Average Error: 13.8 → 0.9
Time: 5.2s
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 r206983 = wj;
        double r206984 = exp(r206983);
        double r206985 = r206983 * r206984;
        double r206986 = x;
        double r206987 = r206985 - r206986;
        double r206988 = r206984 + r206985;
        double r206989 = r206987 / r206988;
        double r206990 = r206983 - r206989;
        return r206990;
}
↓
double f(double wj, double x) {
        double r206991 = x;
        double r206992 = wj;
        double r206993 = 1.0;
        double r206994 = r206992 + r206993;
        double r206995 = r206991 / r206994;
        double r206996 = exp(r206992);
        double r206997 = r206995 / r206996;
        double r206998 = 4.0;
        double r206999 = pow(r206992, r206998);
        double r207000 = 3.0;
        double r207001 = pow(r206992, r207000);
        double r207002 = r206999 - r207001;
        double r207003 = fma(r206992, r206992, r207002);
        double r207004 = r206997 + r207003;
        return r207004;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.2
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Simplified13.2

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

  4. Applied associate--l+6.9

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

  5. Taylor expanded around 0 0.9

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

  6. Simplified0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2020056 +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))))))