Average Error: 14.0 → 0.9
Time: 30.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 5.769639389097466454165273082440798524395 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 5.769639389097466454165273082440798524395 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r171927 = wj;
        double r171928 = exp(r171927);
        double r171929 = r171927 * r171928;
        double r171930 = x;
        double r171931 = r171929 - r171930;
        double r171932 = r171928 + r171929;
        double r171933 = r171931 / r171932;
        double r171934 = r171927 - r171933;
        return r171934;
}
↓
double f(double wj, double x) {
        double r171935 = wj;
        double r171936 = 5.7696393890974665e-09;
        bool r171937 = r171935 <= r171936;
        double r171938 = x;
        double r171939 = 2.0;
        double r171940 = r171939 * r171938;
        double r171941 = r171935 - r171940;
        double r171942 = r171935 * r171941;
        double r171943 = r171938 + r171942;
        double r171944 = 1.0;
        double r171945 = r171935 + r171944;
        double r171946 = r171935 / r171945;
        double r171947 = r171935 - r171946;
        double r171948 = exp(r171935);
        double r171949 = r171938 / r171948;
        double r171950 = r171949 / r171945;
        double r171951 = r171947 + r171950;
        double r171952 = r171937 ? r171943 : r171951;
        return r171952;
}

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
Target13.3
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 5.7696393890974665e-09

    1. Initial program 13.7

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

    2. Simplified13.6

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

    3. Taylor expanded around 0 0.9

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

    4. Simplified0.9

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

    if 5.7696393890974665e-09 < wj

    1. Initial program 25.4

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

    2. Simplified2.3

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

    4. Applied div-sub2.3

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

    5. Applied associate--r-2.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 5.769639389097466454165273082440798524395 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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))))))