Average Error: 13.5 → 1.1
Time: 23.6s
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 r10052478 = wj;
        double r10052479 = exp(r10052478);
        double r10052480 = r10052478 * r10052479;
        double r10052481 = x;
        double r10052482 = r10052480 - r10052481;
        double r10052483 = r10052479 + r10052480;
        double r10052484 = r10052482 / r10052483;
        double r10052485 = r10052478 - r10052484;
        return r10052485;
}
↓
double f(double wj, double x) {
        double r10052486 = x;
        double r10052487 = wj;
        double r10052488 = exp(r10052487);
        double r10052489 = r10052488 * r10052487;
        double r10052490 = r10052488 + r10052489;
        double r10052491 = r10052486 / r10052490;
        double r10052492 = r10052487 * r10052487;
        double r10052493 = r10052492 * r10052492;
        double r10052494 = r10052487 * r10052492;
        double r10052495 = r10052493 - r10052494;
        double r10052496 = r10052492 + r10052495;
        double r10052497 = r10052491 + r10052496;
        return r10052497;
}

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

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

  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 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))))))