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Target

Derivation

1. Initial program 13.6

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

2. Initial simplification6.9

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

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 add-cube-cbrt1.2

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

7. Applied add-sqr-sqrt1.2

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

8. Applied associate-*r*1.2

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

9. Final simplification1.2

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

Reproduce

herbie shell --seed 2018365 
(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: 18.8s)Debug log