Average Error: 13.1 → 1.2
Time: 21.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\mathsf{fma}\left(wj \cdot wj, wj \cdot wj, wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj} \cdot e^{wj} - \left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right)} \cdot \left(e^{wj} - wj \cdot e^{wj}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\mathsf{fma}\left(wj \cdot wj, wj \cdot wj, wj \cdot wj\right) - wj \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 r8257747 = wj;
        double r8257748 = exp(r8257747);
        double r8257749 = r8257747 * r8257748;
        double r8257750 = x;
        double r8257751 = r8257749 - r8257750;
        double r8257752 = r8257748 + r8257749;
        double r8257753 = r8257751 / r8257752;
        double r8257754 = r8257747 - r8257753;
        return r8257754;
}
↓
double f(double wj, double x) {
        double r8257755 = wj;
        double r8257756 = r8257755 * r8257755;
        double r8257757 = fma(r8257756, r8257756, r8257756);
        double r8257758 = r8257755 * r8257756;
        double r8257759 = r8257757 - r8257758;
        double r8257760 = x;
        double r8257761 = exp(r8257755);
        double r8257762 = r8257761 * r8257761;
        double r8257763 = r8257755 * r8257761;
        double r8257764 = r8257763 * r8257763;
        double r8257765 = r8257762 - r8257764;
        double r8257766 = r8257760 / r8257765;
        double r8257767 = r8257761 - r8257763;
        double r8257768 = r8257766 * r8257767;
        double r8257769 = r8257759 + r8257768;
        return r8257769;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.1
Target12.4
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.1

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

  3. Applied div-sub13.1

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

  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

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

  8. Applied flip-+1.2

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

  9. Applied associate-/r/1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))))))