Average Error: 13.9 → 1.0
Time: 23.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\

\end{array}
double f(double wj, double x) {
        double r225222 = wj;
        double r225223 = exp(r225222);
        double r225224 = r225222 * r225223;
        double r225225 = x;
        double r225226 = r225224 - r225225;
        double r225227 = r225223 + r225224;
        double r225228 = r225226 / r225227;
        double r225229 = r225222 - r225228;
        return r225229;
}
↓
double f(double wj, double x) {
        double r225230 = wj;
        double r225231 = 8.133938014656618e-09;
        bool r225232 = r225230 <= r225231;
        double r225233 = x;
        double r225234 = 2.0;
        double r225235 = r225233 * r225234;
        double r225236 = r225230 - r225235;
        double r225237 = r225230 * r225236;
        double r225238 = r225233 + r225237;
        double r225239 = exp(r225230);
        double r225240 = sqrt(r225239);
        double r225241 = r225233 / r225240;
        double r225242 = r225241 / r225240;
        double r225243 = r225230 - r225242;
        double r225244 = 1.0;
        double r225245 = r225230 + r225244;
        double r225246 = r225243 / r225245;
        double r225247 = r225230 - r225246;
        double r225248 = r225232 ? r225238 : r225247;
        return r225248;
}

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
Target13.2
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 < 8.133938014656618e-09

    1. Initial program 13.5

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

    2. Simplified13.5

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

    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 8.133938014656618e-09 < wj

    1. Initial program 27.6

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

    2. Simplified2.3

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

    4. Applied add-sqr-sqrt2.4

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

    5. Applied associate-/r*2.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))))