



"""simple docstring"""

from __future__ import annotations

from math import ceil, floor, sqrt




def __lowercase			(							_a = 2_000_000    ):
    snake_case_ :     list[int]				=	[0]
    snake_case_ :     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
    snake_case_ :     int				=	0
    # the area corresponding to the grid that gives the product closest to target
    snake_case_ :     int				=	0
    # an estimate of b, using the quadratic formula
    snake_case_ :     float
    # the largest integer less than b_estimate
    snake_case_ :     int
    # the largest integer less than b_estimate
    snake_case_ :     int
    # the triangle number corresponding to b_floor
    snake_case_ :     int
    # the triangle number corresponding to b_ceil
    snake_case_ :     int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]		,			1    ):
        snake_case_ :     Union[str, Any]				=	(-1 + sqrt(1 + 8 * target / triangle_a    )) / 2
        snake_case_ :     Tuple				=	floor(_a    )
        snake_case_ :     List[str]				=	ceil(_a    )
        snake_case_ :     Union[str, Any]				=	triangle_numbers[b_floor]
        snake_case_ :     Any				=	triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a    ) < abs(
            target - best_product    ):
            snake_case_ :     List[Any]				=	triangle_b_first_guess * triangle_a
            snake_case_ :     Optional[int]				=	idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a    ) < abs(
            target - best_product    ):
            snake_case_ :     Tuple				=	triangle_b_second_guess * triangle_a
            snake_case_ :     Dict				=	idx_a * b_ceil

    return area


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










"""simple docstring"""

from . import __version__

# Backward compatibility imports, to make sure all those objects can be found in file_utils
from .utils import (
    CLOUDFRONT_DISTRIB_PREFIX,
    CONFIG_NAME,
    DISABLE_TELEMETRY,
    DUMMY_INPUTS,
    DUMMY_MASK,
    ENV_VARS_TRUE_AND_AUTO_VALUES,
    ENV_VARS_TRUE_VALUES,
    FEATURE_EXTRACTOR_NAME,
    FLAX_WEIGHTS_NAME,
    HF_MODULES_CACHE,
    HUGGINGFACE_CO_PREFIX,
    HUGGINGFACE_CO_RESOLVE_ENDPOINT,
    MODEL_CARD_NAME,
    MULTIPLE_CHOICE_DUMMY_INPUTS,
    PYTORCH_PRETRAINED_BERT_CACHE,
    PYTORCH_TRANSFORMERS_CACHE,
    S3_BUCKET_PREFIX,
    SENTENCEPIECE_UNDERLINE,
    SPIECE_UNDERLINE,
    TF2_WEIGHTS_NAME,
    TF_WEIGHTS_NAME,
    TORCH_FX_REQUIRED_VERSION,
    TRANSFORMERS_CACHE,
    TRANSFORMERS_DYNAMIC_MODULE_NAME,
    USE_JAX,
    USE_TF,
    USE_TORCH,
    WEIGHTS_INDEX_NAME,
    WEIGHTS_NAME,
    ContextManagers,
    DummyObject,
    EntryNotFoundError,
    ExplicitEnum,
    ModelOutput,
    PaddingStrategy,
    PushToHubMixin,
    RepositoryNotFoundError,
    RevisionNotFoundError,
    TensorType,
    _LazyModule,
    add_code_sample_docstrings,
    add_end_docstrings,
    add_start_docstrings,
    add_start_docstrings_to_model_forward,
    cached_property,
    copy_func,
    default_cache_path,
    define_sagemaker_information,
    get_cached_models,
    get_file_from_repo,
    get_full_repo_name,
    get_torch_version,
    has_file,
    http_user_agent,
    is_apex_available,
    is_bsa_available,
    is_coloredlogs_available,
    is_datasets_available,
    is_detectrona_available,
    is_faiss_available,
    is_flax_available,
    is_ftfy_available,
    is_in_notebook,
    is_ipex_available,
    is_librosa_available,
    is_offline_mode,
    is_onnx_available,
    is_pandas_available,
    is_phonemizer_available,
    is_protobuf_available,
    is_psutil_available,
    is_pyanvml_available,
    is_pyctcdecode_available,
    is_pytesseract_available,
    is_pytorch_quantization_available,
    is_rjieba_available,
    is_sagemaker_dp_enabled,
    is_sagemaker_mp_enabled,
    is_scipy_available,
    is_sentencepiece_available,
    is_seqio_available,
    is_sklearn_available,
    is_soundfile_availble,
    is_spacy_available,
    is_speech_available,
    is_tensor,
    is_tensorflow_probability_available,
    is_tfaonnx_available,
    is_tf_available,
    is_timm_available,
    is_tokenizers_available,
    is_torch_available,
    is_torch_bfaa_available,
    is_torch_cuda_available,
    is_torch_fx_available,
    is_torch_fx_proxy,
    is_torch_mps_available,
    is_torch_tfaa_available,
    is_torch_tpu_available,
    is_torchaudio_available,
    is_training_run_on_sagemaker,
    is_vision_available,
    replace_return_docstrings,
    requires_backends,
    to_numpy,
    to_py_obj,
    torch_only_method,
)





