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Derivation

1. Initial program 13.4

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

2. Initial simplification7.0

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

3. Taylor expanded around 0 1.0

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

5. Applied *-un-lft-identity1.0

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

6. Applied div-inv1.0

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

7. Applied times-frac1.0

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

8. Simplified1.0

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

9. Taylor expanded around inf 1.0

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

10. Simplified1.0

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

11. Final simplification1.0

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

Runtime

Time bar (total: 32.8s)Debug logProfile

herbie shell --seed 2018351 
(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))))))