Average Error: 13.5 → 1.1
Time: 4.8s
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 r213544 = wj;
        double r213545 = exp(r213544);
        double r213546 = r213544 * r213545;
        double r213547 = x;
        double r213548 = r213546 - r213547;
        double r213549 = r213545 + r213546;
        double r213550 = r213548 / r213549;
        double r213551 = r213544 - r213550;
        return r213551;
}
↓
double f(double wj, double x) {
        double r213552 = x;
        double r213553 = wj;
        double r213554 = 1.0;
        double r213555 = r213553 + r213554;
        double r213556 = r213552 / r213555;
        double r213557 = exp(r213553);
        double r213558 = r213556 / r213557;
        double r213559 = 4.0;
        double r213560 = pow(r213553, r213559);
        double r213561 = 3.0;
        double r213562 = pow(r213553, r213561);
        double r213563 = r213560 - r213562;
        double r213564 = fma(r213553, r213553, r213563);
        double r213565 = r213558 + r213564;
        return r213565;
}

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))))))