Average Error: 14.0 → 1.1
Time: 5.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 2.1610528876044702 \cdot 10^{-5}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 2.1610528876044702 \cdot 10^{-5}:\\
\;\;\;\;\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 r269619 = wj;
        double r269620 = exp(r269619);
        double r269621 = r269619 * r269620;
        double r269622 = x;
        double r269623 = r269621 - r269622;
        double r269624 = r269620 + r269621;
        double r269625 = r269623 / r269624;
        double r269626 = r269619 - r269625;
        return r269626;
}
↓
double f(double wj, double x) {
        double r269627 = wj;
        double r269628 = 2.1610528876044702e-05;
        bool r269629 = r269627 <= r269628;
        double r269630 = x;
        double r269631 = 2.0;
        double r269632 = pow(r269627, r269631);
        double r269633 = r269630 + r269632;
        double r269634 = r269627 * r269630;
        double r269635 = r269631 * r269634;
        double r269636 = r269633 - r269635;
        double r269637 = 1.0;
        double r269638 = r269627 + r269637;
        double r269639 = r269630 / r269638;
        double r269640 = exp(r269627);
        double r269641 = r269639 / r269640;
        double r269642 = r269641 + r269627;
        double r269643 = r269642 * r269642;
        double r269644 = r269627 / r269638;
        double r269645 = r269644 * r269644;
        double r269646 = r269643 - r269645;
        double r269647 = r269642 + r269644;
        double r269648 = r269646 / r269647;
        double r269649 = r269629 ? r269636 : r269648;
        return r269649;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.4
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 2.1610528876044702e-05

    1. Initial program 13.7

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

    2. Simplified13.7

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

    3. Taylor expanded around 0 0.9

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

    if 2.1610528876044702e-05 < wj

    1. Initial program 27.8

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

    2. Simplified1.6

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

    4. Applied flip--9.9

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 2.1610528876044702 \cdot 10^{-5}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))))))