Average Error: 14.0 → 1.2
Time: 4.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 2.1610528876044702 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - x \cdot 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]
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}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj - x \cdot 2, 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 r249830 = wj;
        double r249831 = exp(r249830);
        double r249832 = r249830 * r249831;
        double r249833 = x;
        double r249834 = r249832 - r249833;
        double r249835 = r249831 + r249832;
        double r249836 = r249834 / r249835;
        double r249837 = r249830 - r249836;
        return r249837;
}
↓
double f(double wj, double x) {
        double r249838 = wj;
        double r249839 = 2.1610528876044702e-05;
        bool r249840 = r249838 <= r249839;
        double r249841 = x;
        double r249842 = 2.0;
        double r249843 = r249841 * r249842;
        double r249844 = r249838 - r249843;
        double r249845 = fma(r249838, r249844, r249841);
        double r249846 = 1.0;
        double r249847 = r249838 + r249846;
        double r249848 = r249841 / r249847;
        double r249849 = exp(r249838);
        double r249850 = r249848 / r249849;
        double r249851 = r249850 + r249838;
        double r249852 = r249851 * r249851;
        double r249853 = r249838 / r249847;
        double r249854 = r249853 * r249853;
        double r249855 = r249852 - r249854;
        double r249856 = r249851 + r249853;
        double r249857 = r249855 / r249856;
        double r249858 = r249840 ? r249845 : r249857;
        return r249858;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.4
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 < 2.1610528876044702e-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

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

    4. Taylor expanded around 0 0.9

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

    5. Simplified1.0

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

    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 simplification1.2

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(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))))))