Average Error: 14.1 → 2.0
Time: 26.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2
double f(double wj, double x) {
        double r10214302 = wj;
        double r10214303 = exp(r10214302);
        double r10214304 = r10214302 * r10214303;
        double r10214305 = x;
        double r10214306 = r10214304 - r10214305;
        double r10214307 = r10214303 + r10214304;
        double r10214308 = r10214306 / r10214307;
        double r10214309 = r10214302 - r10214308;
        return r10214309;
}
↓
double f(double wj, double x) {
        double r10214310 = x;
        double r10214311 = wj;
        double r10214312 = r10214311 * r10214311;
        double r10214313 = r10214310 + r10214312;
        double r10214314 = r10214311 * r10214310;
        double r10214315 = -2.0;
        double r10214316 = r10214314 * r10214315;
        double r10214317 = r10214313 + r10214316;
        return r10214317;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.1
Target13.5
Herbie2.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

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

  2. Taylor expanded around 0 2.0

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

  3. Simplified2.0

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

  4. Final simplification2.0

    \[\leadsto \left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]

Reproduce

herbie shell --seed 2019138 
(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))))))