Average Error: 13.7 → 1.0
Time: 23.2s
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 r8319861 = wj;
        double r8319862 = exp(r8319861);
        double r8319863 = r8319861 * r8319862;
        double r8319864 = x;
        double r8319865 = r8319863 - r8319864;
        double r8319866 = r8319862 + r8319863;
        double r8319867 = r8319865 / r8319866;
        double r8319868 = r8319861 - r8319867;
        return r8319868;
}
↓
double f(double wj, double x) {
        double r8319869 = x;
        double r8319870 = wj;
        double r8319871 = exp(r8319870);
        double r8319872 = r8319871 * r8319870;
        double r8319873 = r8319871 + r8319872;
        double r8319874 = r8319869 / r8319873;
        double r8319875 = r8319870 * r8319870;
        double r8319876 = -r8319870;
        double r8319877 = r8319876 * r8319870;
        double r8319878 = r8319870 * r8319877;
        double r8319879 = fma(r8319870, r8319870, r8319878);
        double r8319880 = fma(r8319875, r8319875, r8319879);
        double r8319881 = r8319874 + r8319880;
        return r8319881;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  3. Applied div-sub13.7

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

  4. Applied associate--r-7.3

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

  5. Taylor expanded around 0 1.0

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

  6. Simplified1.0

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

  8. Applied fma-neg1.0

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

  9. Final simplification1.0

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

Reproduce

herbie shell --seed 2019170 +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))))))