






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






def  UpperCAmelCase__						(    lowerCamelCase	:			float    ):
   return (
       (1 - 2 * point) * sqrt(point - point**2    ) + 2 * point + asin(sqrt(1 - point    )    )
   ) / 4






def  UpperCAmelCase__						(    lowerCamelCase	:			int    ):
   lowerCamelCase									=(circles_number + 1 - sqrt(2 * circles_number    )) / (
       2 * (circles_number**2 + 1)
   )
   lowerCamelCase									=circles_number * intersection_y

   lowerCamelCase									=intersection_x * intersection_y / 2
   lowerCamelCase									=circle_bottom_arc_integral(
       1 / 2    ) - circle_bottom_arc_integral(lowerCamelCase    )

   return triangle_area + concave_region_area






def  UpperCAmelCase__						(    lowerCamelCase	:			float = 1 / 10_00    ):
   lowerCamelCase									=(1 - pi / 4) / 4

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

   return -1


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







import unittest

import numpy as np

from transformers.testing_utils import require_torch, require_vision
from transformers.utils import is_torch_available, is_vision_available

from ...test_image_processing_common import ImageProcessingSavingTestMixin


if is_torch_available():
   import torch

if is_vision_available():
   from PIL import Image

   from transformers import SwinaSRImageProcessor
   from transformers.image_transforms import get_image_size


class 		lowerCAmelCase_				(  unittest.TestCase ):





     def __init__(	self							, _UpperCAmelCase							, _UpperCAmelCase=7							, _UpperCAmelCase=3							, _UpperCAmelCase=18							, _UpperCAmelCase=30							, _UpperCAmelCase=400							, _UpperCAmelCase=True							, _UpperCAmelCase=1 / 255							, _UpperCAmelCase=True							, _UpperCAmelCase=8							, )		->				Tuple:
        lowerCamelCase									=parent
        lowerCamelCase									=batch_size
        lowerCamelCase									=num_channels
        lowerCamelCase									=image_size
        lowerCamelCase									=min_resolution
        lowerCamelCase									=max_resolution
        lowerCamelCase									=do_rescale
        lowerCamelCase									=rescale_factor
        lowerCamelCase									=do_pad
        lowerCamelCase									=pad_size





     def 						lowerCamelCase							(	self  )		->				Optional[Any]:
        return {
            "do_rescale": self.do_rescale,
            "rescale_factor": self.rescale_factor,
            "do_pad": self.do_pad,
            "pad_size": self.pad_size,
        }





     def 						lowerCamelCase							(	self							, _UpperCAmelCase=False							, _UpperCAmelCase=False							, _UpperCAmelCase=False  )		->				int:
        assert not (numpify and torchify), "You cannot specify both numpy and PyTorch tensors at the same time"

        if equal_resolution:
           lowerCamelCase									=[]
           for i in range(self.batch_size  ):
              image_inputs.append(
                  np.random.randint(
                      255							, size=(self.num_channels, self.max_resolution, self.max_resolution)							, dtype=np.uinta  )  )
        else:
           lowerCamelCase									=[]
           for i in range(self.batch_size  ):
              lowerCamelCase				,       lowerCamelCase									=np.random.choice(np.arange(self.min_resolution							, self.max_resolution  )							, 2  )
              image_inputs.append(np.random.randint(255							, size=(self.num_channels, width, height)							, dtype=np.uinta  )  )

        if not numpify and not torchify:
           # PIL expects the channel dimension as last dimension
           lowerCamelCase									=[Image.fromarray(np.moveaxis(_UpperCAmelCase							, 0							, -1  )  ) for x in image_inputs]

        if torchify:
           lowerCamelCase									=[torch.from_numpy(_UpperCAmelCase  ) for x in image_inputs]

        return image_inputs




@require_torch
@require_vision
class 		lowerCAmelCase_				(  UpperCamelCase				,     unittest.TestCase ):
     __lowerCamelCase       :    Optional[Any] 	=     SwinaSRImageProcessor if is_vision_available() else None





     def 						lowerCamelCase							(	self  )		->				Dict:
        lowerCamelCase									=SwinaSRImageProcessingTester(self  )





     @property
     def 						lowerCamelCase							(	self  )		->				Any:
        return self.image_processor_tester.prepare_image_processor_dict()





     def 						lowerCamelCase							(	self  )		->				Any:
        lowerCamelCase									=self.image_processing_class(**self.image_processor_dict  )
        self.assertTrue(hasattr(_UpperCAmelCase							, """do_rescale"""  )  )
        self.assertTrue(hasattr(_UpperCAmelCase							, """rescale_factor"""  )  )
        self.assertTrue(hasattr(_UpperCAmelCase							, """do_pad"""  )  )
        self.assertTrue(hasattr(_UpperCAmelCase							, """pad_size"""  )  )





     def 						lowerCamelCase							(	self  )		->				Any:
        pass





     def 						lowerCamelCase							(	self							, _UpperCAmelCase  )		->				int:
        lowerCamelCase				,       lowerCamelCase									=get_image_size(_UpperCAmelCase  )
        lowerCamelCase									=self.image_processor_tester.pad_size

        lowerCamelCase									=(old_height // size + 1) * size - old_height
        lowerCamelCase									=(old_width // size + 1) * size - old_width
        return old_height + pad_height, old_width + pad_width





     def 						lowerCamelCase							(	self  )		->				Optional[Any]:
        # Initialize image_processor
        lowerCamelCase									=self.image_processing_class(**self.image_processor_dict  )
        # create random PIL images
        lowerCamelCase									=self.image_processor_tester.prepare_inputs(equal_resolution=_UpperCAmelCase  )
        for image in image_inputs:
           self.assertIsInstance(_UpperCAmelCase							, Image.Image  )

        # Test not batched input
        lowerCamelCase									=image_processor(image_inputs[0]							, return_tensors="""pt"""  ).pixel_values
        lowerCamelCase				,       lowerCamelCase									=self.calculate_expected_size(np.array(image_inputs[0]  )  )
        self.assertEqual(
            encoded_images.shape							, (
                1,
                self.image_processor_tester.num_channels,
                expected_height,
                expected_width,
            )							, )





     def 						lowerCamelCase							(	self  )		->				Any:
        # Initialize image_processor
        lowerCamelCase									=self.image_processing_class(**self.image_processor_dict  )
        # create random numpy tensors
        lowerCamelCase									=self.image_processor_tester.prepare_inputs(equal_resolution=_UpperCAmelCase							, numpify=_UpperCAmelCase  )
        for image in image_inputs:
           self.assertIsInstance(_UpperCAmelCase							, np.ndarray  )

        # Test not batched input
        lowerCamelCase									=image_processor(image_inputs[0]							, return_tensors="""pt"""  ).pixel_values
        lowerCamelCase				,       lowerCamelCase									=self.calculate_expected_size(image_inputs[0]  )
        self.assertEqual(
            encoded_images.shape							, (
                1,
                self.image_processor_tester.num_channels,
                expected_height,
                expected_width,
            )							, )





     def 						lowerCamelCase							(	self  )		->				Dict:
        # Initialize image_processor
        lowerCamelCase									=self.image_processing_class(**self.image_processor_dict  )
        # create random PyTorch tensors
        lowerCamelCase									=self.image_processor_tester.prepare_inputs(equal_resolution=_UpperCAmelCase							, torchify=_UpperCAmelCase  )
        for image in image_inputs:
           self.assertIsInstance(_UpperCAmelCase							, torch.Tensor  )

        # Test not batched input
        lowerCamelCase									=image_processor(image_inputs[0]							, return_tensors="""pt"""  ).pixel_values
        lowerCamelCase				,       lowerCamelCase									=self.calculate_expected_size(image_inputs[0]  )
        self.assertEqual(
            encoded_images.shape							, (
                1,
                self.image_processor_tester.num_channels,
                expected_height,
                expected_width,
            )							, )