Average Error: 13.7 → 1.5
Time: 45.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -8.922429223447696661376437514499176950622 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(1, wj, \left(\left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{1 + wj}\right) \cdot \frac{-1}{e^{wj}}\right) + \mathsf{fma}\left(\frac{-1}{e^{wj} + e^{wj} \cdot wj}, e^{wj} \cdot wj - x, \left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{e^{wj} + e^{wj} \cdot wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -8.922429223447696661376437514499176950622 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(1, wj, \left(\left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{1 + wj}\right) \cdot \frac{-1}{e^{wj}}\right) + \mathsf{fma}\left(\frac{-1}{e^{wj} + e^{wj} \cdot wj}, e^{wj} \cdot wj - x, \left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{e^{wj} + e^{wj} \cdot wj}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

\end{array}
double f(double wj, double x) {
        double r11840580 = wj;
        double r11840581 = exp(r11840580);
        double r11840582 = r11840580 * r11840581;
        double r11840583 = x;
        double r11840584 = r11840582 - r11840583;
        double r11840585 = r11840581 + r11840582;
        double r11840586 = r11840584 / r11840585;
        double r11840587 = r11840580 - r11840586;
        return r11840587;
}
↓
double f(double wj, double x) {
        double r11840588 = wj;
        double r11840589 = -8.922429223447697e-09;
        bool r11840590 = r11840588 <= r11840589;
        double r11840591 = 1.0;
        double r11840592 = exp(r11840588);
        double r11840593 = r11840592 * r11840588;
        double r11840594 = x;
        double r11840595 = r11840593 - r11840594;
        double r11840596 = r11840591 + r11840588;
        double r11840597 = r11840591 / r11840596;
        double r11840598 = r11840595 * r11840597;
        double r11840599 = -1.0;
        double r11840600 = r11840599 / r11840592;
        double r11840601 = r11840598 * r11840600;
        double r11840602 = fma(r11840591, r11840588, r11840601);
        double r11840603 = r11840592 + r11840593;
        double r11840604 = r11840599 / r11840603;
        double r11840605 = r11840591 / r11840603;
        double r11840606 = r11840595 * r11840605;
        double r11840607 = fma(r11840604, r11840595, r11840606);
        double r11840608 = r11840602 + r11840607;
        double r11840609 = r11840594 * r11840588;
        double r11840610 = -2.0;
        double r11840611 = fma(r11840588, r11840588, r11840594);
        double r11840612 = fma(r11840609, r11840610, r11840611);
        double r11840613 = r11840590 ? r11840608 : r11840612;
        return r11840613;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -8.922429223447697e-09

    1. Initial program 5.6

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied div-inv5.8

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

    4. Applied *-un-lft-identity5.8

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

    5. Applied prod-diff5.9

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

    7. Applied *-un-lft-identity5.9

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

    8. Applied distribute-rgt-out5.8

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

    9. Applied *-un-lft-identity5.8

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

    10. Applied times-frac6.0

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

    11. Applied associate-*l*6.0

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

    if -8.922429223447697e-09 < wj

    1. Initial program 13.9

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

    2. Taylor expanded around 0 1.4

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

    3. Simplified1.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -8.922429223447696661376437514499176950622 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(1, wj, \left(\left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{1 + wj}\right) \cdot \frac{-1}{e^{wj}}\right) + \mathsf{fma}\left(\frac{-1}{e^{wj} + e^{wj} \cdot wj}, e^{wj} \cdot wj - x, \left(e^{wj} \cdot wj - x\right) \cdot \frac{1}{e^{wj} + e^{wj} \cdot wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))