Average Error: 14.2 → 2.1
Time: 16.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)
double f(double wj, double x) {
        double r8907739 = wj;
        double r8907740 = exp(r8907739);
        double r8907741 = r8907739 * r8907740;
        double r8907742 = x;
        double r8907743 = r8907741 - r8907742;
        double r8907744 = r8907740 + r8907741;
        double r8907745 = r8907743 / r8907744;
        double r8907746 = r8907739 - r8907745;
        return r8907746;
}
↓
double f(double wj, double x) {
        double r8907747 = wj;
        double r8907748 = x;
        double r8907749 = r8907747 * r8907748;
        double r8907750 = -2.0;
        double r8907751 = fma(r8907747, r8907747, r8907748);
        double r8907752 = fma(r8907749, r8907750, r8907751);
        return r8907752;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.2
Target13.6
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.2

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019171 +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))))))