Average Error: 13.8 → 1.2
Time: 9.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 2.114553828895713897623393276448133804024 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 2.114553828895713897623393276448133804024 \cdot 10^{-6}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r261521 = wj;
        double r261522 = exp(r261521);
        double r261523 = r261521 * r261522;
        double r261524 = x;
        double r261525 = r261523 - r261524;
        double r261526 = r261522 + r261523;
        double r261527 = r261525 / r261526;
        double r261528 = r261521 - r261527;
        return r261528;
}
↓
double f(double wj, double x) {
        double r261529 = wj;
        double r261530 = 2.114553828895714e-06;
        bool r261531 = r261529 <= r261530;
        double r261532 = x;
        double r261533 = 2.0;
        double r261534 = pow(r261529, r261533);
        double r261535 = r261532 + r261534;
        double r261536 = r261529 * r261532;
        double r261537 = r261533 * r261536;
        double r261538 = r261535 - r261537;
        double r261539 = 1.0;
        double r261540 = r261529 + r261539;
        double r261541 = r261532 / r261540;
        double r261542 = exp(r261529);
        double r261543 = r261541 / r261542;
        double r261544 = r261543 + r261529;
        double r261545 = r261544 * r261544;
        double r261546 = r261529 / r261540;
        double r261547 = r261546 * r261546;
        double r261548 = r261545 - r261547;
        double r261549 = r261544 + r261546;
        double r261550 = r261548 / r261549;
        double r261551 = r261531 ? r261538 : r261550;
        return r261551;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.2
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 2.114553828895714e-06

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 1.0

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

    if 2.114553828895714e-06 < wj

    1. Initial program 26.2

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

    2. Simplified1.9

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

    4. Applied flip--10.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 2.114553828895713897623393276448133804024 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))))))