Error

Target

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 4.904870601809956e-10

   1. Initial program 18.0

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

   3. Applied distribute-rgt1-in18.0

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

   4. Applied *-un-lft-identity18.0

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

   5. Applied times-frac18.0

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

   6. Simplified18.0

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

   8. Applied sub-neg18.0

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

   9. Applied distribute-rgt-in18.0

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

   10. Applied associate--r+9.5

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

   11. Simplified9.5

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

   12. Taylor expanded around 0 0.2

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

   13. Simplified0.2

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

   if 4.904870601809956e-10 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 2.5

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

   3. Applied distribute-rgt1-in2.5

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

   4. Applied *-un-lft-identity2.5

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

   5. Applied times-frac2.5

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

   6. Simplified0.3

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 4.904870601809956 \cdot 10^{-10}:\\ \;\;\;\;(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_* - \frac{-x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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))))))

Details

Time bar (total: 23.6s)Debug log