Average Error: 13.8 → 1.1
Time: 16.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(wj \cdot wj + \left(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(wj \cdot wj + \left(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right)\right)
double f(double wj, double x) {
        double r4053732 = wj;
        double r4053733 = exp(r4053732);
        double r4053734 = r4053732 * r4053733;
        double r4053735 = x;
        double r4053736 = r4053734 - r4053735;
        double r4053737 = r4053733 + r4053734;
        double r4053738 = r4053736 / r4053737;
        double r4053739 = r4053732 - r4053738;
        return r4053739;
}
↓
double f(double wj, double x) {
        double r4053740 = x;
        double r4053741 = wj;
        double r4053742 = exp(r4053741);
        double r4053743 = r4053742 * r4053741;
        double r4053744 = r4053742 + r4053743;
        double r4053745 = r4053740 / r4053744;
        double r4053746 = r4053741 * r4053741;
        double r4053747 = r4053746 * r4053746;
        double r4053748 = r4053741 * r4053746;
        double r4053749 = r4053747 - r4053748;
        double r4053750 = r4053746 + r4053749;
        double r4053751 = r4053745 + r4053750;
        return r4053751;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.1
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  3. Applied div-sub13.8

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

  4. 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 1.1

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

  6. Simplified1.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2019156 
(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))))))