Average Error: 13.7 → 1.4
Time: 27.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.444452482119928874498494494404307012143 \cdot 10^{-11}:\\ \;\;\;\;x - \left(\left(x \cdot wj\right) \cdot 2 - wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj \cdot e^{wj}}{wj \cdot e^{wj} + e^{wj}}\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.444452482119928874498494494404307012143 \cdot 10^{-11}:\\
\;\;\;\;x - \left(\left(x \cdot wj\right) \cdot 2 - wj \cdot wj\right)\\

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

\end{array}
double f(double wj, double x) {
        double r11184386 = wj;
        double r11184387 = exp(r11184386);
        double r11184388 = r11184386 * r11184387;
        double r11184389 = x;
        double r11184390 = r11184388 - r11184389;
        double r11184391 = r11184387 + r11184388;
        double r11184392 = r11184390 / r11184391;
        double r11184393 = r11184386 - r11184392;
        return r11184393;
}
↓
double f(double wj, double x) {
        double r11184394 = wj;
        double r11184395 = 1.4444524821199289e-11;
        bool r11184396 = r11184394 <= r11184395;
        double r11184397 = x;
        double r11184398 = r11184397 * r11184394;
        double r11184399 = 2.0;
        double r11184400 = r11184398 * r11184399;
        double r11184401 = r11184394 * r11184394;
        double r11184402 = r11184400 - r11184401;
        double r11184403 = r11184397 - r11184402;
        double r11184404 = exp(r11184394);
        double r11184405 = r11184394 * r11184404;
        double r11184406 = r11184405 + r11184404;
        double r11184407 = r11184405 / r11184406;
        double r11184408 = r11184394 - r11184407;
        double r11184409 = r11184397 / r11184406;
        double r11184410 = r11184408 + r11184409;
        double r11184411 = r11184396 ? r11184403 : r11184410;
        return r11184411;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.4444524821199289e-11

    1. Initial program 13.4

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if 1.4444524821199289e-11 < wj

    1. Initial program 24.2

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied div-sub24.2

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

    4. Applied associate--r-24.2

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.444452482119928874498494494404307012143 \cdot 10^{-11}:\\ \;\;\;\;x - \left(\left(x \cdot wj\right) \cdot 2 - wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj \cdot e^{wj}}{wj \cdot e^{wj} + e^{wj}}\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))