Average Error: 13.4 → 1.3
Time: 4.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r338717 = wj;
        double r338718 = exp(r338717);
        double r338719 = r338717 * r338718;
        double r338720 = x;
        double r338721 = r338719 - r338720;
        double r338722 = r338718 + r338719;
        double r338723 = r338721 / r338722;
        double r338724 = r338717 - r338723;
        return r338724;
}
↓
double f(double wj, double x) {
        double r338725 = wj;
        double r338726 = 7.787217220559818e-07;
        bool r338727 = r338725 <= r338726;
        double r338728 = x;
        double r338729 = fma(r338725, r338725, r338728);
        double r338730 = 2.0;
        double r338731 = r338725 * r338728;
        double r338732 = r338730 * r338731;
        double r338733 = r338729 - r338732;
        double r338734 = 1.0;
        double r338735 = r338725 + r338734;
        double r338736 = r338728 / r338735;
        double r338737 = exp(r338725);
        double r338738 = r338736 / r338737;
        double r338739 = r338738 * r338738;
        double r338740 = r338725 * r338725;
        double r338741 = r338739 - r338740;
        double r338742 = r338741 * r338735;
        double r338743 = r338738 - r338725;
        double r338744 = r338743 * r338725;
        double r338745 = r338742 - r338744;
        double r338746 = r338743 * r338735;
        double r338747 = r338745 / r338746;
        double r338748 = r338727 ? r338733 : r338747;
        return r338748;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.7
Herbie1.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 7.787217220559818e-07

    1. Initial program 13.1

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

    2. Simplified13.0

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

    3. Taylor expanded around 0 1.0

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

    4. Taylor expanded around 0 1.0

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

    5. Simplified1.0

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

    if 7.787217220559818e-07 < wj

    1. Initial program 25.3

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

    2. Simplified2.0

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

    4. Applied flip-+11.0

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

    5. Applied frac-sub11.2

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +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))))))