Average Error: 13.9 → 1.6
Time: 22.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.072499520511824 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - x \cdot \left(wj + wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{wj}, \sqrt{wj}, -\frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 3.072499520511824 \cdot 10^{-09}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - x \cdot \left(wj + wj\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{wj}, \sqrt{wj}, -\frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}}\right)\\

\end{array}
double f(double wj, double x) {
        double r11313727 = wj;
        double r11313728 = exp(r11313727);
        double r11313729 = r11313727 * r11313728;
        double r11313730 = x;
        double r11313731 = r11313729 - r11313730;
        double r11313732 = r11313728 + r11313729;
        double r11313733 = r11313731 / r11313732;
        double r11313734 = r11313727 - r11313733;
        return r11313734;
}
↓
double f(double wj, double x) {
        double r11313735 = wj;
        double r11313736 = 3.072499520511824e-09;
        bool r11313737 = r11313735 <= r11313736;
        double r11313738 = x;
        double r11313739 = fma(r11313735, r11313735, r11313738);
        double r11313740 = r11313735 + r11313735;
        double r11313741 = r11313738 * r11313740;
        double r11313742 = r11313739 - r11313741;
        double r11313743 = sqrt(r11313735);
        double r11313744 = exp(r11313735);
        double r11313745 = r11313744 * r11313735;
        double r11313746 = r11313745 - r11313738;
        double r11313747 = r11313745 + r11313744;
        double r11313748 = r11313746 / r11313747;
        double r11313749 = -r11313748;
        double r11313750 = fma(r11313743, r11313743, r11313749);
        double r11313751 = r11313737 ? r11313742 : r11313750;
        return r11313751;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.2
Herbie1.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 3.072499520511824e-09

    1. Initial program 13.6

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.8

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

    if 3.072499520511824e-09 < wj

    1. Initial program 25.1

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

    3. Applied add-sqr-sqrt25.4

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

    4. Applied fma-neg25.4

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.072499520511824 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - x \cdot \left(wj + wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{wj}, \sqrt{wj}, -\frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))))))