Average Error: 13.4 → 2.2
Time: 12.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[wj \cdot \left(wj - x \cdot 2\right) + x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
wj \cdot \left(wj - x \cdot 2\right) + x
double f(double wj, double x) {
        double r223486 = wj;
        double r223487 = exp(r223486);
        double r223488 = r223486 * r223487;
        double r223489 = x;
        double r223490 = r223488 - r223489;
        double r223491 = r223487 + r223488;
        double r223492 = r223490 / r223491;
        double r223493 = r223486 - r223492;
        return r223493;
}
↓
double f(double wj, double x) {
        double r223494 = wj;
        double r223495 = x;
        double r223496 = 2.0;
        double r223497 = r223495 * r223496;
        double r223498 = r223494 - r223497;
        double r223499 = r223494 * r223498;
        double r223500 = r223499 + r223495;
        return r223500;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

    \[\leadsto \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} + wj}\]

  3. Taylor expanded around 0 2.2

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

  4. Simplified2.2

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

  5. Final simplification2.2

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

Reproduce

herbie shell --seed 2020046 
(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))))))