Average Error: 13.1 → 1.1
Time: 32.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(wj \cdot wj + \left(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(wj \cdot wj + \left(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right) - wj \cdot \left(wj \cdot wj\right)\right)\right)
double f(double wj, double x) {
        double r11456611 = wj;
        double r11456612 = exp(r11456611);
        double r11456613 = r11456611 * r11456612;
        double r11456614 = x;
        double r11456615 = r11456613 - r11456614;
        double r11456616 = r11456612 + r11456613;
        double r11456617 = r11456615 / r11456616;
        double r11456618 = r11456611 - r11456617;
        return r11456618;
}
↓
double f(double wj, double x) {
        double r11456619 = x;
        double r11456620 = wj;
        double r11456621 = exp(r11456620);
        double r11456622 = r11456621 * r11456620;
        double r11456623 = r11456621 + r11456622;
        double r11456624 = r11456619 / r11456623;
        double r11456625 = r11456620 * r11456620;
        double r11456626 = r11456625 * r11456625;
        double r11456627 = r11456620 * r11456625;
        double r11456628 = r11456626 - r11456627;
        double r11456629 = r11456625 + r11456628;
        double r11456630 = r11456624 + r11456629;
        return r11456630;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2019168 
(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))))))