According to the daylight intensity table in this wikipedia page, the illuminace of a few conditions are:
Brightest sunlight - 120,000 lux
Sunrise or sunset on a clear day (ambient illumination) - 400 lux
Moonlight - < 1 lux
However < 1 lux doesn't sound right during the day, otherwise how would the moon be visible during the day since even at sunset the sky illumiance is orders of magnitude stronger than moonlight.
How should I calculate the moon illuminance during the day?
The value of 1 lux for moonlight is optimistic. This Wikipedia article gives a considerably lower value; the main reference (Kyba, et al. 2017) is pretty solid, but I think a slightly greater value is theoretically possible. With a perfect confluence of factors on the equator at high elevation (say, Quito, Ecuador, elevation 9350 ft), the greatest possible illuminance is about 0.45 lux. This assumes both the Sun and Moon are at minimum distances, the Moon is at zenith, and the Moon barely misses an eclipse. Suffice it to say that this is mighty unusual; typical values for moonlight at mid-latitudes are about 0.25–0.3 lux in winter when the Moon reaches its greatest altitude—close to what Kyba found. Values in summer may be half that.
So the contribution of moonlight to daylight is negligible.
Why can you see the Moon during the day? This is determined luminance (“brightness”) rather than illuminance. Values of Moon and sky illuminance vary with positions of the Sun and Moon; with the Sun at 60° altitude and a full moon at 60°, and the Moon’s azimuth 180° from the Sun’s, the sky luminance could be around 3,000 cd/m^2 and the Moon’s luminance might be around 3800 cd/m^2. For a quarter moon at 90° from the Sun and the same Sun and Moon altitudes, the values might be 4600 cd/m^2 and 674 cd/m^2; that’s why a quarter moon is pretty faint. Why can you still see it? Color contrast would be my guess, but I don’t really know.
I’ve calculated sky luminance using the CIE model developed by Darula and Kittler; Moon luminance and illuminance were calculated using an adaptation of the method in US Naval Observatory Circular No. 171 by Janiczek and DeYoung. The resulting numbers are rough estimates that assume perfect weather; I generally trust calculated values of illuminance and luminance as far as I can throw them.
