That thing calculates the luminous flux difference. A 800 lumen lamp is technically twice as bright as a 400lm lamp, but it will not appear that way to us. The eye's response is non-linear, while luminous flux is linear. For a spot emitter, it's roughly the root of the difference.
1/√13 is 1/3.6, which is between ⅓ and ¼ as you have noticed.