Average Error: 13.4 → 1.0
Time: 12.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1} \cdot \left(wj \cdot wj + \left(1 - wj\right)\right) + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1} \cdot \left(wj \cdot wj + \left(1 - wj\right)\right) + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r193759 = wj;
        double r193760 = exp(r193759);
        double r193761 = r193759 * r193760;
        double r193762 = x;
        double r193763 = r193761 - r193762;
        double r193764 = r193760 + r193761;
        double r193765 = r193763 / r193764;
        double r193766 = r193759 - r193765;
        return r193766;
}
↓
double f(double wj, double x) {
        double r193767 = x;
        double r193768 = wj;
        double r193769 = exp(r193768);
        double r193770 = r193767 / r193769;
        double r193771 = 3.0;
        double r193772 = pow(r193768, r193771);
        double r193773 = 1.0;
        double r193774 = r193772 + r193773;
        double r193775 = r193770 / r193774;
        double r193776 = r193768 * r193768;
        double r193777 = r193773 - r193768;
        double r193778 = r193776 + r193777;
        double r193779 = r193775 * r193778;
        double r193780 = 4.0;
        double r193781 = pow(r193768, r193780);
        double r193782 = fma(r193768, r193768, r193781);
        double r193783 = r193782 - r193772;
        double r193784 = r193779 + r193783;
        return r193784;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.8
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

  2. Simplified12.9

    \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\]
  3. Using strategy rm

  4. Applied div-sub12.8

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

  5. Applied associate--r-6.7

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

  6. Taylor expanded around 0 1.0

    \[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

  7. Simplified1.0

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
  8. Using strategy rm

  9. Applied flip3-+1.0

    \[\leadsto \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{\color{blue}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)}}}\]

  10. Applied associate-/r/1.0

    \[\leadsto \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\frac{\frac{x}{e^{wj}}}{{wj}^{3} + {1}^{3}} \cdot \left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right)}\]

  11. Simplified1.0

    \[\leadsto \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1}} \cdot \left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right)\]

  12. Final simplification1.0

    \[\leadsto \frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1} \cdot \left(wj \cdot wj + \left(1 - wj\right)\right) + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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