Average Error: 13.6 → 1.0
Time: 17.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double f(double wj, double x) {
        double r208055 = wj;
        double r208056 = exp(r208055);
        double r208057 = r208055 * r208056;
        double r208058 = x;
        double r208059 = r208057 - r208058;
        double r208060 = r208056 + r208057;
        double r208061 = r208059 / r208060;
        double r208062 = r208055 - r208061;
        return r208062;
}
↓
double f(double wj, double x) {
        double r208063 = wj;
        double r208064 = 4.0;
        double r208065 = pow(r208063, r208064);
        double r208066 = fma(r208063, r208063, r208065);
        double r208067 = 3.0;
        double r208068 = pow(r208063, r208067);
        double r208069 = r208066 - r208068;
        double r208070 = x;
        double r208071 = exp(r208063);
        double r208072 = r208070 / r208071;
        double r208073 = 1.0;
        double r208074 = r208073 + r208063;
        double r208075 = r208072 / r208074;
        double r208076 = r208069 + r208075;
        return r208076;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.1
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.6

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

  2. Simplified13.1

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

  4. Applied div-sub13.1

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

  5. Applied associate--r-7.2

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

  6. Simplified7.2

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

  7. Taylor expanded around 0 1.0

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

  8. Simplified1.0

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

  9. Final simplification1.0

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

Reproduce

herbie shell --seed 2019209 +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))))))