Average Error: 13.4 → 2.3
Time: 18.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot x\right) \cdot wj\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot x\right) \cdot wj
double f(double wj, double x) {
        double r380880 = wj;
        double r380881 = exp(r380880);
        double r380882 = r380880 * r380881;
        double r380883 = x;
        double r380884 = r380882 - r380883;
        double r380885 = r380881 + r380882;
        double r380886 = r380884 / r380885;
        double r380887 = r380880 - r380886;
        return r380887;
}
↓
double f(double wj, double x) {
        double r380888 = wj;
        double r380889 = x;
        double r380890 = fma(r380888, r380888, r380889);
        double r380891 = 2.0;
        double r380892 = r380891 * r380889;
        double r380893 = r380892 * r380888;
        double r380894 = r380890 - r380893;
        return r380894;
}

Error

Bits error versus wj

Bits error versus x

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

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

  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

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

  5. Final simplification2.3

    \[\leadsto \mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot x\right) \cdot wj\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))))