Average Error: 13.8 → 0.9
Time: 4.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r372590 = wj;
        double r372591 = exp(r372590);
        double r372592 = r372590 * r372591;
        double r372593 = x;
        double r372594 = r372592 - r372593;
        double r372595 = r372591 + r372592;
        double r372596 = r372594 / r372595;
        double r372597 = r372590 - r372596;
        return r372597;
}
↓
double f(double wj, double x) {
        double r372598 = x;
        double r372599 = wj;
        double r372600 = 1.0;
        double r372601 = r372599 + r372600;
        double r372602 = r372598 / r372601;
        double r372603 = exp(r372599);
        double r372604 = r372602 / r372603;
        double r372605 = 4.0;
        double r372606 = pow(r372599, r372605);
        double r372607 = 2.0;
        double r372608 = pow(r372599, r372607);
        double r372609 = r372606 + r372608;
        double r372610 = 3.0;
        double r372611 = pow(r372599, r372610);
        double r372612 = r372609 - r372611;
        double r372613 = r372604 + r372612;
        return r372613;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.2
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Simplified13.2

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

  4. Applied associate--l+6.9

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

  5. Taylor expanded around 0 0.9

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

  6. Final simplification0.9

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

Reproduce

herbie shell --seed 2020056 
(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))))))