arcsinc( 1-y ), for small y

[Albert Chan]

Did plots of how good e = sqrt((m + z)z), vs e = sqrt(6 m d) / 4
The new formula is 4 times more accurate !

Because the ratio is so consistent around 4, this is a fast way to estimate e:
Previous example, d = 100 ft:

e ~ (446.5492654 * 4 + 444.9719092) / 5 = 446.2337942 ft

With 6.5X warp or less (d < 5.5 *miles*), accuracy can be improved:
y = 100/5380
w = 4 - 27/56 y = 3.991038237
e ~ (446.5492654 w + 444.9719092) / (w+1) = 446.2332309 ft

This is not far from exact e of 446.233229831394 ...

We could even reverse the process, use good e to estimate arcsinc !
To simplify, let m = 1, so e = 1/2 tan( x/2 )

Example: calculate arcsinc(0.99):

sinc(x) = 0.99 = 1/(1+ d)
d = 1/99
z = 3/8 d = 1/264

e(min) = √(z)         = 0.06154574549
e(max) = √(z + z²) = 0.06166219923

w ~ 4 - 27/56 y = 3.995178571
e ~ (w e(max) + e(min)) / (w + 1) = 0.061638886
x ~ 2 arctan( 2 e ) = 0.2453178089

For comparison, asinc(0.99) = 0.2453178088 54025 ...