Average Error: 13.9 → 1.0
Time: 25.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\

\end{array}
double f(double wj, double x) {
        double r127764 = wj;
        double r127765 = exp(r127764);
        double r127766 = r127764 * r127765;
        double r127767 = x;
        double r127768 = r127766 - r127767;
        double r127769 = r127765 + r127766;
        double r127770 = r127768 / r127769;
        double r127771 = r127764 - r127770;
        return r127771;
}
↓
double f(double wj, double x) {
        double r127772 = wj;
        double r127773 = 8.133938014656618e-09;
        bool r127774 = r127772 <= r127773;
        double r127775 = x;
        double r127776 = r127772 * r127775;
        double r127777 = -2.0;
        double r127778 = fma(r127772, r127772, r127775);
        double r127779 = fma(r127776, r127777, r127778);
        double r127780 = exp(r127772);
        double r127781 = sqrt(r127780);
        double r127782 = r127775 / r127781;
        double r127783 = r127782 / r127781;
        double r127784 = r127772 - r127783;
        double r127785 = 1.0;
        double r127786 = r127785 + r127772;
        double r127787 = r127784 / r127786;
        double r127788 = r127772 - r127787;
        double r127789 = r127774 ? r127779 : r127788;
        return r127789;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.2
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.133938014656618e-09

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 8.133938014656618e-09 < wj

    1. Initial program 27.6

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

    2. Simplified2.3

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

    4. Applied add-sqr-sqrt2.4

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

    5. Applied associate-/r*2.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))))