\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 6.0 s
Input Error: 28.6
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)\)
  1. Started with
    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    28.6
  2. Applied taylor to get
    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leadsto \left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)\]
    0.0
  3. Taylor expanded around 0 to get
    \[\color{red}{\left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)} \leadsto \color{blue}{\left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)}\]
    0.0

Original test:


(lambda ((wj default) (x default))
  #:name "Jmat.Real.lambertw, newton loop step"
  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
  #:target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj)))))))