algorithm behind the generation of the reverse bits lookup table(8 bit)

Question

I found the lookup table here. The table is generated as a reverse bits table of 8 bits.

I can not figure out why it works. Please explain the theory behind it. Thanks<

Answer 1

If you are working with Python and happen to end up here, this is how the lookup table would look like. Nevertheless, Bernd Elkemann's explanation still stands.

# Generating the REVERSE_BIT_LUT while pre-processing
# LUT is shorthand for lookuptable
def R2(n, REVERSE_BIT_LUT):
    REVERSE_BIT_LUT.extend([n, n + 2 * 64, n + 1 * 64, n + 3 * 64])


def R4(n, REVERSE_BIT_LUT):
    return (
        R2(n, REVERSE_BIT_LUT),
        R2(n + 2 * 16, REVERSE_BIT_LUT),
        R2(n + 1 * 16, REVERSE_BIT_LUT),
        R2(n + 3 * 16, REVERSE_BIT_LUT),
    )


def R6(n, REVERSE_BIT_LUT):
    return (
        R4(n, REVERSE_BIT_LUT),
        R4(n + 2 * 4, REVERSE_BIT_LUT),
        R4(n + 1 * 4, REVERSE_BIT_LUT),
        R4(n + 3 * 4, REVERSE_BIT_LUT),
    )


def LOOK_UP(REVERSE_BIT_LUT):
    return (
        R6(0, REVERSE_BIT_LUT),
        R6(2, REVERSE_BIT_LUT),
        R6(1, REVERSE_BIT_LUT),
        R6(3, REVERSE_BIT_LUT),
    )


# LOOK_UP is the function to generate the REVERSE_BIT_LUT
REVERSE_BIT_LUT = list()
LOOK_UP(REVERSE_BIT_LUT)

I am working on providing code snippets of Sean's bit hacks in Python here.