Error

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Target

Derivation

1. Split input into 2 regimes

2. if wj < 7.274594866092093e-10

   1. Initial program 13.5

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

   2. Taylor expanded around 0 1.0

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

   if 7.274594866092093e-10 < wj

   1. Initial program 21.3

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

   3. Applied div-sub21.3

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

   4. Simplified3.2

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

   6. Applied div-inv3.1

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.274594866092093 \cdot 10^{-10}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{1}{wj + 1} \cdot wj - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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: 18.2s)Debug log