\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 18.0 s
Input Error: 9.3
Output Error: 1.9
Log: ⚲
Profile: 🕒
\(wj - \frac{1}{{wj}^2 - {1}^2} \cdot \left(\left(wj - 1\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)\right)\)
  1. Started with
    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    9.3
  2. Using strategy rm
    9.3
  3. Applied distribute-rgt1-in to get
    \[wj - \frac{wj \cdot e^{wj} - x}{\color{red}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
    10.6
  4. Applied *-un-lft-identity to get
    \[wj - \frac{\color{red}{wj \cdot e^{wj} - x}}{\left(wj + 1\right) \cdot e^{wj}} \leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
    10.6
  5. Applied times-frac to get
    \[wj - \color{red}{\frac{1 \cdot \left(wj \cdot e^{wj} - x\right)}{\left(wj + 1\right) \cdot e^{wj}}} \leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
    10.6
  6. Applied simplify to get
    \[wj - \frac{1}{wj + 1} \cdot \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj}}} \leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
    1.9
  7. Using strategy rm
    1.9
  8. Applied flip-+ to get
    \[wj - \frac{1}{\color{red}{wj + 1}} \cdot \left(wj - \frac{x}{e^{wj}}\right) \leadsto wj - \frac{1}{\color{blue}{\frac{{wj}^2 - {1}^2}{wj - 1}}} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]
    1.9
  9. Applied associate-/r/ to get
    \[wj - \color{red}{\frac{1}{\frac{{wj}^2 - {1}^2}{wj - 1}}} \cdot \left(wj - \frac{x}{e^{wj}}\right) \leadsto wj - \color{blue}{\left(\frac{1}{{wj}^2 - {1}^2} \cdot \left(wj - 1\right)\right)} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]
    1.9
  10. Applied associate-*l* to get
    \[wj - \color{red}{\left(\frac{1}{{wj}^2 - {1}^2} \cdot \left(wj - 1\right)\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)} \leadsto wj - \color{blue}{\frac{1}{{wj}^2 - {1}^2} \cdot \left(\left(wj - 1\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)\right)}\]
    1.9

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)))))))