Average Error: 14.1 → 2.1
Time: 18.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)
double f(double wj, double x) {
        double r356496 = wj;
        double r356497 = exp(r356496);
        double r356498 = r356496 * r356497;
        double r356499 = x;
        double r356500 = r356498 - r356499;
        double r356501 = r356497 + r356498;
        double r356502 = r356500 / r356501;
        double r356503 = r356496 - r356502;
        return r356503;
}
↓
double f(double wj, double x) {
        double r356504 = x;
        double r356505 = wj;
        double r356506 = 2.0;
        double r356507 = pow(r356505, r356506);
        double r356508 = r356504 + r356507;
        double r356509 = r356505 * r356504;
        double r356510 = r356506 * r356509;
        double r356511 = r356508 - r356510;
        return r356511;
}

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
Target13.6
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

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

  2. Simplified13.6

    \[\leadsto \color{blue}{wj - \frac{\frac{wj}{1} - \frac{x}{e^{wj}}}{1 + wj}}\]

  3. Taylor expanded around 0 2.1

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019198 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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