Average Error: 14.1 → 1.2
Time: 28.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj + \mathsf{log1p}\left(wj\right)}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj + \mathsf{log1p}\left(wj\right)}}
double f(double wj, double x) {
        double r4316027 = wj;
        double r4316028 = exp(r4316027);
        double r4316029 = r4316027 * r4316028;
        double r4316030 = x;
        double r4316031 = r4316029 - r4316030;
        double r4316032 = r4316028 + r4316029;
        double r4316033 = r4316031 / r4316032;
        double r4316034 = r4316027 - r4316033;
        return r4316034;
}
↓
double f(double wj, double x) {
        double r4316035 = wj;
        double r4316036 = r4316035 * r4316035;
        double r4316037 = r4316036 - r4316035;
        double r4316038 = r4316037 * r4316036;
        double r4316039 = r4316036 + r4316038;
        double r4316040 = x;
        double r4316041 = log1p(r4316035);
        double r4316042 = r4316035 + r4316041;
        double r4316043 = exp(r4316042);
        double r4316044 = r4316040 / r4316043;
        double r4316045 = r4316039 + r4316044;
        return r4316045;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.5
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub14.1

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

  4. Applied associate--r-7.6

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

  5. Taylor expanded around 0 1.0

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

  6. Simplified1.0

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

  8. Applied add-exp-log1.2

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

  9. Simplified1.2

    \[\leadsto \left(wj \cdot wj + \left(wj \cdot wj\right) \cdot \left(wj \cdot wj - wj\right)\right) + \frac{x}{e^{\color{blue}{wj + \mathsf{log1p}\left(wj\right)}}}\]

  10. Final simplification1.2

    \[\leadsto \left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{e^{wj + \mathsf{log1p}\left(wj\right)}}\]

Reproduce

herbie shell --seed 2019142 +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))))))