Average Error: 13.3 → 1.6
Time: 37.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.4491834582195234 \cdot 10^{-17}:\\ \;\;\;\;x + \left(wj - 2 \cdot x\right) \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.4491834582195234 \cdot 10^{-17}:\\
\;\;\;\;x + \left(wj - 2 \cdot x\right) \cdot wj\\

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

\end{array}
double f(double wj, double x) {
        double r8034552 = wj;
        double r8034553 = exp(r8034552);
        double r8034554 = r8034552 * r8034553;
        double r8034555 = x;
        double r8034556 = r8034554 - r8034555;
        double r8034557 = r8034553 + r8034554;
        double r8034558 = r8034556 / r8034557;
        double r8034559 = r8034552 - r8034558;
        return r8034559;
}
↓
double f(double wj, double x) {
        double r8034560 = wj;
        double r8034561 = exp(r8034560);
        double r8034562 = r8034560 * r8034561;
        double r8034563 = x;
        double r8034564 = r8034562 - r8034563;
        double r8034565 = r8034561 + r8034562;
        double r8034566 = r8034564 / r8034565;
        double r8034567 = r8034560 - r8034566;
        double r8034568 = 1.4491834582195234e-17;
        bool r8034569 = r8034567 <= r8034568;
        double r8034570 = 2.0;
        double r8034571 = r8034570 * r8034563;
        double r8034572 = r8034560 - r8034571;
        double r8034573 = r8034572 * r8034560;
        double r8034574 = r8034563 + r8034573;
        double r8034575 = r8034563 / r8034565;
        double r8034576 = r8034562 / r8034565;
        double r8034577 = r8034560 - r8034576;
        double r8034578 = r8034575 + r8034577;
        double r8034579 = r8034569 ? r8034574 : r8034578;
        return r8034579;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target12.6
Herbie1.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 1.4491834582195234e-17

    1. Initial program 17.7

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

    2. Taylor expanded around 0 1.0

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

    3. Simplified1.0

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

    if 1.4491834582195234e-17 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 3.0

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

    3. Applied div-sub3.0

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

    4. Applied associate--r-3.0

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.4491834582195234 \cdot 10^{-17}:\\ \;\;\;\;x + \left(wj - 2 \cdot x\right) \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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))))))