Average Error: 13.8 → 2.2
Time: 22.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + x\right) - \left(wj + wj\right) \cdot x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + x\right) - \left(wj + wj\right) \cdot x
double f(double wj, double x) {
        double r9317314 = wj;
        double r9317315 = exp(r9317314);
        double r9317316 = r9317314 * r9317315;
        double r9317317 = x;
        double r9317318 = r9317316 - r9317317;
        double r9317319 = r9317315 + r9317316;
        double r9317320 = r9317318 / r9317319;
        double r9317321 = r9317314 - r9317320;
        return r9317321;
}
↓
double f(double wj, double x) {
        double r9317322 = wj;
        double r9317323 = r9317322 * r9317322;
        double r9317324 = x;
        double r9317325 = r9317323 + r9317324;
        double r9317326 = r9317322 + r9317322;
        double r9317327 = r9317326 * r9317324;
        double r9317328 = r9317325 - r9317327;
        return r9317328;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.8
Target13.2
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  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

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

  4. Final simplification2.2

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

Reproduce

herbie shell --seed 2019164 
(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))))))