\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 13.8 s
Input Error: 28.4
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \left({wj}^2 - {wj}^3\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}} & \text{when } wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.8050401405631674 \cdot 10^{-17} \\ wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) & \text{otherwise} \end{cases}\)

    if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 3.8050401405631674e-17

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      17.7
    2. Using strategy rm
      17.7
    3. Applied div-sub to get
      \[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      17.7
    4. Applied simplify to get
      \[wj - \left(\color{red}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \leadsto wj - \left(\color{blue}{\frac{wj}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
      17.7
    5. Applied simplify to get
      \[wj - \left(\frac{wj}{wj + 1} - \color{red}{\frac{x}{e^{wj} + wj \cdot e^{wj}}}\right) \leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{1 + wj}}{e^{wj}}}\right)\]
      17.7
    6. Applied taylor to get
      \[wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) \leadsto wj - \left(\left(\left({wj}^{3} + wj\right) - {wj}^2\right) - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\]
      17.5
    7. Taylor expanded around 0 to get
      \[wj - \left(\color{red}{\left(\left({wj}^{3} + wj\right) - {wj}^2\right)} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) \leadsto wj - \left(\color{blue}{\left(\left({wj}^{3} + wj\right) - {wj}^2\right)} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\]
      17.5
    8. Applied simplify to get
      \[\color{red}{wj - \left(\left(\left({wj}^{3} + wj\right) - {wj}^2\right) - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)} \leadsto \color{blue}{\left({wj}^2 - {wj}^3\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}}}\]
      0.0

    if 3.8050401405631674e-17 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      38.5
    2. Using strategy rm
      38.5
    3. Applied div-sub to get
      \[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      38.5
    4. Applied simplify to get
      \[wj - \left(\color{red}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \leadsto wj - \left(\color{blue}{\frac{wj}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
      0.2
    5. Applied simplify to get
      \[wj - \left(\frac{wj}{wj + 1} - \color{red}{\frac{x}{e^{wj} + wj \cdot e^{wj}}}\right) \leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{1 + wj}}{e^{wj}}}\right)\]
      0.2
  1. Removed slow pow expressions

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)))))))