\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 10.4 s
Input Error: 10.9
Output Error: 1.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right) & \text{when } \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.1906728f-06 \\ wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) & \text{otherwise} \end{cases}\)

    if (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))) < 1.1906728f-06

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      4.5
    2. Applied taylor to get
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leadsto \left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)\]
      0.1
    3. Taylor expanded around 0 to get
      \[\color{red}{\left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)} \leadsto \color{blue}{\left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)}\]
      0.1

    if 1.1906728f-06 < (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      16.5
    2. Using strategy rm
      16.5
    3. Applied div-sub to get
      \[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      16.5
    4. Applied simplify to get
      \[wj - \left(\color{red}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \leadsto wj - \left(\color{blue}{\frac{wj}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
      1.8
    5. Applied simplify to get
      \[wj - \left(\frac{wj}{wj + 1} - \color{red}{\frac{x}{e^{wj} + wj \cdot e^{wj}}}\right) \leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{1 + wj}}{e^{wj}}}\right)\]
      1.8
  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)))))))