Average Error: 14.0 → 2.1
Time: 21.5s
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 r3958627 = wj;
        double r3958628 = exp(r3958627);
        double r3958629 = r3958627 * r3958628;
        double r3958630 = x;
        double r3958631 = r3958629 - r3958630;
        double r3958632 = r3958628 + r3958629;
        double r3958633 = r3958631 / r3958632;
        double r3958634 = r3958627 - r3958633;
        return r3958634;
}
↓
double f(double wj, double x) {
        double r3958635 = wj;
        double r3958636 = x;
        double r3958637 = fma(r3958635, r3958635, r3958636);
        double r3958638 = r3958635 + r3958635;
        double r3958639 = r3958638 * r3958636;
        double r3958640 = r3958637 - r3958639;
        return r3958640;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.3
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.0

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019153 +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))))))