[HP41] Lambert function RPN; question

[Albert Chan]

(04-10-2022 04:03 AM)Albert Chan Wrote:  I borrowed lyuka e^W estimation formula, replace 0.3 by 1/3, for Puiseux series.

e^W(y) ≈ 1/e + sqrt ((2/e)*(y+1/e)) + (y+1/e)/3

If we replace +sqrt (for 0 branch), with -sqrt (for -1 branch), formula also work (good enough for guess)

Formula is exact where 2 branches intersect, e^W(-1/e) = 1/e = 0.367879 441171 ...

e^W₀ (-0.367879) ≈ 0.368449 321330
e^W_-1(-0.367879) ≈ 0.367309 855127
e^W₀ (-0.3678) ≈ 0.375551 151939
e^W_-1(-0.3678) ≈ 0.360260 691185

Update: unfortunately, it does not work well for argument close to 0
Branch -1 with argument clos to 0, e^W guess may return negative number.
Too bad ...