
"""simple docstring"""





from __future__ import annotations

from math import ceil, floor, sqrt



def 				lowerCamelCase__		(      __snake_case = 2_00_00_00				)							->		int:


					"""simple docstring"""





					_UpperCamelCase										=  [0]
					_UpperCamelCase										=  42

					for idx in range(1,					ceil(sqrt(target * 2				) * 1.1				)				):
										triangle_numbers.append(triangle_numbers[-1] + idx				)

					# we want this to be as close as possible to target
					_UpperCamelCase										=  0
					# the area corresponding to the grid that gives the product closest to target
					_UpperCamelCase										=  0
					# an estimate of b, using the quadratic formula
					_UpperCamelCase										=  42
					# the largest integer less than b_estimate
					_UpperCamelCase										=  42
					# the largest integer less than b_estimate
					_UpperCamelCase										=  42
					# the triangle number corresponding to b_floor
					_UpperCamelCase										=  42
					# the triangle number corresponding to b_ceil
					_UpperCamelCase										=  42

					for idx_a, triangle_a in enumerate(triangle_numbers[1:],					1				):
										_UpperCamelCase										=  (-1 + sqrt(1 + 8 * target / triangle_a				)) / 2
										_UpperCamelCase										=  floor(__snake_case				)
										_UpperCamelCase										=  ceil(__snake_case				)
										_UpperCamelCase										=  triangle_numbers[b_floor]
										_UpperCamelCase										=  triangle_numbers[b_ceil]

										if abs(target - triangle_b_first_guess * triangle_a				) < abs(
										    target - best_product				):
															_UpperCamelCase										=  triangle_b_first_guess * triangle_a
															_UpperCamelCase										=  idx_a * b_floor

										if abs(target - triangle_b_second_guess * triangle_a				) < abs(
										    target - best_product				):
															_UpperCamelCase										=  triangle_b_second_guess * triangle_a
															_UpperCamelCase										=  idx_a * b_ceil

					return area


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








"""simple docstring"""





def 				lowerCamelCase__		(      __snake_case,					__snake_case     )							->		str:


					"""simple docstring"""





					if number < 0 or shift_amount < 0:
										raise ValueError('''both inputs must be positive integers'''     )

					_UpperCamelCase										=  str(bin(__snake_case     )     )
					binary_number += "0" * shift_amount
					return binary_number



def 				lowerCamelCase__		(      __snake_case,					__snake_case     )							->		str:


					"""simple docstring"""





					if number < 0 or shift_amount < 0:
										raise ValueError('''both inputs must be positive integers'''     )

					_UpperCamelCase										=  str(bin(__snake_case     )     )[2:]
					if shift_amount >= len(__snake_case     ):
										return "0b0"
					_UpperCamelCase										=  binary_number[: len(__snake_case     ) - shift_amount]
					return "0b" + shifted_binary_number



def 				lowerCamelCase__		(      __snake_case,					__snake_case     )							->		str:


					"""simple docstring"""





					if number >= 0:  # Get binary representation of positive number
										_UpperCamelCase										=  '''0''' + str(bin(__snake_case     )     ).strip('''-'''     )[2:]
					else:  # Get binary (2's complement) representation of negative number
										_UpperCamelCase										=  len(bin(__snake_case     )[3:]     )  # Find 2's complement of number
										_UpperCamelCase										=  bin(abs(__snake_case     ) - (1 << binary_number_length)     )[3:]
										_UpperCamelCase										=  (
										    '''1''' + '''0''' * (binary_number_length - len(__snake_case     )) + binary_number
										)

					if shift_amount >= len(__snake_case     ):
										return "0b" + binary_number[0] * len(__snake_case     )
					return (
					    "0b"
					    + binary_number[0] * shift_amount
					    + binary_number[: len(__snake_case     ) - shift_amount]
					)


if __name__ == "__main__":
				import doctest

				doctest.testmod()






