Average Error: 13.3 → 1.1
Time: 23.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left({wj}^{4} + {wj}^{2}\right) + \left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left({wj}^{4} + {wj}^{2}\right) + \left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} - {wj}^{3}\right)
double f(double wj, double x) {
        double r137718 = wj;
        double r137719 = exp(r137718);
        double r137720 = r137718 * r137719;
        double r137721 = x;
        double r137722 = r137720 - r137721;
        double r137723 = r137719 + r137720;
        double r137724 = r137722 / r137723;
        double r137725 = r137718 - r137724;
        return r137725;
}
↓
double f(double wj, double x) {
        double r137726 = wj;
        double r137727 = 4.0;
        double r137728 = pow(r137726, r137727);
        double r137729 = 2.0;
        double r137730 = pow(r137726, r137729);
        double r137731 = r137728 + r137730;
        double r137732 = x;
        double r137733 = 1.0;
        double r137734 = r137733 + r137726;
        double r137735 = exp(r137726);
        double r137736 = r137734 * r137735;
        double r137737 = r137732 / r137736;
        double r137738 = 3.0;
        double r137739 = pow(r137726, r137738);
        double r137740 = r137737 - r137739;
        double r137741 = r137731 + r137740;
        return r137741;
}

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

Original13.3
Target12.7
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.3

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

  3. Applied div-sub13.3

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

  4. Applied associate--r-7.0

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

  5. Simplified6.4

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

  6. Taylor expanded around 0 1.1

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

  8. Applied sub-neg1.1

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

  9. Applied associate-+l+1.1

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

  10. Simplified1.1

    \[\leadsto \left({wj}^{4} + {wj}^{2}\right) + \color{blue}{\left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} - {wj}^{3}\right)}\]

  11. Final simplification1.1

    \[\leadsto \left({wj}^{4} + {wj}^{2}\right) + \left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2019322 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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