Average Error: 13.2 → 2.0
Time: 20.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[wj \cdot wj + \left(x - \left(wj \cdot x\right) \cdot 2\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
wj \cdot wj + \left(x - \left(wj \cdot x\right) \cdot 2\right)
double f(double wj, double x) {
        double r10347192 = wj;
        double r10347193 = exp(r10347192);
        double r10347194 = r10347192 * r10347193;
        double r10347195 = x;
        double r10347196 = r10347194 - r10347195;
        double r10347197 = r10347193 + r10347194;
        double r10347198 = r10347196 / r10347197;
        double r10347199 = r10347192 - r10347198;
        return r10347199;
}
↓
double f(double wj, double x) {
        double r10347200 = wj;
        double r10347201 = r10347200 * r10347200;
        double r10347202 = x;
        double r10347203 = r10347200 * r10347202;
        double r10347204 = 2.0;
        double r10347205 = r10347203 * r10347204;
        double r10347206 = r10347202 - r10347205;
        double r10347207 = r10347201 + r10347206;
        return r10347207;
}

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
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

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

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

  4. Final simplification2.0

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

Reproduce

herbie shell --seed 2019165 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))