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Target

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 6.214893319422228e-18

   1. Initial program 18.0

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

   2. Initial simplification9.2

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

   3. Taylor expanded around 0 0.1

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

   if 6.214893319422228e-18 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 3.1

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

   2. Initial simplification0.7

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

   4. Applied flip--0.7

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.214893319422228 \cdot 10^{-18}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1} + \frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\frac{wj}{wj + 1} + wj}\\ \end{array}\]

Runtime

Time bar (total: 52.5s)Debug logProfile

herbie shell --seed 2018349 
(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))))))