Average Error: 13.4 → 1.2
Time: 7.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r685808 = wj;
        double r685809 = exp(r685808);
        double r685810 = r685808 * r685809;
        double r685811 = x;
        double r685812 = r685810 - r685811;
        double r685813 = r685809 + r685810;
        double r685814 = r685812 / r685813;
        double r685815 = r685808 - r685814;
        return r685815;
}
↓
double f(double wj, double x) {
        double r685816 = wj;
        double r685817 = 3.1088434258944978e-06;
        bool r685818 = r685816 <= r685817;
        double r685819 = x;
        double r685820 = 2.0;
        double r685821 = pow(r685816, r685820);
        double r685822 = r685819 + r685821;
        double r685823 = r685816 * r685819;
        double r685824 = r685820 * r685823;
        double r685825 = r685822 - r685824;
        double r685826 = 1.0;
        double r685827 = r685816 + r685826;
        double r685828 = r685819 / r685827;
        double r685829 = exp(r685816);
        double r685830 = r685828 / r685829;
        double r685831 = r685830 * r685830;
        double r685832 = r685816 * r685816;
        double r685833 = r685831 - r685832;
        double r685834 = r685833 * r685827;
        double r685835 = r685830 - r685816;
        double r685836 = r685835 * r685816;
        double r685837 = r685834 - r685836;
        double r685838 = r685835 * r685827;
        double r685839 = r685837 / r685838;
        double r685840 = r685818 ? r685825 : r685839;
        return r685840;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 3.1088434258944978e-06

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 1.0

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

    if 3.1088434258944978e-06 < wj

    1. Initial program 27.7

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

    2. Simplified1.7

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

    4. Applied flip-+10.6

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

    5. Applied frac-sub10.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))))