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Target

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 1.8403877061482827e-12

   1. Initial program 17.6

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

   2. Initial simplification8.8

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

   3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
   4. Using strategy rm

   5. Applied associate--l+0.2

      \[\leadsto \color{blue}{\left({wj}^{2} + \left({wj}^{4} - {wj}^{3}\right)\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

   if 1.8403877061482827e-12 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 2.3

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

   2. Initial simplification0.3

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

   4. Applied flip--0.3

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.8403877061482827 \cdot 10^{-12}:\\ \;\;\;\;\left({wj}^{2} + \left({wj}^{4} - {wj}^{3}\right)\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1} + \frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\frac{wj}{wj + 1} + wj}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018248 
(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))))))