Average Error: 13.5 → 1.3
Time: 5.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left(\sqrt[3]{{wj}^{4} + {wj}^{2}} \cdot \sqrt[3]{{wj}^{4} + {wj}^{2}}\right) \cdot \sqrt[3]{{wj}^{4} + {wj}^{2}} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left(\sqrt[3]{{wj}^{4} + {wj}^{2}} \cdot \sqrt[3]{{wj}^{4} + {wj}^{2}}\right) \cdot \sqrt[3]{{wj}^{4} + {wj}^{2}} - {wj}^{3}\right)
double f(double wj, double x) {
        double r281682 = wj;
        double r281683 = exp(r281682);
        double r281684 = r281682 * r281683;
        double r281685 = x;
        double r281686 = r281684 - r281685;
        double r281687 = r281683 + r281684;
        double r281688 = r281686 / r281687;
        double r281689 = r281682 - r281688;
        return r281689;
}
↓
double f(double wj, double x) {
        double r281690 = x;
        double r281691 = wj;
        double r281692 = 1.0;
        double r281693 = r281691 + r281692;
        double r281694 = r281690 / r281693;
        double r281695 = exp(r281691);
        double r281696 = r281694 / r281695;
        double r281697 = 4.0;
        double r281698 = pow(r281691, r281697);
        double r281699 = 2.0;
        double r281700 = pow(r281691, r281699);
        double r281701 = r281698 + r281700;
        double r281702 = cbrt(r281701);
        double r281703 = r281702 * r281702;
        double r281704 = r281703 * r281702;
        double r281705 = 3.0;
        double r281706 = pow(r281691, r281705);
        double r281707 = r281704 - r281706;
        double r281708 = r281696 + r281707;
        return r281708;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target12.9
Herbie1.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.9

    \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
  3. Using strategy rm

  4. Applied associate--l+6.6

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

  5. Taylor expanded around 0 1.2

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.3

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

  8. Final simplification1.3

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

Reproduce

herbie shell --seed 2019354 
(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))))))