Average Error: 14.1 → 1.0
Time: 25.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 5.620068417277854487526887316151819051768 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{{wj}^{2} - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 5.620068417277854487526887316151819051768 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{{wj}^{2} - 1} \cdot \left(wj - 1\right)\\

\end{array}
double f(double wj, double x) {
        double r174279 = wj;
        double r174280 = exp(r174279);
        double r174281 = r174279 * r174280;
        double r174282 = x;
        double r174283 = r174281 - r174282;
        double r174284 = r174280 + r174281;
        double r174285 = r174283 / r174284;
        double r174286 = r174279 - r174285;
        return r174286;
}
↓
double f(double wj, double x) {
        double r174287 = wj;
        double r174288 = 5.6200684172778545e-09;
        bool r174289 = r174287 <= r174288;
        double r174290 = x;
        double r174291 = 2.0;
        double r174292 = r174290 * r174291;
        double r174293 = r174287 - r174292;
        double r174294 = r174287 * r174293;
        double r174295 = r174290 + r174294;
        double r174296 = exp(r174287);
        double r174297 = r174290 / r174296;
        double r174298 = r174287 - r174297;
        double r174299 = pow(r174287, r174291);
        double r174300 = 1.0;
        double r174301 = r174299 - r174300;
        double r174302 = r174298 / r174301;
        double r174303 = r174287 - r174300;
        double r174304 = r174302 * r174303;
        double r174305 = r174287 - r174304;
        double r174306 = r174289 ? r174295 : r174305;
        return r174306;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.6
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 < 5.6200684172778545e-09

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 5.6200684172778545e-09 < wj

    1. Initial program 23.0

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

    2. Simplified3.1

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

    4. Applied flip-+3.2

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

    5. Applied associate-/r/3.1

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

    6. Simplified3.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 5.620068417277854487526887316151819051768 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{{wj}^{2} - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))))