Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if wj < 0.0002922266443431827

   1. Initial program 13.2

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

   2. Initial simplification6.8

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

   3. Taylor expanded around 0 0.3

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

   5. Applied add-cube-cbrt0.3

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

   6. Applied *-un-lft-identity0.3

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

   7. Applied times-frac0.3

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

   9. Applied pow1/30.5

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

   10. Applied pow1/30.5

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

   11. Applied pow-prod-down0.3

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

   if 0.0002922266443431827 < wj

   1. Initial program 28.4

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

   2. Initial simplification0.9

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

   4. Applied add-sqr-sqrt1.0

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

   5. Applied associate-/r*1.0

      \[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\color{blue}{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}}{wj + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 0.0002922266443431827:\\ \;\;\;\;\frac{1}{{\left(\left(wj + 1\right) \cdot \left(wj + 1\right)\right)}^{\frac{1}{3}}} \cdot \frac{\frac{x}{e^{wj}}}{\sqrt[3]{wj + 1}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Runtime

Time bar (total: 26.6s)Debug logProfile

herbie shell --seed 2018235 
(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))))))