Average Error: 13.5 → 1.1
Time: 24.2s
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 r8613616 = wj;
        double r8613617 = exp(r8613616);
        double r8613618 = r8613616 * r8613617;
        double r8613619 = x;
        double r8613620 = r8613618 - r8613619;
        double r8613621 = r8613617 + r8613618;
        double r8613622 = r8613620 / r8613621;
        double r8613623 = r8613616 - r8613622;
        return r8613623;
}
↓
double f(double wj, double x) {
        double r8613624 = x;
        double r8613625 = wj;
        double r8613626 = exp(r8613625);
        double r8613627 = r8613626 * r8613625;
        double r8613628 = r8613626 + r8613627;
        double r8613629 = r8613624 / r8613628;
        double r8613630 = r8613625 * r8613625;
        double r8613631 = -r8613625;
        double r8613632 = r8613631 * r8613625;
        double r8613633 = r8613625 * r8613632;
        double r8613634 = fma(r8613625, r8613625, r8613633);
        double r8613635 = fma(r8613630, r8613630, r8613634);
        double r8613636 = r8613629 + r8613635;
        return r8613636;
}

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))))))