\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 11.1 s
Input Error: 28.7
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} wj + \frac{-\left(e^{wj} \cdot wj - x\right)}{\left(1 + wj\right) \cdot e^{wj}} & \text{when } wj \le -2.63216061572775 \cdot 10^{-13} \\ \left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right) & \text{when } wj \le 1.522317513311434 \cdot 10^{-06} \\ \left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}} & \text{otherwise} \end{cases}\)

    if wj < -2.63216061572775e-13

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      6.3
    2. Using strategy rm
      6.3
    3. Applied sub-neg to get
      \[\color{red}{wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      6.3
    4. Applied simplify to get
      \[wj + \color{red}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \leadsto wj + \color{blue}{\frac{-\left(e^{wj} \cdot wj - x\right)}{\left(1 + wj\right) \cdot e^{wj}}}\]
      6.4

    if -2.63216061572775e-13 < wj < 1.522317513311434e-06

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      13.5
    2. Applied taylor to get
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leadsto \left({wj}^2 + x\right) - 2 \cdot \left(wj \cdot x\right)\]
      0.0
    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

    if 1.522317513311434e-06 < wj

    1. Started with
      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
      58.9
    2. Using strategy rm
      58.9
    3. Applied div-sub to get
      \[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
      58.9
    4. Applied associate--r- to get
      \[\color{red}{wj - \left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \leadsto \color{blue}{\left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}}\]
      58.9
    5. Applied simplify to get
      \[\color{red}{\left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}} \leadsto \color{blue}{\left(wj - \frac{wj}{1 + wj}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
      0.1
  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)))))))