Average Error: 13.4 → 1.1
Time: 5.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)
double f(double wj, double x) {
        double r209685 = wj;
        double r209686 = exp(r209685);
        double r209687 = r209685 * r209686;
        double r209688 = x;
        double r209689 = r209687 - r209688;
        double r209690 = r209686 + r209687;
        double r209691 = r209689 / r209690;
        double r209692 = r209685 - r209691;
        return r209692;
}
↓
double f(double wj, double x) {
        double r209693 = x;
        double r209694 = wj;
        double r209695 = 1.0;
        double r209696 = r209694 + r209695;
        double r209697 = r209693 / r209696;
        double r209698 = exp(r209694);
        double r209699 = r209697 / r209698;
        double r209700 = 4.0;
        double r209701 = pow(r209694, r209700);
        double r209702 = 3.0;
        double r209703 = pow(r209694, r209702);
        double r209704 = r209701 - r209703;
        double r209705 = fma(r209694, r209694, r209704);
        double r209706 = r209699 + r209705;
        return r209706;
}

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

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

  4. Applied associate--l+6.9

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2020034 +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))))))