Average Error: 13.5 → 0.4
Time: 25.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -6.6836729835936655 \cdot 10^{-09}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{1 + wj}\\ \mathbf{elif}\;wj \le 6.3842652986747504 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{1 + wj}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -6.6836729835936655 \cdot 10^{-09}:\\
\;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{1 + wj}\\

\mathbf{elif}\;wj \le 6.3842652986747504 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\

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

\end{array}
double f(double wj, double x) {
        double r55209571 = wj;
        double r55209572 = exp(r55209571);
        double r55209573 = r55209571 * r55209572;
        double r55209574 = x;
        double r55209575 = r55209573 - r55209574;
        double r55209576 = r55209572 + r55209573;
        double r55209577 = r55209575 / r55209576;
        double r55209578 = r55209571 - r55209577;
        return r55209578;
}
↓
double f(double wj, double x) {
        double r55209579 = wj;
        double r55209580 = -6.6836729835936655e-09;
        bool r55209581 = r55209579 <= r55209580;
        double r55209582 = x;
        double r55209583 = exp(r55209579);
        double r55209584 = r55209582 / r55209583;
        double r55209585 = r55209579 - r55209584;
        double r55209586 = 1.0;
        double r55209587 = r55209586 + r55209579;
        double r55209588 = r55209586 / r55209587;
        double r55209589 = r55209585 * r55209588;
        double r55209590 = r55209579 - r55209589;
        double r55209591 = 6.3842652986747504e-09;
        bool r55209592 = r55209579 <= r55209591;
        double r55209593 = -2.0;
        double r55209594 = r55209593 * r55209582;
        double r55209595 = r55209594 + r55209579;
        double r55209596 = r55209579 * r55209595;
        double r55209597 = r55209582 + r55209596;
        double r55209598 = r55209592 ? r55209597 : r55209590;
        double r55209599 = r55209581 ? r55209590 : r55209598;
        return r55209599;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target13.0
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -6.6836729835936655e-09 or 6.3842652986747504e-09 < wj

    1. Initial program 15.1

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

    3. Applied distribute-rgt1-in15.2

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

    4. Applied *-un-lft-identity15.2

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

    5. Applied times-frac15.1

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

    6. Simplified3.9

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

    if -6.6836729835936655e-09 < wj < 6.3842652986747504e-09

    1. Initial program 13.4

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -6.6836729835936655 \cdot 10^{-09}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{1 + wj}\\ \mathbf{elif}\;wj \le 6.3842652986747504 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{1 + wj}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(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))))))