I'd say there are basically three levels of approximation. The charted value is 2.4 nm.
1) HAT divided by 300. In the LUK 3R case, 799/300 = 2.66 nm
2) "Flat earth math": For this you need to know the threshold elevation, which is not published on the chart. But it is on the airport diagram as 481. In this case that also happens to be the touchdown zone elevation, but that's not always the case.
We need the Threshold Crossing Height of the VGSI, which from the chart is 44. Note that if the TCH of the VGSI and of the procedure are within 3 feet, the non-coincident note containing the TCH of the VGSI will not be published, so we'd either have to look in the Chart Supplement or just estimate again using the procedural TCH. (The actual value here is probably not 44 feet, but 43.7 or something like that, so it's >3 feet different.)
We have the glidepath angle of 3 degrees.
We could use "tangent" here, but since it's an approximation anyway, I'll use 318 ft/nm for a 3 deg glidepath.
Take the MDA, subtract the VGSI TCH and the threshold elevation, and divide by 318.
(1280-44-481)/318 = 2.37
3) The round-earth formula contained in the 8260.3E, which takes the same inputs as 2) but accounts for the curvature of the earth.
View attachment 106694
Inputting the numbers above, I get 2.36 nm.
HOWEVER, even this is an approximation because the values for TCH and THRe were rounded to the nearest foot because that's what's published. The real values may have a decimal place or two. Finding the exact numbers isn't something the casual user could really do. They're available on form 8260-9 for each procedure, but this is generally only easily available when a new procedure is on the IFP Gateway's Coordination tab, for a few months before it's published.
Note that the difference between the easy estimate in #1 and the almost-precise estimate in #3 is 0.3 nm. That's 12 seconds at 90 knots, not a big difference. That's about how much time it's really going to take to start a descent anyway. So in practice, the simple "HAT/300" is pretty good.