Average Error: 14.0 → 2.2
Time: 19.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[x + wj \cdot \left(wj - x \cdot 2\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
x + wj \cdot \left(wj - x \cdot 2\right)
double f(double wj, double x) {
        double r123533 = wj;
        double r123534 = exp(r123533);
        double r123535 = r123533 * r123534;
        double r123536 = x;
        double r123537 = r123535 - r123536;
        double r123538 = r123534 + r123535;
        double r123539 = r123537 / r123538;
        double r123540 = r123533 - r123539;
        return r123540;
}
↓
double f(double wj, double x) {
        double r123541 = x;
        double r123542 = wj;
        double r123543 = 2.0;
        double r123544 = r123541 * r123543;
        double r123545 = r123542 - r123544;
        double r123546 = r123542 * r123545;
        double r123547 = r123541 + r123546;
        return r123547;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.4
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.0

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto x + wj \cdot \left(wj - x \cdot 2\right)\]

Reproduce

herbie shell --seed 2019297 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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