Average Error: 13.4 → 1.1
Time: 23.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double f(double wj, double x) {
        double r164762 = wj;
        double r164763 = exp(r164762);
        double r164764 = r164762 * r164763;
        double r164765 = x;
        double r164766 = r164764 - r164765;
        double r164767 = r164763 + r164764;
        double r164768 = r164766 / r164767;
        double r164769 = r164762 - r164768;
        return r164769;
}
↓
double f(double wj, double x) {
        double r164770 = wj;
        double r164771 = 4.0;
        double r164772 = pow(r164770, r164771);
        double r164773 = 3.0;
        double r164774 = pow(r164770, r164773);
        double r164775 = r164772 - r164774;
        double r164776 = fma(r164770, r164770, r164775);
        double r164777 = x;
        double r164778 = exp(r164770);
        double r164779 = r164777 / r164778;
        double r164780 = 1.0;
        double r164781 = r164780 + r164770;
        double r164782 = r164779 / r164781;
        double r164783 = r164776 + r164782;
        return r164783;
}

Error

Bits error versus wj

Bits error versus x

Target

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

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  4. Applied div-sub12.8

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

  5. Applied associate--r-6.6

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

  6. Taylor expanded around 0 1.1

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

  7. Simplified1.1

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

  8. Taylor expanded around 0 1.1

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

  9. Simplified1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2019323 +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))))))