Average Error: 13.1 → 1.2
Time: 40.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\mathsf{fma}\left(wj \cdot wj, wj \cdot wj, wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj} \cdot e^{wj} - \left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right)} \cdot \left(e^{wj} - wj \cdot e^{wj}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\mathsf{fma}\left(wj \cdot wj, wj \cdot wj, wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj} \cdot e^{wj} - \left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right)} \cdot \left(e^{wj} - wj \cdot e^{wj}\right)
double f(double wj, double x) {
        double r8502411 = wj;
        double r8502412 = exp(r8502411);
        double r8502413 = r8502411 * r8502412;
        double r8502414 = x;
        double r8502415 = r8502413 - r8502414;
        double r8502416 = r8502412 + r8502413;
        double r8502417 = r8502415 / r8502416;
        double r8502418 = r8502411 - r8502417;
        return r8502418;
}
↓
double f(double wj, double x) {
        double r8502419 = wj;
        double r8502420 = r8502419 * r8502419;
        double r8502421 = fma(r8502420, r8502420, r8502420);
        double r8502422 = r8502419 * r8502420;
        double r8502423 = r8502421 - r8502422;
        double r8502424 = x;
        double r8502425 = exp(r8502419);
        double r8502426 = r8502425 * r8502425;
        double r8502427 = r8502419 * r8502425;
        double r8502428 = r8502427 * r8502427;
        double r8502429 = r8502426 - r8502428;
        double r8502430 = r8502424 / r8502429;
        double r8502431 = r8502425 - r8502427;
        double r8502432 = r8502430 * r8502431;
        double r8502433 = r8502423 + r8502432;
        return r8502433;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.1
Target12.4
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.1

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

  3. Applied div-sub13.1

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

  4. Applied associate--r-7.2

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

  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

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

  8. Applied flip-+1.2

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

  9. Applied associate-/r/1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2019168 +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))))))