ["epsilon", "v_prediction"]:
                self.check_over_configs(prediction_type=a   )


    def        __a							(					self   ):
          for algorithm_type in ["dpmsolver", "dpmsolver++", "sde-dpmsolver", "sde-dpmsolver++"]:
                for solver_type in ["midpoint", "heun"]:
                      for order in [1, 2, 3]:
                            for prediction_type in ["epsilon", "sample"]:
                                  if algorithm_type in ["sde-dpmsolver", "sde-dpmsolver++"]:
                                        if order == 3:
                                              continue
                                  else:
                                        self.check_over_configs(
                                            solver_order=a  , solver_type=a  , prediction_type=a  , algorithm_type=a  , )
                                  UpperCamelCase__  =    self.full_loop(
                                      solver_order=a  , solver_type=a  , prediction_type=a  , algorithm_type=a  , )
                                  assert not torch.isnan(a   ).any(), "Samples have nan numbers"


    def        __a							(					self   ):
          self.check_over_configs(lower_order_final=a   )
          self.check_over_configs(lower_order_final=a   )


    def        __a							(					self   ):
          self.check_over_configs(lambda_min_clipped=-float("inf"   )   )
          self.check_over_configs(lambda_min_clipped=-5.1   )


    def        __a							(					self   ):
          self.check_over_configs(variance_type=a   )
          self.check_over_configs(variance_type="learned_range"   )


    def        __a							(					self   ):
          for num_inference_steps in [1, 2, 3, 5, 10, 50, 1_00, 9_99, 10_00]:
                self.check_over_forward(num_inference_steps=a  , time_step=0   )


    def        __a							(					self   ):
          UpperCamelCase__  =    self.full_loop()
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 0.3301   ) < 1e-3


    def        __a							(					self   ):
          UpperCamelCase__  =    self.full_loop(thresholding=a  , dynamic_thresholding_ratio=0.87  , sample_max_value=0.5   )
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 1.1364   ) < 1e-3


    def        __a							(					self   ):
          UpperCamelCase__  =    self.full_loop(prediction_type="v_prediction"   )
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 0.2251   ) < 1e-3


    def        __a							(					self   ):
          UpperCamelCase__  =    self.full_loop(prediction_type="v_prediction"  , use_karras_sigmas=a   )
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 0.2096   ) < 1e-3


    def        __a							(					self   ):
          # make sure that iterating over schedulers with same config names gives same results
          # for defaults
          UpperCamelCase__  =    DPMSolverMultistepScheduler(**self.get_scheduler_config()   )
          UpperCamelCase__  =    self.full_loop(scheduler=a   )
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 0.3301   ) < 1e-3

          UpperCamelCase__  =    DPMSolverSinglestepScheduler.from_config(scheduler.config   )
          UpperCamelCase__  =    UniPCMultistepScheduler.from_config(scheduler.config   )
          UpperCamelCase__  =    DEISMultistepScheduler.from_config(scheduler.config   )
          UpperCamelCase__  =    DPMSolverMultistepScheduler.from_config(scheduler.config   )

          UpperCamelCase__  =    self.full_loop(scheduler=a   )
          UpperCamelCase__  =    torch.mean(torch.abs(a   )   )

          assert abs(result_mean.item() - 0.3301   ) < 1e-3


    def        __a							(					self   ):
          UpperCamelCase__  =    self.scheduler_classes[0]
          UpperCamelCase__  =    self.get_scheduler_config(thresholding=a  , dynamic_thresholding_ratio=0   )
          UpperCamelCase__  =    scheduler_class(**a   )

          UpperCamelCase__  =    10
          UpperCamelCase__  =    self.dummy_model()
          UpperCamelCase__  =    self.dummy_sample_deter.half()
          scheduler.set_timesteps(a   )

          for i, t in enumerate(scheduler.timesteps   ):
                UpperCamelCase__  =    model(a  , a   )
                UpperCamelCase__  =    scheduler.step(a  , a  , a   ).prev_sample

          assert sample.dtype == torch.floataa


    def        __a							(					self  , **a   ):
          for scheduler_class in self.scheduler_classes:
                UpperCamelCase__  =    self.get_scheduler_config(**a   )
                UpperCamelCase__  =    scheduler_class(**a   )

                scheduler.set_timesteps(scheduler.config.num_train_timesteps   )
                assert len(scheduler.timesteps.unique()   ) == scheduler.num_inference_steps





