Average Error: 13.4 → 2.3
Time: 18.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2
double f(double wj, double x) {
        double r9518873 = wj;
        double r9518874 = exp(r9518873);
        double r9518875 = r9518873 * r9518874;
        double r9518876 = x;
        double r9518877 = r9518875 - r9518876;
        double r9518878 = r9518874 + r9518875;
        double r9518879 = r9518877 / r9518878;
        double r9518880 = r9518873 - r9518879;
        return r9518880;
}
↓
double f(double wj, double x) {
        double r9518881 = x;
        double r9518882 = wj;
        double r9518883 = r9518882 * r9518882;
        double r9518884 = r9518881 + r9518883;
        double r9518885 = r9518882 * r9518881;
        double r9518886 = -2.0;
        double r9518887 = r9518885 * r9518886;
        double r9518888 = r9518884 + r9518887;
        return r9518888;
}

Error

Bits error versus wj

Bits error versus x

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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. Taylor expanded around 0 2.3

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

  3. Simplified2.3

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

  4. Final simplification2.3

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

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))))))