Average Error: 14.0 → 0.9
Time: 25.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 5.801891356124452416919000006471224262228 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, -wj, 1\right)}, wj\right) + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, -wj, 1\right)} \cdot \left(\left(wj - 1\right) + \left(1 - wj\right)\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 5.801891356124452416919000006471224262228 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

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

\end{array}
double f(double wj, double x) {
        double r259661 = wj;
        double r259662 = exp(r259661);
        double r259663 = r259661 * r259662;
        double r259664 = x;
        double r259665 = r259663 - r259664;
        double r259666 = r259662 + r259663;
        double r259667 = r259665 / r259666;
        double r259668 = r259661 - r259667;
        return r259668;
}
↓
double f(double wj, double x) {
        double r259669 = wj;
        double r259670 = 5.8018913561244524e-09;
        bool r259671 = r259669 <= r259670;
        double r259672 = x;
        double r259673 = r259669 * r259672;
        double r259674 = -2.0;
        double r259675 = fma(r259669, r259669, r259672);
        double r259676 = fma(r259673, r259674, r259675);
        double r259677 = 1.0;
        double r259678 = r259669 - r259677;
        double r259679 = exp(r259669);
        double r259680 = r259672 / r259679;
        double r259681 = r259669 - r259680;
        double r259682 = -r259669;
        double r259683 = fma(r259669, r259682, r259677);
        double r259684 = r259681 / r259683;
        double r259685 = fma(r259678, r259684, r259669);
        double r259686 = r259677 - r259669;
        double r259687 = r259678 + r259686;
        double r259688 = r259684 * r259687;
        double r259689 = r259685 + r259688;
        double r259690 = r259671 ? r259676 : r259689;
        return r259690;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.4
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 5.8018913561244524e-09

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded around 0 0.8

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

    4. Simplified0.8

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

    if 5.8018913561244524e-09 < wj

    1. Initial program 26.5

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

    2. Simplified3.9

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

    4. Applied flip-+4.1

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

    5. Applied associate-/r/4.0

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

    6. Applied add-sqr-sqrt4.4

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

    7. Applied prod-diff4.3

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

    8. Simplified3.8

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

    9. Simplified3.8

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 5.801891356124452416919000006471224262228 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, -wj, 1\right)}, wj\right) + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, -wj, 1\right)} \cdot \left(\left(wj - 1\right) + \left(1 - wj\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))))