Average Error: 13.7 → 2.1
Time: 1.8m
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)
double f(double wj, double x) {
        double r51946488 = wj;
        double r51946489 = exp(r51946488);
        double r51946490 = r51946488 * r51946489;
        double r51946491 = x;
        double r51946492 = r51946490 - r51946491;
        double r51946493 = r51946489 + r51946490;
        double r51946494 = r51946492 / r51946493;
        double r51946495 = r51946488 - r51946494;
        return r51946495;
}
↓
double f(double wj, double x) {
        double r51946496 = x;
        double r51946497 = -2.0;
        double r51946498 = wj;
        double r51946499 = fma(r51946496, r51946497, r51946498);
        double r51946500 = fma(r51946499, r51946498, r51946496);
        return r51946500;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019124 +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))))))