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Target

Derivation

1. Split input into 2 regimes

2. if wj < -4.312359683013498e-09 or 5.2518683439657065e-09 < wj

   1. Initial program 17.4

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

   3. Applied *-un-lft-identity17.4

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

   4. Applied *-un-lft-identity17.4

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

   5. Applied times-frac17.4

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

   6. Simplified17.4

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

   7. Simplified4.1

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

   if -4.312359683013498e-09 < wj < 5.2518683439657065e-09

   1. Initial program 13.1

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

   2. Taylor expanded around 0 0.2

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -4.312359683013498 \cdot 10^{-09} \lor \neg \left(wj \le 5.2518683439657065 \cdot 10^{-09}\right):\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(x \cdot wj\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(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

Time bar (total: 19.2s)Debug log