Average Error: 13.8 → 2.0
Time: 22.6s
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 r13120694 = wj;
        double r13120695 = exp(r13120694);
        double r13120696 = r13120694 * r13120695;
        double r13120697 = x;
        double r13120698 = r13120696 - r13120697;
        double r13120699 = r13120695 + r13120696;
        double r13120700 = r13120698 / r13120699;
        double r13120701 = r13120694 - r13120700;
        return r13120701;
}
↓
double f(double wj, double x) {
        double r13120702 = x;
        double r13120703 = wj;
        double r13120704 = r13120703 * r13120703;
        double r13120705 = r13120702 + r13120704;
        double r13120706 = r13120703 * r13120702;
        double r13120707 = -2.0;
        double r13120708 = r13120706 * r13120707;
        double r13120709 = r13120705 + r13120708;
        return r13120709;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.2
Herbie2.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Taylor expanded around 0 2.0

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

  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 2019192 
(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))))))