Average Error: 13.4 → 1.5
Time: 29.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{else}:\\
\;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r140528 = wj;
        double r140529 = exp(r140528);
        double r140530 = r140528 * r140529;
        double r140531 = x;
        double r140532 = r140530 - r140531;
        double r140533 = r140529 + r140530;
        double r140534 = r140532 / r140533;
        double r140535 = r140528 - r140534;
        return r140535;
}
↓
double f(double wj, double x) {
        double r140536 = wj;
        double r140537 = -5.00771993997508e-09;
        bool r140538 = r140536 <= r140537;
        double r140539 = exp(r140536);
        double r140540 = r140536 * r140539;
        double r140541 = x;
        double r140542 = r140540 - r140541;
        double r140543 = r140539 + r140540;
        double r140544 = r140542 / r140543;
        double r140545 = r140536 - r140544;
        double r140546 = 2.0;
        double r140547 = r140546 * r140541;
        double r140548 = r140536 - r140547;
        double r140549 = r140536 * r140548;
        double r140550 = r140541 + r140549;
        double r140551 = r140538 ? r140545 : r140550;
        return r140551;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
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 < -5.00771993997508e-09

    1. Initial program 5.8

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

    if -5.00771993997508e-09 < wj

    1. Initial program 13.6

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

    2. Simplified12.9

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

    3. Taylor expanded around 0 1.4

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

    4. Simplified1.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))