Average Error: 14.2 → 2.1
Time: 22.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[wj \cdot wj + \left(x - x \cdot \left(wj + wj\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
wj \cdot wj + \left(x - x \cdot \left(wj + wj\right)\right)
double f(double wj, double x) {
        double r9319322 = wj;
        double r9319323 = exp(r9319322);
        double r9319324 = r9319322 * r9319323;
        double r9319325 = x;
        double r9319326 = r9319324 - r9319325;
        double r9319327 = r9319323 + r9319324;
        double r9319328 = r9319326 / r9319327;
        double r9319329 = r9319322 - r9319328;
        return r9319329;
}
↓
double f(double wj, double x) {
        double r9319330 = wj;
        double r9319331 = r9319330 * r9319330;
        double r9319332 = x;
        double r9319333 = r9319330 + r9319330;
        double r9319334 = r9319332 * r9319333;
        double r9319335 = r9319332 - r9319334;
        double r9319336 = r9319331 + r9319335;
        return r9319336;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.2
Target13.6
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.2

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019171 
(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))))))