



from __future__ import annotations

import math

A__	      =       """2020.9.26"""
A__	      =       """xcodz-dot, cclaus, dhruvmanila"""


def 		_UpperCAmelCase				(      snake_case	,						snake_case	,						snake_case	,						snake_case	,						snake_case			):





					"""simple docstring"""

					if not all(isinstance(snake_case	,						(float, int)			) for val in locals().values()			):
										_lowerCAmelCase						=    F'Input values must either be float or int: {list(locals().values()			)}'
										raise TypeError(snake_case			)
					_lowerCAmelCase						=    ((x * distance) / (z + distance)) * scale
					_lowerCAmelCase						=    ((y * distance) / (z + distance)) * scale
					return projected_x, projected_y


def 		_UpperCAmelCase				(      snake_case	,						snake_case	,						snake_case	,						snake_case	,						snake_case			):





					"""simple docstring"""

					if not isinstance(snake_case	,						snake_case			):
										raise TypeError("""Axis must be a str"""			)
					_lowerCAmelCase						=    locals()
					del input_variables["axis"]
					if not all(isinstance(snake_case	,						(float, int)			) for val in input_variables.values()			):
										_lowerCAmelCase						=    (
										    """Input values except axis must either be float or int: """
										    F'{list(input_variables.values()			)}'
										)
										raise TypeError(snake_case			)
					_lowerCAmelCase						=    (angle % 3_60) / 4_50 * 1_80 / math.pi
					if axis == "z":
										_lowerCAmelCase						=    x * math.cos(snake_case			) - y * math.sin(snake_case			)
										_lowerCAmelCase						=    y * math.cos(snake_case			) + x * math.sin(snake_case			)
										_lowerCAmelCase						=    z
					elif axis == "x":
										_lowerCAmelCase						=    y * math.cos(snake_case			) - z * math.sin(snake_case			)
										_lowerCAmelCase						=    z * math.cos(snake_case			) + y * math.sin(snake_case			)
										_lowerCAmelCase						=    x
					elif axis == "y":
										_lowerCAmelCase						=    x * math.cos(snake_case			) - z * math.sin(snake_case			)
										_lowerCAmelCase						=    z * math.cos(snake_case			) + x * math.sin(snake_case			)
										_lowerCAmelCase						=    y
					else:
										raise ValueError("""not a valid axis, choose one of 'x', 'y', 'z'"""			)

					return new_x, new_y, new_z


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	print(f"{convert_to_ad(1.0, 2.0, 3.0, 1_0.0, 1_0.0) = }")
	print(f"{rotate(1.0, 2.0, 3.0, 'y', 9_0.0) = }")







from __future__ import annotations

from math import ceil, floor, sqrt



def 				UpperCamelCase_(      lowerCamelCase_ = 200_0000 )    ->						int:
    _lowercase       :							list[int]      		=		[0]
    _lowercase       :							int

    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       :							int      		=		0
    # the area corresponding to the grid that gives the product closest to target
    _lowercase       :							int      		=		0
    # an estimate of b, using the quadratic formula
    _lowercase       :							float
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the triangle number corresponding to b_floor
    _lowercase       :							int
    # the triangle number corresponding to b_ceil
    _lowercase       :							int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							, 1 ):
        _lowercase       :							Optional[int]      		=		(-1 + sqrt(1 + 8 * target / triangle_a )) / 2
        _lowercase       :							List[str]      		=		floor(lowerCamelCase_ )
        _lowercase       :							Dict      		=		ceil(lowerCamelCase_ )
        _lowercase       :							List[str]      		=		triangle_numbers[b_floor]
        _lowercase       :							List[str]      		=		triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Union[str, Any]      		=		triangle_b_first_guess * triangle_a
            _lowercase       :							Union[str, Any]      		=		idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Any      		=		triangle_b_second_guess * triangle_a
            _lowercase       :							Optional[Any]      		=		idx_a * b_ceil

    return area


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


