Average Error: 13.7 → 1.2
Time: 7.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.0018755083909558 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.0018755083909558 \cdot 10^{-9}:\\
\;\;\;\;\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}} + 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 r220693 = wj;
        double r220694 = exp(r220693);
        double r220695 = r220693 * r220694;
        double r220696 = x;
        double r220697 = r220695 - r220696;
        double r220698 = r220694 + r220695;
        double r220699 = r220697 / r220698;
        double r220700 = r220693 - r220699;
        return r220700;
}
↓
double f(double wj, double x) {
        double r220701 = wj;
        double r220702 = 1.0018755083909558e-09;
        bool r220703 = r220701 <= r220702;
        double r220704 = x;
        double r220705 = fma(r220701, r220701, r220704);
        double r220706 = 2.0;
        double r220707 = r220701 * r220704;
        double r220708 = r220706 * r220707;
        double r220709 = r220705 - r220708;
        double r220710 = 1.0;
        double r220711 = r220701 + r220710;
        double r220712 = r220704 / r220711;
        double r220713 = exp(r220701);
        double r220714 = r220712 / r220713;
        double r220715 = r220714 + r220701;
        double r220716 = r220715 * r220715;
        double r220717 = r220701 / r220711;
        double r220718 = r220717 * r220717;
        double r220719 = r220716 - r220718;
        double r220720 = r220715 + r220717;
        double r220721 = r220719 / r220720;
        double r220722 = r220703 ? r220709 : r220721;
        return r220722;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
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 < 1.0018755083909558e-09

    1. Initial program 13.4

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

    2. Simplified13.4

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

    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. Simplified0.9

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

    if 1.0018755083909558e-09 < wj

    1. Initial program 25.7

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

    2. Simplified2.8

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

    4. Applied flip--11.3

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.0018755083909558 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) + \frac{wj}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))))))