Average Error: 13.8 → 2.1
Time: 18.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + wj \cdot wj\right) + -2 \cdot \left(x \cdot wj\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + wj \cdot wj\right) + -2 \cdot \left(x \cdot wj\right)
double f(double wj, double x) {
        double r203654 = wj;
        double r203655 = exp(r203654);
        double r203656 = r203654 * r203655;
        double r203657 = x;
        double r203658 = r203656 - r203657;
        double r203659 = r203655 + r203656;
        double r203660 = r203658 / r203659;
        double r203661 = r203654 - r203660;
        return r203661;
}
↓
double f(double wj, double x) {
        double r203662 = x;
        double r203663 = wj;
        double r203664 = r203663 * r203663;
        double r203665 = r203662 + r203664;
        double r203666 = -2.0;
        double r203667 = r203662 * r203663;
        double r203668 = r203666 * r203667;
        double r203669 = r203665 + r203668;
        return r203669;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 13.8

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

  2. Simplified13.2

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

  3. Taylor expanded around 0 2.1

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

  4. Simplified2.1

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

  5. Final simplification2.1

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

Reproduce

herbie shell --seed 2019194 
(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))))))