(28/48/50) Lambert W Function

[John Keith]

(03-31-2023 04:06 PM)Albert Chan Wrote:  lua> x, lny = -0.9793607152032429, -0.9793607152084147
lua> (x + x*lny) / (1+x)
-0.9793607149578298

Simply extrapolate for 0 error, we get good W(-0.3678) estimate.
Actually, extrapolate for 0 error is equivalent to a Newton's step!

Thanks for your help and insights, this is all quite interesting if a bit beyond my level of knowledge.

Unfortunately, I am still not able to achieve that level of accuracy with the 12-digit precision of HP calculators.

For an input of -.3678, your program for e^W(x) returns .375551106194 and -.979360715069

Applying your correction above returns -.979360714941 which is a noticeable improvement but still off by 16 ULP's. My program from post #1 returns -.979360714903 which is a bit worse (off by 54 ULP's) but with no correction applied.

What I am hoping for is a way to achieve 11-digit accuracy for values very close to -1/e, such as -.36787944117. I still don't think this will be possible without extended precision but I am hoping to be pleasantly surprised.

I am close to completing a new version of my program which covers the entire complex plane in all branches. It seems to be accurate to within 2 ULP's except for a circle of radius ~.001 around -1/e. It is only that area that i am concerned about.