Average Error: 13.4 → 2.3
Time: 15.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[wj \cdot wj + \left(x - \left(x \cdot 2\right) \cdot wj\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
wj \cdot wj + \left(x - \left(x \cdot 2\right) \cdot wj\right)
double f(double wj, double x) {
        double r184115 = wj;
        double r184116 = exp(r184115);
        double r184117 = r184115 * r184116;
        double r184118 = x;
        double r184119 = r184117 - r184118;
        double r184120 = r184116 + r184117;
        double r184121 = r184119 / r184120;
        double r184122 = r184115 - r184121;
        return r184122;
}
↓
double f(double wj, double x) {
        double r184123 = wj;
        double r184124 = r184123 * r184123;
        double r184125 = x;
        double r184126 = 2.0;
        double r184127 = r184125 * r184126;
        double r184128 = r184127 * r184123;
        double r184129 = r184125 - r184128;
        double r184130 = r184124 + r184129;
        return r184130;
}

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
Herbie2.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  3. Taylor expanded around 0 2.3

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

  4. Simplified2.3

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

  5. Final simplification2.3

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

Reproduce

herbie shell --seed 2019174 
(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))))))