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Target

Derivation

1. Split input into 2 regimes

2. if wj < 1.7333967120862666e-08

   1. Initial program 13.1

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

   2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]

   if 1.7333967120862666e-08 < wj

   1. Initial program 27.2

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity27.2

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}}\]

   4. Applied *-un-lft-identity27.2

      \[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{1 \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\]

   5. Applied times-frac27.2

      \[\leadsto wj - \color{blue}{\frac{1}{1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\]

   6. Simplified27.2

      \[\leadsto wj - \color{blue}{1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

   7. Simplified3.0

      \[\leadsto wj - 1 \cdot \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\]
   8. Using strategy rm

   9. Applied flip-+3.2

      \[\leadsto wj - 1 \cdot \frac{wj - \frac{x}{e^{wj}}}{\color{blue}{\frac{wj \cdot wj - 1 \cdot 1}{wj - 1}}}\]

   10. Applied associate-/r/3.0

       \[\leadsto wj - 1 \cdot \color{blue}{\left(\frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)\right)}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube12.8

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(wj - 1 \cdot \left(\frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)\right)\right) \cdot \left(wj - 1 \cdot \left(\frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)\right)\right)\right) \cdot \left(wj - 1 \cdot \left(\frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.7333967120862666 \cdot 10^{-08}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(wj - \left(wj - 1\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1}\right) \cdot \left(\left(wj - \left(wj - 1\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1}\right) \cdot \left(wj - \left(wj - 1\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{wj \cdot wj - 1}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

Details

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