Average Error: 14.1 → 1.5
Time: 24.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \left(\left(\frac{5}{2} \cdot \left(wj \cdot wj\right) - \left(wj + wj\right)\right) \cdot x + x\right)\]
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) + \left(\left(\frac{5}{2} \cdot \left(wj \cdot wj\right) - \left(wj + wj\right)\right) \cdot x + x\right)
double f(double wj, double x) {
        double r6794871 = wj;
        double r6794872 = exp(r6794871);
        double r6794873 = r6794871 * r6794872;
        double r6794874 = x;
        double r6794875 = r6794873 - r6794874;
        double r6794876 = r6794872 + r6794873;
        double r6794877 = r6794875 / r6794876;
        double r6794878 = r6794871 - r6794877;
        return r6794878;
}
↓
double f(double wj, double x) {
        double r6794879 = wj;
        double r6794880 = r6794879 * r6794879;
        double r6794881 = r6794880 - r6794879;
        double r6794882 = r6794881 * r6794880;
        double r6794883 = r6794880 + r6794882;
        double r6794884 = 2.5;
        double r6794885 = r6794884 * r6794880;
        double r6794886 = r6794879 + r6794879;
        double r6794887 = r6794885 - r6794886;
        double r6794888 = x;
        double r6794889 = r6794887 * r6794888;
        double r6794890 = r6794889 + r6794888;
        double r6794891 = r6794883 + r6794890;
        return r6794891;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.5
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

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

  3. Applied div-sub14.1

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

  4. Applied associate--r-7.6

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

  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

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

  7. Taylor expanded around 0 1.5

    \[\leadsto \left(wj \cdot wj + \left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right)\right) + \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left(x \cdot {wj}^{2}\right)\right) - 2 \cdot \left(x \cdot wj\right)\right)}\]

  8. Simplified1.5

    \[\leadsto \left(wj \cdot wj + \left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right)\right) + \color{blue}{\left(x + x \cdot \left(\frac{5}{2} \cdot \left(wj \cdot wj\right) - \left(wj + wj\right)\right)\right)}\]

  9. Final simplification1.5

    \[\leadsto \left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \left(\left(\frac{5}{2} \cdot \left(wj \cdot wj\right) - \left(wj + wj\right)\right) \cdot x + x\right)\]

Reproduce

herbie shell --seed 2019142 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))