Average Error: 13.4 → 2.2
Time: 6.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)
double f(double wj, double x) {
        double r231526 = wj;
        double r231527 = exp(r231526);
        double r231528 = r231526 * r231527;
        double r231529 = x;
        double r231530 = r231528 - r231529;
        double r231531 = r231527 + r231528;
        double r231532 = r231530 / r231531;
        double r231533 = r231526 - r231532;
        return r231533;
}
↓
double f(double wj, double x) {
        double r231534 = wj;
        double r231535 = x;
        double r231536 = fma(r231534, r231534, r231535);
        double r231537 = 2.0;
        double r231538 = r231534 * r231535;
        double r231539 = r231537 * r231538;
        double r231540 = r231536 - r231539;
        return r231540;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.8
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  3. Taylor expanded around 0 2.2

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

  4. Taylor expanded around 0 2.2

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

  5. Simplified2.2

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

  6. Final simplification2.2

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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