Average Error: 13.6 → 0.4
Time: 17.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.365019380480616014925110413003039866453 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;wj - \sqrt{\frac{wj - \frac{x}{e^{wj}}}{1 + wj}} \cdot \sqrt{\frac{wj - \frac{x}{e^{wj}}}{1 + wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.365019380480616014925110413003039866453 \cdot 10^{-5}:\\
\;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\

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

\end{array}
double f(double wj, double x) {
        double r204639 = wj;
        double r204640 = exp(r204639);
        double r204641 = r204639 * r204640;
        double r204642 = x;
        double r204643 = r204641 - r204642;
        double r204644 = r204640 + r204641;
        double r204645 = r204643 / r204644;
        double r204646 = r204639 - r204645;
        return r204646;
}
↓
double f(double wj, double x) {
        double r204647 = wj;
        double r204648 = 9.365019380480616e-05;
        bool r204649 = r204647 <= r204648;
        double r204650 = 4.0;
        double r204651 = pow(r204647, r204650);
        double r204652 = 2.0;
        double r204653 = pow(r204647, r204652);
        double r204654 = r204651 + r204653;
        double r204655 = 3.0;
        double r204656 = pow(r204647, r204655);
        double r204657 = r204654 - r204656;
        double r204658 = x;
        double r204659 = exp(r204647);
        double r204660 = r204658 / r204659;
        double r204661 = 1.0;
        double r204662 = r204661 + r204647;
        double r204663 = r204660 / r204662;
        double r204664 = r204657 + r204663;
        double r204665 = r204647 - r204660;
        double r204666 = r204665 / r204662;
        double r204667 = sqrt(r204666);
        double r204668 = r204667 * r204667;
        double r204669 = r204647 - r204668;
        double r204670 = r204649 ? r204664 : r204669;
        return r204670;
}

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
Target12.8
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.365019380480616e-05

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied div-sub13.0

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

    5. Applied associate--r-7.0

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

    6. Simplified7.0

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

    7. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

    if 9.365019380480616e-05 < wj

    1. Initial program 36.0

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

    2. Simplified1.0

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

    4. Applied add-sqr-sqrt4.7

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

    5. Simplified4.7

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

    6. Simplified4.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.365019380480616014925110413003039866453 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;wj - \sqrt{\frac{wj - \frac{x}{e^{wj}}}{1 + wj}} \cdot \sqrt{\frac{wj - \frac{x}{e^{wj}}}{1 + wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))))