Average Error: 13.5 → 1.1
Time: 22.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{1}{e^{wj}} \cdot \frac{x}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{1}{e^{wj}} \cdot \frac{x}{1 + wj}
double f(double wj, double x) {
        double r9186646 = wj;
        double r9186647 = exp(r9186646);
        double r9186648 = r9186646 * r9186647;
        double r9186649 = x;
        double r9186650 = r9186648 - r9186649;
        double r9186651 = r9186647 + r9186648;
        double r9186652 = r9186650 / r9186651;
        double r9186653 = r9186646 - r9186652;
        return r9186653;
}
↓
double f(double wj, double x) {
        double r9186654 = wj;
        double r9186655 = r9186654 * r9186654;
        double r9186656 = r9186655 - r9186654;
        double r9186657 = r9186656 * r9186655;
        double r9186658 = r9186655 + r9186657;
        double r9186659 = 1.0;
        double r9186660 = exp(r9186654);
        double r9186661 = r9186659 / r9186660;
        double r9186662 = x;
        double r9186663 = r9186659 + r9186654;
        double r9186664 = r9186662 / r9186663;
        double r9186665 = r9186661 * r9186664;
        double r9186666 = r9186658 + r9186665;
        return r9186666;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  8. Applied *-un-lft-identity1.1

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

  9. Applied distribute-rgt-out1.1

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

  10. Applied *-un-lft-identity1.1

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

  11. Applied times-frac1.1

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

  12. Final simplification1.1

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

Reproduce

herbie shell --seed 2019172 
(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))))))