Average Error: 13.3 → 2.2
Time: 24.1s
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 r10928959 = wj;
        double r10928960 = exp(r10928959);
        double r10928961 = r10928959 * r10928960;
        double r10928962 = x;
        double r10928963 = r10928961 - r10928962;
        double r10928964 = r10928960 + r10928961;
        double r10928965 = r10928963 / r10928964;
        double r10928966 = r10928959 - r10928965;
        return r10928966;
}
↓
double f(double wj, double x) {
        double r10928967 = wj;
        double r10928968 = r10928967 * r10928967;
        double r10928969 = x;
        double r10928970 = r10928968 + r10928969;
        double r10928971 = r10928967 + r10928967;
        double r10928972 = r10928971 * r10928969;
        double r10928973 = r10928970 - r10928972;
        return r10928973;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.3
Target12.7
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.3

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

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

Reproduce

herbie shell --seed 2019158 
(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))))))