Average Error: 13.9 → 1.0
Time: 23.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\
\;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\

\end{array}
double f(double wj, double x) {
        double r186659 = wj;
        double r186660 = exp(r186659);
        double r186661 = r186659 * r186660;
        double r186662 = x;
        double r186663 = r186661 - r186662;
        double r186664 = r186660 + r186661;
        double r186665 = r186663 / r186664;
        double r186666 = r186659 - r186665;
        return r186666;
}
↓
double f(double wj, double x) {
        double r186667 = wj;
        double r186668 = 8.133938014656618e-09;
        bool r186669 = r186667 <= r186668;
        double r186670 = x;
        double r186671 = 2.0;
        double r186672 = r186670 * r186671;
        double r186673 = r186667 - r186672;
        double r186674 = r186667 * r186673;
        double r186675 = r186670 + r186674;
        double r186676 = exp(r186667);
        double r186677 = sqrt(r186676);
        double r186678 = r186670 / r186677;
        double r186679 = r186678 / r186677;
        double r186680 = r186667 - r186679;
        double r186681 = 1.0;
        double r186682 = r186667 + r186681;
        double r186683 = r186680 / r186682;
        double r186684 = r186667 - r186683;
        double r186685 = r186669 ? r186675 : r186684;
        return r186685;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
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 < 8.133938014656618e-09

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 8.133938014656618e-09 < wj

    1. Initial program 27.6

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

    2. Simplified2.3

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

    4. Applied add-sqr-sqrt2.4

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

    5. Applied associate-/r*2.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))))