Average Error: 13.5 → 1.1
Time: 20.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} + e^{wj} \cdot wj} + \mathsf{fma}\left(wj \cdot wj, wj \cdot wj, \mathsf{fma}\left(wj, wj, wj \cdot \left(\left(-wj\right) \cdot wj\right)\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} + e^{wj} \cdot wj} + \mathsf{fma}\left(wj \cdot wj, wj \cdot wj, \mathsf{fma}\left(wj, wj, wj \cdot \left(\left(-wj\right) \cdot wj\right)\right)\right)
double f(double wj, double x) {
        double r7012691 = wj;
        double r7012692 = exp(r7012691);
        double r7012693 = r7012691 * r7012692;
        double r7012694 = x;
        double r7012695 = r7012693 - r7012694;
        double r7012696 = r7012692 + r7012693;
        double r7012697 = r7012695 / r7012696;
        double r7012698 = r7012691 - r7012697;
        return r7012698;
}
↓
double f(double wj, double x) {
        double r7012699 = x;
        double r7012700 = wj;
        double r7012701 = exp(r7012700);
        double r7012702 = r7012701 * r7012700;
        double r7012703 = r7012701 + r7012702;
        double r7012704 = r7012699 / r7012703;
        double r7012705 = r7012700 * r7012700;
        double r7012706 = -r7012700;
        double r7012707 = r7012706 * r7012700;
        double r7012708 = r7012700 * r7012707;
        double r7012709 = fma(r7012700, r7012700, r7012708);
        double r7012710 = fma(r7012705, r7012705, r7012709);
        double r7012711 = r7012704 + r7012710;
        return r7012711;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  3. Applied div-sub13.5

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

  4. Applied associate--r-7.5

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

  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

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

  8. Applied fma-neg1.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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