\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 8.8 s
Input Error: 31.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\begin{cases} (wj * wj + x)_* & \text{when } \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 2.899075745548247 \cdot 10^{-08} \\ wj - \left(\frac{wj}{wj + 1} - \frac{x}{(wj * \left(e^{wj}\right) + \left(e^{wj}\right))_*}\right) & \text{otherwise} \end{cases}\)

    if (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))) < 2.899075745548247e-08

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      23.9
    2. Applied taylor to get
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leadsto wj - \left(wj - \left({wj}^2 + x\right)\right)\]
      14.7
    3. Taylor expanded around 0 to get
      \[wj - \color{red}{\left(wj - \left({wj}^2 + x\right)\right)} \leadsto wj - \color{blue}{\left(wj - \left({wj}^2 + x\right)\right)}\]
      14.7
    4. Applied simplify to get
      \[\color{red}{wj - \left(wj - \left({wj}^2 + x\right)\right)} \leadsto \color{blue}{(wj * wj + x)_*}\]
      0

    if 2.899075745548247e-08 < (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      39.6
    2. Using strategy rm
      39.6
    3. Applied div-sub to get
      \[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      39.6
    4. Applied simplify to get
      \[wj - \left(\color{red}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \leadsto wj - \left(\color{blue}{\frac{wj}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
      0.5
    5. Applied simplify to get
      \[wj - \left(\frac{wj}{wj + 1} - \color{red}{\frac{x}{e^{wj} + wj \cdot e^{wj}}}\right) \leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{x}{(wj * \left(e^{wj}\right) + \left(e^{wj}\right))_*}}\right)\]
      0.5
  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)))))))