Average Error: 14.0 → 0.9
Time: 26.1s
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 r225534 = wj;
        double r225535 = exp(r225534);
        double r225536 = r225534 * r225535;
        double r225537 = x;
        double r225538 = r225536 - r225537;
        double r225539 = r225535 + r225536;
        double r225540 = r225538 / r225539;
        double r225541 = r225534 - r225540;
        return r225541;
}
↓
double f(double wj, double x) {
        double r225542 = wj;
        double r225543 = 5.8018913561244524e-09;
        bool r225544 = r225542 <= r225543;
        double r225545 = x;
        double r225546 = r225542 * r225545;
        double r225547 = -2.0;
        double r225548 = fma(r225542, r225542, r225545);
        double r225549 = fma(r225546, r225547, r225548);
        double r225550 = 1.0;
        double r225551 = r225542 - r225550;
        double r225552 = exp(r225542);
        double r225553 = r225545 / r225552;
        double r225554 = r225542 - r225553;
        double r225555 = -r225542;
        double r225556 = fma(r225542, r225555, r225550);
        double r225557 = r225554 / r225556;
        double r225558 = fma(r225551, r225557, r225542);
        double r225559 = r225550 - r225542;
        double r225560 = r225551 + r225559;
        double r225561 = r225557 * r225560;
        double r225562 = r225558 + r225561;
        double r225563 = r225544 ? r225549 : r225562;
        return r225563;
}

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))))))