Average Error: 13.4 → 1.1
Time: 18.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}
double f(double wj, double x) {
        double r120447 = wj;
        double r120448 = exp(r120447);
        double r120449 = r120447 * r120448;
        double r120450 = x;
        double r120451 = r120449 - r120450;
        double r120452 = r120448 + r120449;
        double r120453 = r120451 / r120452;
        double r120454 = r120447 - r120453;
        return r120454;
}
↓
double f(double wj, double x) {
        double r120455 = wj;
        double r120456 = 4.0;
        double r120457 = pow(r120455, r120456);
        double r120458 = 3.0;
        double r120459 = pow(r120455, r120458);
        double r120460 = r120457 - r120459;
        double r120461 = fma(r120455, r120455, r120460);
        double r120462 = x;
        double r120463 = exp(r120455);
        double r120464 = 1.0;
        double r120465 = r120455 + r120464;
        double r120466 = r120463 * r120465;
        double r120467 = r120462 / r120466;
        double r120468 = r120461 + r120467;
        return r120468;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.8
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  4. Applied div-sub12.8

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

  5. Applied associate--r-6.7

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

  6. Simplified6.7

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

  7. Taylor expanded around 0 1.1

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

  8. Simplified1.1

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

  10. Applied div-inv1.1

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

  11. Applied associate-/l*1.1

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

  12. Simplified1.1

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

  13. Final simplification1.1

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

Reproduce

herbie shell --seed 2019195 +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))))))