Average Error: 13.9 → 1.2
Time: 26.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}
double f(double wj, double x) {
        double r5249629 = wj;
        double r5249630 = exp(r5249629);
        double r5249631 = r5249629 * r5249630;
        double r5249632 = x;
        double r5249633 = r5249631 - r5249632;
        double r5249634 = r5249630 + r5249631;
        double r5249635 = r5249633 / r5249634;
        double r5249636 = r5249629 - r5249635;
        return r5249636;
}
↓
double f(double wj, double x) {
        double r5249637 = wj;
        double r5249638 = r5249637 * r5249637;
        double r5249639 = r5249638 - r5249637;
        double r5249640 = r5249639 * r5249638;
        double r5249641 = r5249638 + r5249640;
        double r5249642 = x;
        double r5249643 = exp(r5249637);
        double r5249644 = r5249637 * r5249643;
        double r5249645 = r5249644 + r5249643;
        double r5249646 = r5249642 / r5249645;
        double r5249647 = r5249641 + r5249646;
        return r5249647;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.9
Target13.2
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  3. Applied div-sub13.9

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

  4. Applied associate--r-7.6

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

  5. Taylor expanded around 0 1.2

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

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019151 
(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))))))