Average Error: 13.7 → 1.5
Time: 1.2m
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 r10198622 = wj;
        double r10198623 = exp(r10198622);
        double r10198624 = r10198622 * r10198623;
        double r10198625 = x;
        double r10198626 = r10198624 - r10198625;
        double r10198627 = r10198623 + r10198624;
        double r10198628 = r10198626 / r10198627;
        double r10198629 = r10198622 - r10198628;
        return r10198629;
}
↓
double f(double wj, double x) {
        double r10198630 = wj;
        double r10198631 = -8.922429223447697e-09;
        bool r10198632 = r10198630 <= r10198631;
        double r10198633 = 1.0;
        double r10198634 = exp(r10198630);
        double r10198635 = r10198634 * r10198630;
        double r10198636 = x;
        double r10198637 = r10198635 - r10198636;
        double r10198638 = r10198633 + r10198630;
        double r10198639 = r10198633 / r10198638;
        double r10198640 = r10198637 * r10198639;
        double r10198641 = -1.0;
        double r10198642 = r10198641 / r10198634;
        double r10198643 = r10198640 * r10198642;
        double r10198644 = fma(r10198633, r10198630, r10198643);
        double r10198645 = r10198634 + r10198635;
        double r10198646 = r10198641 / r10198645;
        double r10198647 = r10198633 / r10198645;
        double r10198648 = r10198637 * r10198647;
        double r10198649 = fma(r10198646, r10198637, r10198648);
        double r10198650 = r10198644 + r10198649;
        double r10198651 = r10198636 * r10198630;
        double r10198652 = -2.0;
        double r10198653 = fma(r10198630, r10198630, r10198636);
        double r10198654 = fma(r10198651, r10198652, r10198653);
        double r10198655 = r10198632 ? r10198650 : r10198654;
        return r10198655;
}

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))))))