Average Error: 13.4 → 1.1
Time: 23.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double f(double wj, double x) {
        double r122132 = wj;
        double r122133 = exp(r122132);
        double r122134 = r122132 * r122133;
        double r122135 = x;
        double r122136 = r122134 - r122135;
        double r122137 = r122133 + r122134;
        double r122138 = r122136 / r122137;
        double r122139 = r122132 - r122138;
        return r122139;
}
↓
double f(double wj, double x) {
        double r122140 = wj;
        double r122141 = 4.0;
        double r122142 = pow(r122140, r122141);
        double r122143 = 3.0;
        double r122144 = pow(r122140, r122143);
        double r122145 = r122142 - r122144;
        double r122146 = fma(r122140, r122140, r122145);
        double r122147 = x;
        double r122148 = exp(r122140);
        double r122149 = r122147 / r122148;
        double r122150 = 1.0;
        double r122151 = r122150 + r122140;
        double r122152 = r122149 / r122151;
        double r122153 = r122146 + r122152;
        return r122153;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.8
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  4. Applied div-sub12.8

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

  5. Applied associate--r-6.6

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

  6. Taylor expanded around 0 1.1

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

  7. Simplified1.1

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

  9. Applied add-cube-cbrt1.1

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

  10. Applied unpow-prod-down1.1

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

  11. Applied add-sqr-sqrt1.1

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

  12. Applied prod-diff1.1

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

  13. Simplified1.1

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

  14. Simplified1.1

    \[\leadsto \left(\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \color{blue}{0}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

  15. Final simplification1.1

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

Reproduce

herbie shell --seed 2019323 +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))))))