Average Error: 14.1 → 1.3
Time: 24.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\

\end{array}
double f(double wj, double x) {
        double r131703 = wj;
        double r131704 = exp(r131703);
        double r131705 = r131703 * r131704;
        double r131706 = x;
        double r131707 = r131705 - r131706;
        double r131708 = r131704 + r131705;
        double r131709 = r131707 / r131708;
        double r131710 = r131703 - r131709;
        return r131710;
}
↓
double f(double wj, double x) {
        double r131711 = wj;
        double r131712 = 1.4258272867532584e-07;
        bool r131713 = r131711 <= r131712;
        double r131714 = x;
        double r131715 = r131711 * r131714;
        double r131716 = -2.0;
        double r131717 = fma(r131711, r131711, r131714);
        double r131718 = fma(r131715, r131716, r131717);
        double r131719 = 3.0;
        double r131720 = pow(r131711, r131719);
        double r131721 = exp(r131711);
        double r131722 = r131714 / r131721;
        double r131723 = pow(r131722, r131719);
        double r131724 = r131720 - r131723;
        double r131725 = r131711 + r131722;
        double r131726 = r131722 * r131725;
        double r131727 = fma(r131711, r131711, r131726);
        double r131728 = 1.0;
        double r131729 = r131728 + r131711;
        double r131730 = r131727 * r131729;
        double r131731 = r131724 / r131730;
        double r131732 = r131711 - r131731;
        double r131733 = r131713 ? r131718 : r131732;
        return r131733;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.1
Target13.6
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 < 1.4258272867532584e-07

    1. Initial program 13.9

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

    2. Simplified13.9

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

    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 1.4258272867532584e-07 < wj

    1. Initial program 24.7

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

    2. Simplified2.5

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

    4. Applied flip3--15.1

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

    5. Applied associate-/l/15.3

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

    6. Simplified15.3

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))))