Average Error: 14.0 → 0.9
Time: 49.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.3996584940110106 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 + wj}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.3996584940110106 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\

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

\end{array}
double f(double wj, double x) {
        double r50520451 = wj;
        double r50520452 = exp(r50520451);
        double r50520453 = r50520451 * r50520452;
        double r50520454 = x;
        double r50520455 = r50520453 - r50520454;
        double r50520456 = r50520452 + r50520453;
        double r50520457 = r50520455 / r50520456;
        double r50520458 = r50520451 - r50520457;
        return r50520458;
}
↓
double f(double wj, double x) {
        double r50520459 = wj;
        double r50520460 = 4.3996584940110106e-09;
        bool r50520461 = r50520459 <= r50520460;
        double r50520462 = x;
        double r50520463 = -2.0;
        double r50520464 = r50520462 * r50520463;
        double r50520465 = r50520464 + r50520459;
        double r50520466 = r50520459 * r50520465;
        double r50520467 = r50520462 + r50520466;
        double r50520468 = exp(r50520459);
        double r50520469 = r50520462 / r50520468;
        double r50520470 = r50520459 - r50520469;
        double r50520471 = 1.0;
        double r50520472 = r50520471 + r50520459;
        double r50520473 = r50520470 / r50520472;
        double r50520474 = r50520459 - r50520473;
        double r50520475 = r50520461 ? r50520467 : r50520474;
        return r50520475;
}

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
Target13.4
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 < 4.3996584940110106e-09

    1. Initial program 13.7

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.8

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

    if 4.3996584940110106e-09 < wj

    1. Initial program 25.5

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

    3. Applied *-un-lft-identity25.5

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

    4. Applied distribute-rgt-out25.6

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

    5. Applied associate-/r*25.6

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

    6. Simplified2.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.3996584940110106 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 + wj}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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