Average Error: 14.1 → 1.0
Time: 28.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.130902437626314746148313908347127210163 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.130902437626314746148313908347127210163 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\

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

\end{array}
double f(double wj, double x) {
        double r175589 = wj;
        double r175590 = exp(r175589);
        double r175591 = r175589 * r175590;
        double r175592 = x;
        double r175593 = r175591 - r175592;
        double r175594 = r175590 + r175591;
        double r175595 = r175593 / r175594;
        double r175596 = r175589 - r175595;
        return r175596;
}
↓
double f(double wj, double x) {
        double r175597 = wj;
        double r175598 = 9.130902437626315e-09;
        bool r175599 = r175597 <= r175598;
        double r175600 = x;
        double r175601 = 2.0;
        double r175602 = r175600 * r175601;
        double r175603 = r175597 - r175602;
        double r175604 = r175597 * r175603;
        double r175605 = r175600 + r175604;
        double r175606 = 1.0;
        double r175607 = r175597 + r175606;
        double r175608 = r175597 / r175607;
        double r175609 = r175597 - r175608;
        double r175610 = exp(r175597);
        double r175611 = r175600 / r175610;
        double r175612 = r175611 / r175607;
        double r175613 = r175609 + r175612;
        double r175614 = r175599 ? r175605 : r175613;
        return r175614;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.6
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.130902437626315e-09

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 9.130902437626315e-09 < wj

    1. Initial program 23.0

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

    2. Simplified3.1

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

    4. Applied div-sub3.1

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

    5. Applied associate--r-3.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.130902437626314746148313908347127210163 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))))