arcsinc( 1-y ), for small y

[Albert Chan]

Forman Acton's book Numerical Method that work (page 68) have a neat solution to the railroad problem:

sinc(x) = 5280/5281 = 1/(1+e), where e=1/5280

--> 1 - x^2/3! + x^4/5! - x^6/7! + ... = 1 - e/(1+e)

Subtract 1 from both side, factor out x^2, we get:

x^2 = 6e / (1+e) / (1 - x^2/20 + ...)

Last denominator is about 1, we get x^2 ~ 6e / (1 + e) = 6/5281
We get x = sqrt(6/5281) = 0.0337068013

Iterate once more:

x^2 = 6e / (1 + e) / (1 - (6e / (1+e)) / 20) = 6e / (1 + 0.7e) = 6/5280.7
We get x = sqrt(6/5280.7) = 0.0337077588

With this x, warp = (5280/2) tan(x/2) = 44.498455 ft (matched rounded exact value)

The book use this example to illustrate two points:
1. Remove large masking quantities, (in this case, the 1.0 term)
2. Do not solve quadratic equation if the quadratic term carry little weight.

Amazing book ...