Error

Target

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 3.560668638667739e-20

   1. Initial program 18.3

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

   2. Taylor expanded around 0 0.8

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

   3. Simplified0.8

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

   if 3.560668638667739e-20 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 2.7

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

   3. Applied div-sub2.8

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

   4. Simplified0.7

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

4. Final simplification0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.560668638667739 \cdot 10^{-20}:\\ \;\;\;\;(\left(-2\right) \cdot \left(x \cdot wj\right) + \left((wj \cdot wj + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{1 + wj} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\\ \end{array}\]

Runtime

Time bar (total: 41.9s)Debug logProfile

herbie shell --seed 2018217 +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))))))