Average Error: 13.5 → 1.2
Time: 5.8s
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 r176694 = wj;
        double r176695 = exp(r176694);
        double r176696 = r176694 * r176695;
        double r176697 = x;
        double r176698 = r176696 - r176697;
        double r176699 = r176695 + r176696;
        double r176700 = r176698 / r176699;
        double r176701 = r176694 - r176700;
        return r176701;
}
↓
double f(double wj, double x) {
        double r176702 = x;
        double r176703 = wj;
        double r176704 = 1.0;
        double r176705 = r176703 + r176704;
        double r176706 = r176702 / r176705;
        double r176707 = exp(r176703);
        double r176708 = r176706 / r176707;
        double r176709 = 4.0;
        double r176710 = pow(r176703, r176709);
        double r176711 = 3.0;
        double r176712 = pow(r176703, r176711);
        double r176713 = r176710 - r176712;
        double r176714 = fma(r176703, r176703, r176713);
        double r176715 = r176708 + r176714;
        return r176715;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.9

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

  4. Applied associate--l+6.6

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

  5. Taylor expanded around 0 1.2

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

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019354 +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))))))