import math


def floor_to_odd_integer(x: float) -> int:
    """
    Round down to the nearest odd number.

    >>> floor_to_odd_integer(3)
    3
    >>> floor_to_odd_integer(4)
    3

    :param x: Value to be rounded.
    :return: Nearest lower odd number no larger than the provided value.
    """
    x_floor = math.floor(x)
    return x_floor if x_floor % 2 == 1 else x_floor - 1