Average Error: 13.8 → 2.1
Time: 18.2s
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 r199813 = wj;
        double r199814 = exp(r199813);
        double r199815 = r199813 * r199814;
        double r199816 = x;
        double r199817 = r199815 - r199816;
        double r199818 = r199814 + r199815;
        double r199819 = r199817 / r199818;
        double r199820 = r199813 - r199819;
        return r199820;
}
↓
double f(double wj, double x) {
        double r199821 = x;
        double r199822 = wj;
        double r199823 = r199822 * r199822;
        double r199824 = r199821 + r199823;
        double r199825 = -2.0;
        double r199826 = r199821 * r199822;
        double r199827 = r199825 * r199826;
        double r199828 = r199824 + r199827;
        return r199828;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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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))))))