Where the shortcut breaks
The entries ranked by one question: how far can you push the quick formula before it is wrong by the amount you care about?
Each entry compares a quick formula with a careful one. This page puts the first two on a common scale: the input at which the quick formula's error reaches a tolerance you choose. The calendar entry is a different kind of shortcut (a division, not an approximation), so it is measured differently below.
Try it
| Entry | Shortcut | Wrong by that much at |
|---|---|---|
| Tall building | gh/c² vs potential difference | |
| Pendulum swing | small-angle period vs exact (AGM) | |
| Pendulum swing | one-term 1+θ²/16 vs exact, error as a share of the correction |
What it shows
At 1%, the tower shortcut holds to about 73 km of height and the small-angle pendulum to about 22.8° of swing. Set against the defaults on those pages (the 828 m Burj Khalifa and a 5° swing), the tower has roughly 88 times headroom and the pendulum about 4.6 times. The ratio is the useful number: a shortcut breaks late if the input you actually use is far below the threshold.
The one-term pendulum correction is a better shortcut than it looks. It misses 1% of the true correction only at about 23.9°, barely beyond where the uncorrected formula misses 1% of the whole period, because the correction is itself small.
The calendar is different
The leap-year entry has no approximation to break; its shortcut is 1 / |average year − tropical year|, which is exact arithmetic on an uncertain input. The right question there is how little the input can move before the answer moves 1%. For the Gregorian rule that is 3.1×10−6 day, about 0.27 s of tropical-year length. For the 31-in-128 rule it is about 0.0013 s. The first is within the spread of published values of the year; the second is far finer than any definition of the year can supply. So the quick calendar claim for Gregorian is a measurement, and for the better rules it is not.
My reading
This site is willing to be wrong about this: a shortcut should be quoted with its threshold, not its value. "gh/c²" with no height limit, or "T = 2π√(L/g)" with no amplitude limit, is incomplete in the same way as a measurement with no unit.
What I did not check
The thresholds come from my own bisection script using the same constants as the entries (g = 9.80665 m/s², GM = 3.986004418×1014 m³/s², R = 6 371 km, a point-mass Earth with no rotation), and the page runs the same bisection in your browser. They inherit every unchecked assumption in those entries. At 73 km the weak-field Newtonian potential is still fine, but I did not check that a higher-order relativistic term stays below the tolerance at 647 km or beyond. "Error" here is the shortcut's relative difference from the careful formula, not from a measurement. The calendar sensitivities use 365.242189 days for the tropical year, as in that entry.