Average Error: 14.0 → 1.5
Time: 7.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.2387214544875811 \cdot 10^{-28}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 3.2387214544875811 \cdot 10^{-28}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r213229 = wj;
        double r213230 = exp(r213229);
        double r213231 = r213229 * r213230;
        double r213232 = x;
        double r213233 = r213231 - r213232;
        double r213234 = r213230 + r213231;
        double r213235 = r213233 / r213234;
        double r213236 = r213229 - r213235;
        return r213236;
}
↓
double f(double wj, double x) {
        double r213237 = wj;
        double r213238 = 3.238721454487581e-28;
        bool r213239 = r213237 <= r213238;
        double r213240 = x;
        double r213241 = 2.0;
        double r213242 = pow(r213237, r213241);
        double r213243 = r213240 + r213242;
        double r213244 = r213237 * r213240;
        double r213245 = r213241 * r213244;
        double r213246 = r213243 - r213245;
        double r213247 = 1.0;
        double r213248 = r213237 + r213247;
        double r213249 = r213240 / r213248;
        double r213250 = exp(r213237);
        double r213251 = r213249 / r213250;
        double r213252 = r213251 + r213237;
        double r213253 = r213248 / r213237;
        double r213254 = r213247 / r213253;
        double r213255 = r213252 - r213254;
        double r213256 = r213239 ? r213246 : r213255;
        return r213256;
}

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
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 < 3.238721454487581e-28

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 0.9

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

    if 3.238721454487581e-28 < wj

    1. Initial program 24.5

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

    2. Simplified11.8

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

    4. Applied clear-num11.8

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.2387214544875811 \cdot 10^{-28}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))))