Error

Target

Derivation

1. Split input into 2 regimes

2. if (- wj (* (log (exp (/ 1 (+ wj 1)))) (- wj (/ x (exp wj))))) < 7.565976676171613e-13

   1. Initial program 17.7

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

   3. Applied distribute-rgt1-in17.8

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

   4. Applied *-un-lft-identity17.8

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

   5. Applied times-frac17.7

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

   6. Applied simplify17.7

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

   8. Applied add-log-exp17.8

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

   9. Taylor expanded around 0 18.3

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

   10. Applied simplify0.6

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

   if 7.565976676171613e-13 < (- wj (* (log (exp (/ 1 (+ wj 1)))) (- wj (/ x (exp wj)))))

   1. Initial program 2.6

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

   3. Applied distribute-rgt1-in2.6

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

   4. Applied *-un-lft-identity2.6

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

   5. Applied times-frac2.6

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

   6. Applied simplify0.4

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

   8. Applied add-log-exp0.5

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

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(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))))))