Average Error: 13.8 → 1.0
Time: 6.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.02314066329064058 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right) + \left(\frac{{\left(\sqrt[3]{wj}\right)}^{3}}{1 \cdot \left(wj + 1\right)} + \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.02314066329064058 \cdot 10^{-8}:\\
\;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right) + \left(\frac{{\left(\sqrt[3]{wj}\right)}^{3}}{1 \cdot \left(wj + 1\right)} + \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)\\

\end{array}
double f(double wj, double x) {
        double r231665 = wj;
        double r231666 = exp(r231665);
        double r231667 = r231665 * r231666;
        double r231668 = x;
        double r231669 = r231667 - r231668;
        double r231670 = r231666 + r231667;
        double r231671 = r231669 / r231670;
        double r231672 = r231665 - r231671;
        return r231672;
}
↓
double f(double wj, double x) {
        double r231673 = wj;
        double r231674 = 1.0231406632906406e-08;
        bool r231675 = r231673 <= r231674;
        double r231676 = 1.0;
        double r231677 = x;
        double r231678 = fma(r231673, r231673, r231677);
        double r231679 = r231676 * r231678;
        double r231680 = 2.0;
        double r231681 = r231673 * r231677;
        double r231682 = r231680 * r231681;
        double r231683 = r231679 - r231682;
        double r231684 = r231673 + r231676;
        double r231685 = r231677 / r231684;
        double r231686 = exp(r231673);
        double r231687 = r231685 / r231686;
        double r231688 = r231687 + r231673;
        double r231689 = cbrt(r231688);
        double r231690 = r231689 * r231689;
        double r231691 = cbrt(r231673);
        double r231692 = 3.0;
        double r231693 = pow(r231691, r231692);
        double r231694 = -r231693;
        double r231695 = r231694 / r231684;
        double r231696 = fma(r231689, r231690, r231695);
        double r231697 = r231676 * r231684;
        double r231698 = r231693 / r231697;
        double r231699 = r231698 + r231695;
        double r231700 = r231696 + r231699;
        double r231701 = r231675 ? r231683 : r231700;
        return r231701;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.2
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.0231406632906406e-08

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity0.9

      \[\leadsto \left(x + \color{blue}{1 \cdot {wj}^{2}}\right) - 2 \cdot \left(wj \cdot x\right)\]

    6. Applied *-un-lft-identity0.9

      \[\leadsto \left(\color{blue}{1 \cdot x} + 1 \cdot {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\]

    7. Applied distribute-lft-out0.9

      \[\leadsto \color{blue}{1 \cdot \left(x + {wj}^{2}\right)} - 2 \cdot \left(wj \cdot x\right)\]

    8. Simplified0.9

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} - 2 \cdot \left(wj \cdot x\right)\]

    if 1.0231406632906406e-08 < wj

    1. Initial program 25.9

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

    2. Simplified2.7

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

    4. Applied *-un-lft-identity2.7

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

    5. Applied add-cube-cbrt3.2

      \[\leadsto \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{\color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}}}{1 \cdot \left(wj + 1\right)}\]

    6. Applied times-frac3.2

      \[\leadsto \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \color{blue}{\frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1} \cdot \frac{\sqrt[3]{wj}}{wj + 1}}\]

    7. Applied add-cube-cbrt3.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}\right) \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}} - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1} \cdot \frac{\sqrt[3]{wj}}{wj + 1}\]

    8. Applied prod-diff3.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, -\frac{\sqrt[3]{wj}}{wj + 1} \cdot \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{wj}}{wj + 1}, \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1}, \frac{\sqrt[3]{wj}}{wj + 1} \cdot \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1}\right)}\]

    9. Simplified3.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{wj}}{wj + 1}, \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1}, \frac{\sqrt[3]{wj}}{wj + 1} \cdot \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{1}\right)\]

    10. Simplified3.7

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.02314066329064058 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right) + \left(\frac{{\left(\sqrt[3]{wj}\right)}^{3}}{1 \cdot \left(wj + 1\right)} + \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))))