HP-11C/15C Code optimization

[Albert Chan]

Hi, jam

It may be simpler to solve Colebrook with lambertw function.

x = -2 * log10(a*x + b)      // {a,b} switched to match https://en.wikipedia.org/wiki/Darcy_fric...r_formulae

let y = a*x+b, k = 2a/ln(10)

(y-b)/a = -k/a * ln(y)
y + k*ln(y) = b

--> y = k * W(z)            , where z = e^(b/k)/k

OP example

lua> a = 2.51/126400
lua> b = 4.2e-5/3.7
lua> k = 2*a/log(10)
lua> z = exp(b/k)/k

lua> z -- guess for exp(W(z))
111963.8779391653
lua> (_ + z) / (log(_) + 1)
17735.543271562103
lua> (_ + z) / (log(_) + 1)
12027.775216761022
lua> (_ + z) / (log(_) + 1)
11928.038961742051
lua> (_ + z) / (log(_) + 1)
11927.999039108403
lua> (_ + z) / (log(_) + 1) -- = exp(W(z))
11927.99903910197
lua> z / _ -- = W(z)
9.386643775886386

lua> (k*_ - b)/a -- = x
7.5814974176865215

Or, slightly more complicated Newton setup, straight for W(z)

lua> log(z) -- guess for W(z)
11.625931579856678
lua> _ - (_ + log(_/z)) / (1+1/_)
9.366995046326272
lua> _ - (_ + log(_/z)) / (1+1/_)
9.386641790802015
lua> _ - (_ + log(_/z)) / (1+1/_)
9.386643775886366
lua> _ - (_ + log(_/z)) / (1+1/_)
9.386643775886386