Average Error: 13.5 → 1.1
Time: 4.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)
double f(double wj, double x) {
        double r250531 = wj;
        double r250532 = exp(r250531);
        double r250533 = r250531 * r250532;
        double r250534 = x;
        double r250535 = r250533 - r250534;
        double r250536 = r250532 + r250533;
        double r250537 = r250535 / r250536;
        double r250538 = r250531 - r250537;
        return r250538;
}
↓
double f(double wj, double x) {
        double r250539 = x;
        double r250540 = wj;
        double r250541 = 1.0;
        double r250542 = r250540 + r250541;
        double r250543 = r250539 / r250542;
        double r250544 = exp(r250540);
        double r250545 = r250543 / r250544;
        double r250546 = 4.0;
        double r250547 = pow(r250540, r250546);
        double r250548 = 3.0;
        double r250549 = pow(r250540, r250548);
        double r250550 = r250547 - r250549;
        double r250551 = fma(r250540, r250540, r250550);
        double r250552 = r250545 + r250551;
        return r250552;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.8
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.8

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

  4. Applied associate--l+6.6

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

  5. Taylor expanded around 0 1.1

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

  6. Simplified1.1

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)}\]

  7. Final simplification1.1

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2020060 +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))))))