Average Error: 14.0 → 1.5
Time: 28.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.944304526105059 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot wj\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, wj, \frac{-1}{1 + wj} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}\right) + \mathsf{fma}\left(-\frac{wj \cdot e^{wj} - x}{e^{wj}}, \frac{1}{1 + wj}, \frac{wj \cdot e^{wj} - x}{e^{wj}} \cdot \frac{1}{1 + wj}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.944304526105059 \cdot 10^{-31}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot wj\right) \cdot x\\

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

\end{array}
double f(double wj, double x) {
        double r4235576 = wj;
        double r4235577 = exp(r4235576);
        double r4235578 = r4235576 * r4235577;
        double r4235579 = x;
        double r4235580 = r4235578 - r4235579;
        double r4235581 = r4235577 + r4235578;
        double r4235582 = r4235580 / r4235581;
        double r4235583 = r4235576 - r4235582;
        return r4235583;
}
↓
double f(double wj, double x) {
        double r4235584 = wj;
        double r4235585 = exp(r4235584);
        double r4235586 = r4235584 * r4235585;
        double r4235587 = x;
        double r4235588 = r4235586 - r4235587;
        double r4235589 = r4235585 + r4235586;
        double r4235590 = r4235588 / r4235589;
        double r4235591 = r4235584 - r4235590;
        double r4235592 = 3.944304526105059e-31;
        bool r4235593 = r4235591 <= r4235592;
        double r4235594 = fma(r4235584, r4235584, r4235587);
        double r4235595 = 2.0;
        double r4235596 = r4235595 * r4235584;
        double r4235597 = r4235596 * r4235587;
        double r4235598 = r4235594 - r4235597;
        double r4235599 = 1.0;
        double r4235600 = -1.0;
        double r4235601 = r4235599 + r4235584;
        double r4235602 = r4235600 / r4235601;
        double r4235603 = r4235588 / r4235585;
        double r4235604 = r4235602 * r4235603;
        double r4235605 = fma(r4235599, r4235584, r4235604);
        double r4235606 = -r4235603;
        double r4235607 = r4235599 / r4235601;
        double r4235608 = r4235603 * r4235607;
        double r4235609 = fma(r4235606, r4235607, r4235608);
        double r4235610 = r4235605 + r4235609;
        double r4235611 = r4235593 ? r4235598 : r4235610;
        return r4235611;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.4
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 (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 3.944304526105059e-31

    1. Initial program 18.3

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

    if 3.944304526105059e-31 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 3.8

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

    3. Applied distribute-rgt1-in3.8

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

    4. Applied *-un-lft-identity3.8

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

    5. Applied times-frac3.8

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

    6. Applied *-un-lft-identity3.8

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

    7. Applied prod-diff3.8

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.944304526105059 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - \left(2 \cdot wj\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, wj, \frac{-1}{1 + wj} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}\right) + \mathsf{fma}\left(-\frac{wj \cdot e^{wj} - x}{e^{wj}}, \frac{1}{1 + wj}, \frac{wj \cdot e^{wj} - x}{e^{wj}} \cdot \frac{1}{1 + wj}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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