Error

Target

Derivation

1. Split input into 2 regimes

2. if wj < 8.218026681193705e-05

   1. Initial program 13.5

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

   3. Applied div-sub13.5

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

   4. Applied associate--r-6.9

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

   5. Simplified6.9

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

   6. Taylor expanded around 0 0.3

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

   7. Simplified0.3

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

   if 8.218026681193705e-05 < wj

   1. Initial program 25.7

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

   3. Applied div-sub25.7

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

   4. Applied associate--r-25.6

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

   5. Simplified25.6

      \[\leadsto \left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \color{blue}{\frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity25.6

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

   8. Applied times-frac25.6

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

   9. Simplified25.6

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

   10. Simplified0.9

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.218026681193705 \cdot 10^{-05}:\\ \;\;\;\;\frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*} + (\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*} + \left(wj - \frac{1}{1 + wj} \cdot wj\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(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: 19.3s)Debug log