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# Built-in Complex Truth Table Search #
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import multiplied as mp
import pandas as pd
import pyarrow as pa

def df_early_bitwidth_cout(df: pd.DataFrame) -> pd.DataFrame:
    """Return rows which carry past operand width during partial product generation"""
    ...

def df_late_bitwidth_cout(df: pd.DataFrame) -> pd.DataFrame:
    """Return rows which carry past operand width during reduction"""
    ...
