Average Error: 13.7 → 2.1
Time: 20.6s
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 r10745775 = wj;
        double r10745776 = exp(r10745775);
        double r10745777 = r10745775 * r10745776;
        double r10745778 = x;
        double r10745779 = r10745777 - r10745778;
        double r10745780 = r10745776 + r10745777;
        double r10745781 = r10745779 / r10745780;
        double r10745782 = r10745775 - r10745781;
        return r10745782;
}
↓
double f(double wj, double x) {
        double r10745783 = x;
        double r10745784 = wj;
        double r10745785 = r10745784 * r10745784;
        double r10745786 = r10745783 + r10745785;
        double r10745787 = r10745784 * r10745783;
        double r10745788 = -2.0;
        double r10745789 = r10745787 * r10745788;
        double r10745790 = r10745786 + r10745789;
        return r10745790;
}

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

Original13.7
Target13.1
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019179 
(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))))))