Average Error: 13.2 → 1.1
Time: 24.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\]
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 r10629756 = wj;
        double r10629757 = exp(r10629756);
        double r10629758 = r10629756 * r10629757;
        double r10629759 = x;
        double r10629760 = r10629758 - r10629759;
        double r10629761 = r10629757 + r10629758;
        double r10629762 = r10629760 / r10629761;
        double r10629763 = r10629756 - r10629762;
        return r10629763;
}
↓
double f(double wj, double x) {
        double r10629764 = wj;
        double r10629765 = r10629764 * r10629764;
        double r10629766 = r10629765 - r10629764;
        double r10629767 = r10629766 * r10629765;
        double r10629768 = r10629765 + r10629767;
        double r10629769 = x;
        double r10629770 = exp(r10629764);
        double r10629771 = r10629764 * r10629770;
        double r10629772 = r10629771 + r10629770;
        double r10629773 = r10629769 / r10629772;
        double r10629774 = r10629768 + r10629773;
        return r10629774;
}

Error

Bits error versus wj

Bits error versus x

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Results

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Target

Original13.2
Target12.5
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.2

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

  3. Applied add-cube-cbrt13.3

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

  4. Applied associate-*l*13.3

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

  6. Applied div-sub13.3

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

  7. Applied associate--r-12.5

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

  8. Simplified6.6

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

  9. Taylor expanded around 0 1.1

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

  10. Simplified1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019149 
(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))))))