Error

Target

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 8.36959831267844e-18

   1. Initial program 18.6

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

   2. Taylor expanded around 0 0.9

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

   3. Simplified0.9

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

   if 8.36959831267844e-18 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (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 div-sub2.6

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

   4. Simplified0.5

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

4. Final simplification0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 8.36959831267844 \cdot 10^{-18}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{1 + wj} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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: 12.9s)Debug log