Average Error: 13.4 → 1.0
Time: 6.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 6.381542263961784 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 6.381542263961784 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r395851 = wj;
        double r395852 = exp(r395851);
        double r395853 = r395851 * r395852;
        double r395854 = x;
        double r395855 = r395853 - r395854;
        double r395856 = r395852 + r395853;
        double r395857 = r395855 / r395856;
        double r395858 = r395851 - r395857;
        return r395858;
}
↓
double f(double wj, double x) {
        double r395859 = wj;
        double r395860 = 6.381542263961784e-09;
        bool r395861 = r395859 <= r395860;
        double r395862 = x;
        double r395863 = 2.0;
        double r395864 = pow(r395859, r395863);
        double r395865 = r395862 + r395864;
        double r395866 = r395859 * r395862;
        double r395867 = r395863 * r395866;
        double r395868 = r395865 - r395867;
        double r395869 = 1.0;
        double r395870 = r395859 + r395869;
        double r395871 = r395862 / r395870;
        double r395872 = exp(r395859);
        double r395873 = r395871 / r395872;
        double r395874 = r395873 + r395859;
        double r395875 = r395859 * r395859;
        double r395876 = r395875 - r395869;
        double r395877 = r395859 / r395876;
        double r395878 = r395859 - r395869;
        double r395879 = r395877 * r395878;
        double r395880 = r395874 - r395879;
        double r395881 = r395861 ? r395868 : r395880;
        return r395881;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
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 < 6.381542263961784e-09

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 0.9

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

    if 6.381542263961784e-09 < wj

    1. Initial program 24.3

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

    2. Simplified3.2

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

    4. Applied flip-+3.4

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

    5. Applied associate-/r/3.3

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

    6. Simplified3.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.381542263961784 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))))