Average Error: 13.9 → 1.1
Time: 11.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.508407581583140375763293699815092985261 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.508407581583140375763293699815092985261 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r318875 = wj;
        double r318876 = exp(r318875);
        double r318877 = r318875 * r318876;
        double r318878 = x;
        double r318879 = r318877 - r318878;
        double r318880 = r318876 + r318877;
        double r318881 = r318879 / r318880;
        double r318882 = r318875 - r318881;
        return r318882;
}
↓
double f(double wj, double x) {
        double r318883 = wj;
        double r318884 = 9.50840758158314e-09;
        bool r318885 = r318883 <= r318884;
        double r318886 = x;
        double r318887 = 2.0;
        double r318888 = pow(r318883, r318887);
        double r318889 = r318886 + r318888;
        double r318890 = r318883 * r318886;
        double r318891 = r318887 * r318890;
        double r318892 = r318889 - r318891;
        double r318893 = exp(r318883);
        double r318894 = 1.0;
        double r318895 = r318883 + r318894;
        double r318896 = r318893 * r318895;
        double r318897 = r318886 / r318896;
        double r318898 = r318897 + r318883;
        double r318899 = r318883 / r318895;
        double r318900 = r318898 - r318899;
        double r318901 = r318885 ? r318892 : r318900;
        return r318901;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.3
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.50840758158314e-09

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded around 0 1.0

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

    if 9.50840758158314e-09 < wj

    1. Initial program 24.1

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

    2. Simplified2.6

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

    4. Applied div-inv2.6

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

    5. Applied associate-/l*2.6

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

    6. Simplified2.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.508407581583140375763293699815092985261 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))))))