New Sum of Powers Log Function

[C.Ret]

The french Silicium Forum have various implementations of Lambert W function for HP-Prime, SHARP PC-1211 and HP-15C

For example:
   

Code:

001- LBL A  CF 8  STO 0  1  +  LN  1  ENTER  e^x  +  LN  ÷  GTO 0
014- LBL B  CF 8  STO 0  1  e^x  ×  2  ENTER  CHS  ENTER  RCL 0  x²  LN   -  √x  ×  -  GTO 0
032- LBL C  STO 0  re↔im  STO 1  re↔im  LN  ENTER  ENTER  LN  -  x↔y  LSTx  ÷  1/x  +            
047- LBL 0
048-     CF 0  PSE  ENTER  ENTER  e^x  ENTER  R↑  ×  +  LSTx  FS? 8  GSB 1
060-     RCL-0  ENTER  ABS  RND  x>0?  SF 0
075-     R↓  x↔y  ÷  -  LSTx  ABS  RND  x=0?  CF 0
075-     R↓  FS? 0  GTO 0
078- RTN         
079- LBL 1  re↔im  RCL-1  re↔im  RTN

Set desired precision with FIX 1~9.
Enter argument x (or complex z) in first stack level X: and press:
f- A for real value of \( W_0(x) \) in main positive branch
f- B for real value of \(W_{-1}(x) \) in negative branch
f- C for complex value of \( W(z) \)