Average Error: 13.8 → 0.5
Time: 6.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 21.9702282043939157:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}\right) \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 21.9702282043939157:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}\right) \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\

\end{array}
double f(double wj, double x) {
        double r289679 = wj;
        double r289680 = exp(r289679);
        double r289681 = r289679 * r289680;
        double r289682 = x;
        double r289683 = r289681 - r289682;
        double r289684 = r289680 + r289681;
        double r289685 = r289683 / r289684;
        double r289686 = r289679 - r289685;
        return r289686;
}
↓
double f(double wj, double x) {
        double r289687 = wj;
        double r289688 = 21.970228204393916;
        bool r289689 = r289687 <= r289688;
        double r289690 = x;
        double r289691 = 1.0;
        double r289692 = r289687 + r289691;
        double r289693 = r289690 / r289692;
        double r289694 = exp(r289687);
        double r289695 = r289693 / r289694;
        double r289696 = 4.0;
        double r289697 = pow(r289687, r289696);
        double r289698 = 2.0;
        double r289699 = pow(r289687, r289698);
        double r289700 = r289697 + r289699;
        double r289701 = 3.0;
        double r289702 = pow(r289687, r289701);
        double r289703 = r289700 - r289702;
        double r289704 = r289695 + r289703;
        double r289705 = cbrt(r289693);
        double r289706 = r289705 * r289705;
        double r289707 = r289705 / r289694;
        double r289708 = r289706 * r289707;
        double r289709 = r289687 / r289692;
        double r289710 = r289687 - r289709;
        double r289711 = r289708 + r289710;
        double r289712 = r289689 ? r289704 : r289711;
        return r289712;
}

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
Herbie0.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 21.970228204393916

    1. Initial program 13.3

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

    2. Simplified13.3

      \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
    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 0.5

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

    if 21.970228204393916 < wj

    1. Initial program 50.7

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

    2. Simplified0.0

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

    4. Applied associate--l+0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied add-cube-cbrt0.1

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

    8. Applied times-frac0.1

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

    9. Simplified0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}\right)} \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 21.9702282043939157:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}\right) \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))))