Average Error: 14.0 → 1.3
Time: 26.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double f(double wj, double x) {
        double r161694 = wj;
        double r161695 = exp(r161694);
        double r161696 = r161694 * r161695;
        double r161697 = x;
        double r161698 = r161696 - r161697;
        double r161699 = r161695 + r161696;
        double r161700 = r161698 / r161699;
        double r161701 = r161694 - r161700;
        return r161701;
}
↓
double f(double wj, double x) {
        double r161702 = wj;
        double r161703 = 4.0;
        double r161704 = pow(r161702, r161703);
        double r161705 = fma(r161702, r161702, r161704);
        double r161706 = 3.0;
        double r161707 = pow(r161702, r161706);
        double r161708 = r161705 - r161707;
        double r161709 = x;
        double r161710 = exp(r161702);
        double r161711 = r161709 / r161710;
        double r161712 = 1.0;
        double r161713 = r161712 + r161702;
        double r161714 = r161711 / r161713;
        double r161715 = r161708 + r161714;
        return r161715;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.3
Herbie1.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.0

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

  2. Simplified13.3

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

  4. Applied div-sub13.3

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

  5. Applied associate--r-7.2

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

  6. Taylor expanded around 0 1.3

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

  7. Simplified1.3

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

  8. Final simplification1.3

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

Reproduce

herbie shell --seed 2019306 +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))))))