Average Error: 13.8 → 1.4
Time: 6.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -1.1501535783428499 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{x}{wj \cdot wj - 1} \cdot \frac{wj - 1}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\frac{wj + 1}{\sqrt[3]{wj}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -1.1501535783428499 \cdot 10^{-8}:\\
\;\;\;\;\left(\frac{x}{wj \cdot wj - 1} \cdot \frac{wj - 1}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\frac{wj + 1}{\sqrt[3]{wj}}}\\

\mathbf{else}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r228856 = wj;
        double r228857 = exp(r228856);
        double r228858 = r228856 * r228857;
        double r228859 = x;
        double r228860 = r228858 - r228859;
        double r228861 = r228857 + r228858;
        double r228862 = r228860 / r228861;
        double r228863 = r228856 - r228862;
        return r228863;
}
↓
double f(double wj, double x) {
        double r228864 = wj;
        double r228865 = -1.1501535783428499e-08;
        bool r228866 = r228864 <= r228865;
        double r228867 = x;
        double r228868 = r228864 * r228864;
        double r228869 = 1.0;
        double r228870 = r228868 - r228869;
        double r228871 = r228867 / r228870;
        double r228872 = r228864 - r228869;
        double r228873 = exp(r228864);
        double r228874 = r228872 / r228873;
        double r228875 = r228871 * r228874;
        double r228876 = r228875 + r228864;
        double r228877 = cbrt(r228864);
        double r228878 = r228877 * r228877;
        double r228879 = r228864 + r228869;
        double r228880 = r228879 / r228877;
        double r228881 = r228878 / r228880;
        double r228882 = r228876 - r228881;
        double r228883 = 2.0;
        double r228884 = pow(r228864, r228883);
        double r228885 = r228867 + r228884;
        double r228886 = r228864 * r228867;
        double r228887 = r228883 * r228886;
        double r228888 = r228885 - r228887;
        double r228889 = r228866 ? r228882 : r228888;
        return r228889;
}

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
Target13.3
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -1.1501535783428499e-08

    1. Initial program 4.4

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

    2. Simplified4.3

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

    4. Applied *-un-lft-identity4.3

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

    5. Applied flip-+4.4

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

    6. Applied associate-/r/4.3

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

    7. Applied times-frac4.4

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

    8. Simplified4.4

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

    10. Applied add-cube-cbrt5.3

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

    11. Applied associate-/l*5.3

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

    if -1.1501535783428499e-08 < wj

    1. Initial program 14.0

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 1.3

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -1.1501535783428499 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{x}{wj \cdot wj - 1} \cdot \frac{wj - 1}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\frac{wj + 1}{\sqrt[3]{wj}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))))