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Target

Derivation

1. Split input into 2 regimes

2. if wj < 4.458584244320131e-09

   1. Initial program 13.6

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

   2. Taylor expanded around 0 0.9

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

   if 4.458584244320131e-09 < wj

   1. Initial program 24.5

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

   3. Applied *-un-lft-identity24.5

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

   4. Applied *-un-lft-identity24.5

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

   5. Applied times-frac24.5

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

   6. Simplified24.5

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

   7. Simplified3.1

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

   9. Applied add-sqr-sqrt3.2

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

   10. Applied associate-/r*3.2

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.458584244320131 \cdot 10^{-09}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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: 19.9s)Debug log