Error

Target

Derivation

1. Split input into 2 regimes

2. if (- (+ (pow wj 2) x) (* 2 (* wj x))) < 7.565976676175648e-13

   1. Initial program 17.9

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

   3. Applied div-sub17.9

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

   4. Applied simplify17.7

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

   5. Applied simplify17.7

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

   6. Taylor expanded around 0 17.9

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

   7. Applied simplify0.3

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

   if 7.565976676175648e-13 < (- (+ (pow wj 2) x) (* 2 (* wj x)))

   1. Initial program 2.1

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

   3. Applied div-sub2.1

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

   4. Applied simplify0.4

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

   5. Applied simplify0.4

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

   7. Applied add-cube-cbrt0.5

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

   8. Applied flip-+0.5

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

   9. Applied associate-/r/0.5

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

   10. Applied times-frac0.5

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

   11. Applied simplify0.5

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

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))))))