Lambert W Function (hp-42s)

[Albert Chan]

We were testing special case x = -r = -1/e
I was curious, is it the same as guess(x) = -x ?

guess(x) = r + 0.3*(x+r) + √(2r*(x+r)) = -x   → √(2r*(x+r)) = -1.3*(x+r)

Let z = x+r, we have √(2/e*z) = -1.3*z           → z = 0 → x = -r

With this, we can simplify the code greatly, moving the test after guess(x).
And, turning repeat until loop into while do loop, remove the test altogether !

Bonus: now it checked both special case, x = -1/e, -1/e + 0j

Code:

00 { 56-Byte Prgm }
01▶LBL "eW"
02 -1
03 E↑X          # r     x
04 ENTER
05 RCL+ ST Z    # x+r   r       x
06 ENTER
07 STO+ ST X    # 2x+2r x+r     r   x
08 RCL× ST Z
09 SQRT         # sqrt  x+r     r   x
10 STO+ ST Z
11 R↓           # x+r   sqrt+r  x   sqrt
12 0.3
13 ×
14 +            # y     x       x   x
15 X<>Y
16 +/-
17 X<>Y         # y     -x      x   x
18▶LBL 01
19 X=Y?         # Convergence check
20 RTN
21 STO ST Y
22 LN
23 1
24 +
25 R↑           
26 RCL+ ST Z
27 X<>Y         
28 ÷            # Newton-Raphson method
29 -            
30 STO× ST X    
31 LASTX        
32 STO+ ST Y    # y'    y'+dy^2 x   x
33 GTO 01
34 END

UPDATE:
There is a flaw in termination test.
I keep this for historical record. Please use corrected version