Average Error: 14.1 → 1.5
Time: 8.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -6.1770903366602869 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj}, \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right) + \left(\frac{{\left(\sqrt[3]{wj}\right)}^{3}}{1 \cdot \left(wj + 1\right)} + \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -6.1770903366602869 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj}, \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{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)\\

\mathbf{else}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r256640 = wj;
        double r256641 = exp(r256640);
        double r256642 = r256640 * r256641;
        double r256643 = x;
        double r256644 = r256642 - r256643;
        double r256645 = r256641 + r256642;
        double r256646 = r256644 / r256645;
        double r256647 = r256640 - r256646;
        return r256647;
}
↓
double f(double wj, double x) {
        double r256648 = wj;
        double r256649 = -6.177090336660287e-09;
        bool r256650 = r256648 <= r256649;
        double r256651 = x;
        double r256652 = 1.0;
        double r256653 = r256648 + r256652;
        double r256654 = r256651 / r256653;
        double r256655 = exp(r256648);
        double r256656 = r256652 / r256655;
        double r256657 = r256654 * r256656;
        double r256658 = r256657 + r256648;
        double r256659 = cbrt(r256658);
        double r256660 = r256659 * r256659;
        double r256661 = cbrt(r256648);
        double r256662 = 3.0;
        double r256663 = pow(r256661, r256662);
        double r256664 = -r256663;
        double r256665 = r256664 / r256653;
        double r256666 = fma(r256659, r256660, r256665);
        double r256667 = r256652 * r256653;
        double r256668 = r256663 / r256667;
        double r256669 = r256668 + r256665;
        double r256670 = r256666 + r256669;
        double r256671 = 2.0;
        double r256672 = pow(r256648, r256671);
        double r256673 = r256651 + r256672;
        double r256674 = r256648 * r256651;
        double r256675 = r256671 * r256674;
        double r256676 = r256673 - r256675;
        double r256677 = r256650 ? r256670 : r256676;
        return r256677;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.1
Target13.5
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 < -6.177090336660287e-09

    1. Initial program 4.9

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

    2. Simplified4.7

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

    4. Applied div-inv4.7

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

    6. Applied *-un-lft-identity4.7

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

    7. Applied add-cube-cbrt5.9

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

    8. Applied times-frac6.0

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

    9. Applied add-cube-cbrt5.7

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

    10. Applied prod-diff5.5

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

    11. Simplified5.6

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

    12. Simplified5.6

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

    if -6.177090336660287e-09 < wj

    1. Initial program 14.2

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

    2. Simplified13.6

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

    3. Taylor expanded around 0 1.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -6.1770903366602869 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj}, \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj} \cdot \sqrt[3]{\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj}, \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right) + \left(\frac{{\left(\sqrt[3]{wj}\right)}^{3}}{1 \cdot \left(wj + 1\right)} + \frac{-{\left(\sqrt[3]{wj}\right)}^{3}}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))))