Error

Target

Derivation

1. Initial program 13.9

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

3. Applied div-sub13.9

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

4. Applied associate--r-7.8

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

5. Simplified7.8

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

6. Taylor expanded around 0 1.1

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

7. Simplified1.1

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

9. Applied add-sqr-sqrt1.1

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

10. Applied fma-def1.1

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

11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019002 +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: 18.5s)Debug log