Average Error: 13.4 → 1.1
Time: 1.2m
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right) + \left(wj \cdot wj\right))_* + \frac{x}{e^{wj} + e^{wj} \cdot wj}\]
double f(double wj, double x) {
        double r22893138 = wj;
        double r22893139 = exp(r22893138);
        double r22893140 = r22893138 * r22893139;
        double r22893141 = x;
        double r22893142 = r22893140 - r22893141;
        double r22893143 = r22893139 + r22893140;
        double r22893144 = r22893142 / r22893143;
        double r22893145 = r22893138 - r22893144;
        return r22893145;
}
↓
double f(double wj, double x) {
        double r22893146 = wj;
        double r22893147 = r22893146 * r22893146;
        double r22893148 = r22893147 - r22893146;
        double r22893149 = fma(r22893147, r22893148, r22893147);
        double r22893150 = x;
        double r22893151 = exp(r22893146);
        double r22893152 = r22893151 * r22893146;
        double r22893153 = r22893151 + r22893152;
        double r22893154 = r22893150 / r22893153;
        double r22893155 = r22893149 + r22893154;
        return r22893155;
}


wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
(\left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right) + \left(wj \cdot wj\right))_* + \frac{x}{e^{wj} + e^{wj} \cdot wj}

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. Using strategy rm

  3. Applied div-sub13.4

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

  4. Applied associate--r-7.3

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  7. Final simplification1.1

    \[\leadsto (\left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right) + \left(wj \cdot wj\right))_* + \frac{x}{e^{wj} + e^{wj} \cdot wj}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))