Average Error: 13.9 → 1.2
Time: 23.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{x \cdot e^{-wj}}{1 + wj}\]
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{x \cdot e^{-wj}}{1 + wj}
double f(double wj, double x) {
        double r113810 = wj;
        double r113811 = exp(r113810);
        double r113812 = r113810 * r113811;
        double r113813 = x;
        double r113814 = r113812 - r113813;
        double r113815 = r113811 + r113812;
        double r113816 = r113814 / r113815;
        double r113817 = r113810 - r113816;
        return r113817;
}
↓
double f(double wj, double x) {
        double r113818 = wj;
        double r113819 = 4.0;
        double r113820 = pow(r113818, r113819);
        double r113821 = fma(r113818, r113818, r113820);
        double r113822 = 3.0;
        double r113823 = pow(r113818, r113822);
        double r113824 = r113821 - r113823;
        double r113825 = x;
        double r113826 = -r113818;
        double r113827 = exp(r113826);
        double r113828 = r113825 * r113827;
        double r113829 = 1.0;
        double r113830 = r113829 + r113818;
        double r113831 = r113828 / r113830;
        double r113832 = r113824 + r113831;
        return r113832;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.3
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  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-7.0

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

  6. Taylor expanded around 0 1.2

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

  7. Simplified1.2

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

  9. Applied div-inv1.2

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

  10. Simplified1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019304 +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))))))