Average Error: 13.9 → 2.2
Time: 23.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x
double f(double wj, double x) {
        double r11316892 = wj;
        double r11316893 = exp(r11316892);
        double r11316894 = r11316892 * r11316893;
        double r11316895 = x;
        double r11316896 = r11316894 - r11316895;
        double r11316897 = r11316893 + r11316894;
        double r11316898 = r11316896 / r11316897;
        double r11316899 = r11316892 - r11316898;
        return r11316899;
}
↓
double f(double wj, double x) {
        double r11316900 = wj;
        double r11316901 = r11316900 * r11316900;
        double r11316902 = x;
        double r11316903 = r11316900 + r11316900;
        double r11316904 = r11316902 * r11316903;
        double r11316905 = r11316901 - r11316904;
        double r11316906 = r11316905 + r11316902;
        return r11316906;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.9
Target13.3
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto \left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x\]

Reproduce

herbie shell --seed 2019162 
(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))))))