Average Error: 14.2 → 1.1
Time: 8.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r183500 = wj;
        double r183501 = exp(r183500);
        double r183502 = r183500 * r183501;
        double r183503 = x;
        double r183504 = r183502 - r183503;
        double r183505 = r183501 + r183502;
        double r183506 = r183504 / r183505;
        double r183507 = r183500 - r183506;
        return r183507;
}
↓
double f(double wj, double x) {
        double r183508 = wj;
        double r183509 = 9.06252678575456e-09;
        bool r183510 = r183508 <= r183509;
        double r183511 = x;
        double r183512 = 2.0;
        double r183513 = pow(r183508, r183512);
        double r183514 = r183511 + r183513;
        double r183515 = r183508 * r183511;
        double r183516 = r183512 * r183515;
        double r183517 = r183514 - r183516;
        double r183518 = 1.0;
        double r183519 = r183508 + r183518;
        double r183520 = r183511 / r183519;
        double r183521 = exp(r183508);
        double r183522 = r183520 / r183521;
        double r183523 = r183522 + r183508;
        double r183524 = r183508 * r183508;
        double r183525 = r183524 - r183518;
        double r183526 = r183508 / r183525;
        double r183527 = r183508 - r183518;
        double r183528 = r183526 * r183527;
        double r183529 = r183523 - r183528;
        double r183530 = r183510 ? r183517 : r183529;
        return r183530;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.2
Target13.5
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.06252678575456e-09

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded around 0 1.0

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

    if 9.06252678575456e-09 < wj

    1. Initial program 26.3

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

    2. Simplified3.1

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

    4. Applied flip-+3.2

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

    5. Applied associate-/r/3.0

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

    6. Simplified3.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))))))