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Target

Derivation

1. Split input into 2 regimes

2. if wj < 8.55419651697416e-05

   1. Initial program 13.1

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

   2. Initial simplification6.8

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

   3. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

   if 8.55419651697416e-05 < wj

   1. Initial program 32.4

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

   2. Initial simplification1.1

      \[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
   3. Using strategy rm

   4. Applied flip-+1.1

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

   5. Applied associate-/r/1.1

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.55419651697416 \cdot 10^{-05}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \end{array}\]

Runtime

Time bar (total: 40.6s)Debug logProfile

herbie shell --seed 2018290 
(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))))))