Average Error: 13.6 → 0.6
Time: 21.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.256044322215378018516810255250648609149 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \left(wj \cdot x + {wj}^{3}\right)\right) - \left(-wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}} \cdot \left(1 - wj\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 3.256044322215378018516810255250648609149 \cdot 10^{-7}:\\
\;\;\;\;\left(x - \left(wj \cdot x + {wj}^{3}\right)\right) - \left(-wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}}\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}} \cdot \left(1 - wj\right)\\

\end{array}
double f(double wj, double x) {
        double r136666 = wj;
        double r136667 = exp(r136666);
        double r136668 = r136666 * r136667;
        double r136669 = x;
        double r136670 = r136668 - r136669;
        double r136671 = r136667 + r136668;
        double r136672 = r136670 / r136671;
        double r136673 = r136666 - r136672;
        return r136673;
}
↓
double f(double wj, double x) {
        double r136674 = wj;
        double r136675 = 3.256044322215378e-07;
        bool r136676 = r136674 <= r136675;
        double r136677 = x;
        double r136678 = r136674 * r136677;
        double r136679 = 3.0;
        double r136680 = pow(r136674, r136679);
        double r136681 = r136678 + r136680;
        double r136682 = r136677 - r136681;
        double r136683 = -r136674;
        double r136684 = exp(r136674);
        double r136685 = r136677 / r136684;
        double r136686 = r136674 - r136685;
        double r136687 = 1.0;
        double r136688 = 2.0;
        double r136689 = pow(r136674, r136688);
        double r136690 = r136687 - r136689;
        double r136691 = r136686 / r136690;
        double r136692 = r136683 * r136691;
        double r136693 = r136682 - r136692;
        double r136694 = r136687 - r136674;
        double r136695 = r136691 * r136694;
        double r136696 = r136674 - r136695;
        double r136697 = r136676 ? r136693 : r136696;
        return r136697;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target12.9
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 3.256044322215378e-07

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied flip-+13.2

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

    5. Applied associate-/r/13.2

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

    6. Simplified13.2

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}}} \cdot \left(1 - wj\right)\]
    7. Using strategy rm

    8. Applied sub-neg13.2

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

    9. Applied distribute-rgt-in13.2

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

    10. Applied associate--r+7.2

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

    11. Simplified7.2

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

    12. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\left(x - \left(wj \cdot x + {wj}^{3}\right)\right)} - \left(-wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}}\]

    if 3.256044322215378e-07 < wj

    1. Initial program 30.3

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

    2. Simplified2.1

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

    4. Applied flip-+2.2

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

    5. Applied associate-/r/2.1

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

    6. Simplified2.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.256044322215378018516810255250648609149 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \left(wj \cdot x + {wj}^{3}\right)\right) - \left(-wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{1 - {wj}^{2}} \cdot \left(1 - wj\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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))))))