Average Error: 13.8 → 2.0
Time: 18.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, x\right) - \left(wj + wj\right) \cdot x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, x\right) - \left(wj + wj\right) \cdot x
double f(double wj, double x) {
        double r3410726 = wj;
        double r3410727 = exp(r3410726);
        double r3410728 = r3410726 * r3410727;
        double r3410729 = x;
        double r3410730 = r3410728 - r3410729;
        double r3410731 = r3410727 + r3410728;
        double r3410732 = r3410730 / r3410731;
        double r3410733 = r3410726 - r3410732;
        return r3410733;
}
↓
double f(double wj, double x) {
        double r3410734 = wj;
        double r3410735 = x;
        double r3410736 = fma(r3410734, r3410734, r3410735);
        double r3410737 = r3410734 + r3410734;
        double r3410738 = r3410737 * r3410735;
        double r3410739 = r3410736 - r3410738;
        return r3410739;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.2
Herbie2.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Taylor expanded around 0 2.0

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

  3. Simplified2.0

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

  4. Final simplification2.0

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

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

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