Average Error: 13.4 → 1.5
Time: 29.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

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

\end{array}
double f(double wj, double x) {
        double r421081 = wj;
        double r421082 = exp(r421081);
        double r421083 = r421081 * r421082;
        double r421084 = x;
        double r421085 = r421083 - r421084;
        double r421086 = r421082 + r421083;
        double r421087 = r421085 / r421086;
        double r421088 = r421081 - r421087;
        return r421088;
}
↓
double f(double wj, double x) {
        double r421089 = wj;
        double r421090 = -5.00771993997508e-09;
        bool r421091 = r421089 <= r421090;
        double r421092 = exp(r421089);
        double r421093 = r421089 * r421092;
        double r421094 = x;
        double r421095 = r421093 - r421094;
        double r421096 = r421092 + r421093;
        double r421097 = r421095 / r421096;
        double r421098 = r421089 - r421097;
        double r421099 = 2.0;
        double r421100 = r421099 * r421094;
        double r421101 = r421089 - r421100;
        double r421102 = r421089 * r421101;
        double r421103 = r421094 + r421102;
        double r421104 = r421091 ? r421098 : r421103;
        return r421104;
}

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
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -5.00771993997508e-09

    1. Initial program 5.8

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

    if -5.00771993997508e-09 < wj

    1. Initial program 13.6

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

    2. Simplified12.9

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

    3. Taylor expanded around 0 1.4

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

    4. Simplified1.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))