Average Error: 13.2 → 2.0
Time: 49.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + x\right) - \left(wj + wj\right) \cdot x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + x\right) - \left(wj + wj\right) \cdot x
double f(double wj, double x) {
        double r6796518 = wj;
        double r6796519 = exp(r6796518);
        double r6796520 = r6796518 * r6796519;
        double r6796521 = x;
        double r6796522 = r6796520 - r6796521;
        double r6796523 = r6796519 + r6796520;
        double r6796524 = r6796522 / r6796523;
        double r6796525 = r6796518 - r6796524;
        return r6796525;
}
↓
double f(double wj, double x) {
        double r6796526 = wj;
        double r6796527 = r6796526 * r6796526;
        double r6796528 = x;
        double r6796529 = r6796527 + r6796528;
        double r6796530 = r6796526 + r6796526;
        double r6796531 = r6796530 * r6796528;
        double r6796532 = r6796529 - r6796531;
        return r6796532;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.2
Target12.6
Herbie2.0
\[wj - \left(\frac{wj}{wj + 1.0} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.2

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

  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

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

  4. Final simplification2.0

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

Reproduce

herbie shell --seed 2019165 
(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))))))