Average Error: 13.4 → 1.1
Time: 22.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{e^{wj}}}{1 + wj} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{e^{wj}}}{1 + wj} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)
double f(double wj, double x) {
        double r130943 = wj;
        double r130944 = exp(r130943);
        double r130945 = r130943 * r130944;
        double r130946 = x;
        double r130947 = r130945 - r130946;
        double r130948 = r130944 + r130945;
        double r130949 = r130947 / r130948;
        double r130950 = r130943 - r130949;
        return r130950;
}
↓
double f(double wj, double x) {
        double r130951 = x;
        double r130952 = wj;
        double r130953 = exp(r130952);
        double r130954 = r130951 / r130953;
        double r130955 = 1.0;
        double r130956 = r130955 + r130952;
        double r130957 = r130954 / r130956;
        double r130958 = 4.0;
        double r130959 = pow(r130952, r130958);
        double r130960 = 3.0;
        double r130961 = pow(r130952, r130960);
        double r130962 = r130959 - r130961;
        double r130963 = fma(r130952, r130952, r130962);
        double r130964 = r130957 + r130963;
        return r130964;
}

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 *-un-lft-identity1.1

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

  10. Applied *-un-lft-identity1.1

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

  11. Applied distribute-lft-out--1.1

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

  12. Simplified1.1

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

  13. Final simplification1.1

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

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))))))