Average Error: 14.0 → 1.0
Time: 26.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.976401343690175460051364946315044512914 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{\sqrt{wj + 1}} \cdot \frac{wj - \frac{x}{e^{wj}}}{\sqrt{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.976401343690175460051364946315044512914 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{1}{\sqrt{wj + 1}} \cdot \frac{wj - \frac{x}{e^{wj}}}{\sqrt{wj + 1}}\\

\end{array}
double f(double wj, double x) {
        double r142775 = wj;
        double r142776 = exp(r142775);
        double r142777 = r142775 * r142776;
        double r142778 = x;
        double r142779 = r142777 - r142778;
        double r142780 = r142776 + r142777;
        double r142781 = r142779 / r142780;
        double r142782 = r142775 - r142781;
        return r142782;
}
↓
double f(double wj, double x) {
        double r142783 = wj;
        double r142784 = 8.976401343690175e-09;
        bool r142785 = r142783 <= r142784;
        double r142786 = x;
        double r142787 = 2.0;
        double r142788 = r142787 * r142786;
        double r142789 = r142783 - r142788;
        double r142790 = r142783 * r142789;
        double r142791 = r142786 + r142790;
        double r142792 = 1.0;
        double r142793 = r142783 + r142792;
        double r142794 = sqrt(r142793);
        double r142795 = r142792 / r142794;
        double r142796 = exp(r142783);
        double r142797 = r142786 / r142796;
        double r142798 = r142783 - r142797;
        double r142799 = r142798 / r142794;
        double r142800 = r142795 * r142799;
        double r142801 = r142783 - r142800;
        double r142802 = r142785 ? r142791 : r142801;
        return r142802;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.4
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.976401343690175e-09

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded around 0 0.8

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

    4. Simplified0.9

      \[\leadsto \color{blue}{x + wj \cdot \left(wj - 2 \cdot x\right)}\]

    if 8.976401343690175e-09 < wj

    1. Initial program 26.6

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

    2. Simplified3.8

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

    4. Applied clear-num3.9

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

    6. Applied *-un-lft-identity3.9

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

    7. Applied add-sqr-sqrt4.3

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

    8. Applied times-frac4.2

      \[\leadsto wj - \frac{1}{\color{blue}{\frac{\sqrt{wj + 1}}{1} \cdot \frac{\sqrt{wj + 1}}{wj - \frac{x}{e^{wj}}}}}\]

    9. Applied add-cube-cbrt4.2

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

    10. Applied times-frac4.2

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

    11. Simplified4.2

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

    12. Simplified4.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.976401343690175460051364946315044512914 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{\sqrt{wj + 1}} \cdot \frac{wj - \frac{x}{e^{wj}}}{\sqrt{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))))