Average Error: 13.7 → 1.1
Time: 23.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}
double f(double wj, double x) {
        double r6938012 = wj;
        double r6938013 = exp(r6938012);
        double r6938014 = r6938012 * r6938013;
        double r6938015 = x;
        double r6938016 = r6938014 - r6938015;
        double r6938017 = r6938013 + r6938014;
        double r6938018 = r6938016 / r6938017;
        double r6938019 = r6938012 - r6938018;
        return r6938019;
}
↓
double f(double wj, double x) {
        double r6938020 = wj;
        double r6938021 = r6938020 * r6938020;
        double r6938022 = r6938021 - r6938020;
        double r6938023 = r6938022 * r6938021;
        double r6938024 = r6938021 + r6938023;
        double r6938025 = x;
        double r6938026 = exp(r6938020);
        double r6938027 = r6938020 * r6938026;
        double r6938028 = r6938027 + r6938026;
        double r6938029 = r6938025 / r6938028;
        double r6938030 = r6938024 + r6938029;
        return r6938030;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.7

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

  4. Applied associate--r-7.7

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

  5. Taylor expanded around 0 1.1

    \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]

  6. Simplified1.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2019152 
(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))))))