arcsinc( 1-y ), for small y

[Albert Chan]

I re-read the Five Minute challenge posts (highly recommended)
I found a simple formula that is slightly better than e = sqrt(6 m d) / 4

Let z = gain in length of of hypotenuse
Using post #30 3/4 rule, z = d/2 * 3/4 = 3/8 d

e^2 = (m/2 + z)^2 - (m/2)^2 = m z + z^2

e = sqrt((m + z) z)

With m=5280, d=1: e = sqrt((5280 + 3/8)(3/8)) = 44.49877105 ~ 44.50 ft

3/4 rule is the upper bound for tiny angle, so z (thus e) is over-estimated.

Let's try with bigger sector angle, say, d = 100 ft:

Old formula, e = sqrt(6 * 5280 * 100) / 4 = 444.9719092 ~ 444.97 ft
New formula, e = sqrt((5280 + 300/8)(300/8)) = 446.5492694 ~ 446.55 ft

We have a nice bound, correct e is between 444.97 ft to 446.55 ft

For correct value of e, with big angle, do arcsinc with corrected k:

y = 1 - 5280/5380 = 100/5380 = 0.01858736
k = 0.15 / (1 - 13/56 y / (1 - 0.19068 y / (1 - 0.21627 y))) = 0.1506523749
x = sqrt(6y) / (1 - k y) = 0.3348901066 (about 19 degree)

e = (m/2) tan( x/2 ) = 446.2332298 ~ 446.23 ft (correct)