Average Error: 13.3 → 1.2
Time: 22.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj} \cdot e^{wj} - \left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right)} \cdot \left(e^{wj} - wj \cdot e^{wj}\right)\]
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}{e^{wj} \cdot e^{wj} - \left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right)} \cdot \left(e^{wj} - wj \cdot e^{wj}\right)
double f(double wj, double x) {
        double r10045977 = wj;
        double r10045978 = exp(r10045977);
        double r10045979 = r10045977 * r10045978;
        double r10045980 = x;
        double r10045981 = r10045979 - r10045980;
        double r10045982 = r10045978 + r10045979;
        double r10045983 = r10045981 / r10045982;
        double r10045984 = r10045977 - r10045983;
        return r10045984;
}
↓
double f(double wj, double x) {
        double r10045985 = wj;
        double r10045986 = r10045985 * r10045985;
        double r10045987 = r10045986 - r10045985;
        double r10045988 = r10045987 * r10045986;
        double r10045989 = r10045986 + r10045988;
        double r10045990 = x;
        double r10045991 = exp(r10045985);
        double r10045992 = r10045991 * r10045991;
        double r10045993 = r10045985 * r10045991;
        double r10045994 = r10045993 * r10045993;
        double r10045995 = r10045992 - r10045994;
        double r10045996 = r10045990 / r10045995;
        double r10045997 = r10045991 - r10045993;
        double r10045998 = r10045996 * r10045997;
        double r10045999 = r10045989 + r10045998;
        return r10045999;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target12.7
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.3

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

  3. Applied div-sub13.3

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

  4. Applied associate--r-7.2

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  8. Applied flip-+1.2

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

  9. Applied associate-/r/1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2019169 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))