

'''simple docstring'''

from __future__ import annotations

from math import ceil, floor, sqrt

def 					__UpperCamelCase				(						lowercase__    :	int = 2_00_00_00			):






     '''simple docstring'''





     __lowercase      		=[0]
     __lowercase      		=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
     __lowercase      		=0
     # the area corresponding to the grid that gives the product closest to target
     __lowercase      		=0
     # an estimate of b, using the quadratic formula
     __lowercase      		=42
     # the largest integer less than b_estimate
     __lowercase      		=42
     # the largest integer less than b_estimate
     __lowercase      		=42
     # the triangle number corresponding to b_floor
     __lowercase      		=42
     # the triangle number corresponding to b_ceil
     __lowercase      		=42

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

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

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

     return area


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




"""simple docstring"""




def     __magic_name__		(					UpperCamelCase     :		float							,		UpperCamelCase     :		float							,		UpperCamelCase     :		float							,		UpperCamelCase     :		float							,		UpperCamelCase     :		float							,		)     ->						float:
    a__		=			[redshift, radiation_density, matter_density, dark_energy]
    if any(p < 0 for p in parameters     ):
        raise ValueError('All input parameters must be positive'     )

    if any(p > 1 for p in parameters[1:4]     ):
        raise ValueError('Relative densities cannot be greater than one'     )
    else:
        a__		=			1 - (matter_density + radiation_density + dark_energy)

        a__		=			(
            radiation_density * (redshift + 1) ** 4
            + matter_density * (redshift + 1) ** 3
            + curvature * (redshift + 1) ** 2
            + dark_energy
        )

        a__		=			hubble_constant * e_a ** (1 / 2)
        return hubble


if __name__ == "__main__":
    import doctest

    # run doctest
    doctest.testmod()

    # demo LCDM approximation
    a		:       Any     						=       0.3

    print(
        hubble_parameter(
            hubble_constant=68.3,
            radiation_density=1e-4,
            matter_density=matter_density,
            dark_energy=1 - matter_density,
            redshift=0,
        )
    )


