Error

Target

Derivation

1. Initial program 13.4

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

3. Applied div-sub13.4

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

4. Applied associate--r-7.3

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

5. Applied simplify6.7

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

6. Taylor expanded around 0 1.0

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

7. Applied simplify1.0

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

9. Applied add-sqr-sqrt1.0

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018166 +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))))))