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Target

Derivation

1. Initial program 14.4

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

2. Initial simplification7.4

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

3. Taylor expanded around 0 1.1

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

5. Applied div-inv1.1

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

6. Applied associate-/l*1.1

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

7. Taylor expanded around 0 1.6

   \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left(x \cdot {wj}^{2}\right)\right) - 2 \cdot \left(x \cdot wj\right)\right)}\]

8. Simplified1.6

   \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(wj \cdot \frac{5}{2} + -2\right) \cdot \left(x \cdot wj\right) + x\right)}\]

9. Final simplification1.6

   \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \left(\left(-2 + \frac{5}{2} \cdot wj\right) \cdot \left(x \cdot wj\right) + x\right)\]

Runtime

Time bar (total: 28.6s)Debug logProfile

herbie shell --seed 2018263 
(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))))))