Average Error: 13.6 → 1.1
Time: 22.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1} \cdot \left(1 + \left(wj \cdot wj - wj\right)\right) + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{e^{wj}}}{{wj}^{3} + 1} \cdot \left(1 + \left(wj \cdot wj - wj\right)\right) + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r186985 = wj;
        double r186986 = exp(r186985);
        double r186987 = r186985 * r186986;
        double r186988 = x;
        double r186989 = r186987 - r186988;
        double r186990 = r186986 + r186987;
        double r186991 = r186989 / r186990;
        double r186992 = r186985 - r186991;
        return r186992;
}
↓
double f(double wj, double x) {
        double r186993 = x;
        double r186994 = wj;
        double r186995 = exp(r186994);
        double r186996 = r186993 / r186995;
        double r186997 = 3.0;
        double r186998 = pow(r186994, r186997);
        double r186999 = 1.0;
        double r187000 = r186998 + r186999;
        double r187001 = r186996 / r187000;
        double r187002 = r186994 * r186994;
        double r187003 = r187002 - r186994;
        double r187004 = r186999 + r187003;
        double r187005 = r187001 * r187004;
        double r187006 = 4.0;
        double r187007 = pow(r186994, r187006);
        double r187008 = fma(r186994, r186994, r187007);
        double r187009 = r187008 - r186998;
        double r187010 = r187005 + r187009;
        return r187010;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target12.9
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.6

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

  2. Simplified12.9

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

  4. Applied div-sub12.9

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

  5. Applied associate--r-6.8

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

  6. Simplified6.8

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

  7. Taylor expanded around 0 1.1

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

  8. Simplified1.1

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

  10. Applied flip3-+1.1

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

  11. Applied associate-/r/1.1

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

  12. Simplified1.1

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

  13. Final simplification1.1

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

Reproduce

herbie shell --seed 2019212 +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))))))