Average Error: 14.1 → 1.0
Time: 21.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le \frac{2262931210409171}{302231454903657293676544}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{wj}}} \cdot \frac{\frac{1}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le \frac{2262931210409171}{302231454903657293676544}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r115741 = wj;
        double r115742 = exp(r115741);
        double r115743 = r115741 * r115742;
        double r115744 = x;
        double r115745 = r115743 - r115744;
        double r115746 = r115742 + r115743;
        double r115747 = r115745 / r115746;
        double r115748 = r115741 - r115747;
        return r115748;
}
↓
double f(double wj, double x) {
        double r115749 = wj;
        double r115750 = 2262931210409171.0;
        double r115751 = 3.022314549036573e+23;
        double r115752 = r115750 / r115751;
        bool r115753 = r115749 <= r115752;
        double r115754 = x;
        double r115755 = 2.0;
        double r115756 = pow(r115749, r115755);
        double r115757 = r115754 + r115756;
        double r115758 = r115749 * r115754;
        double r115759 = r115755 * r115758;
        double r115760 = r115757 - r115759;
        double r115761 = exp(r115749);
        double r115762 = sqrt(r115761);
        double r115763 = r115754 / r115762;
        double r115764 = 1.0;
        double r115765 = r115749 + r115764;
        double r115766 = r115764 / r115765;
        double r115767 = r115766 / r115762;
        double r115768 = r115763 * r115767;
        double r115769 = r115768 + r115749;
        double r115770 = r115749 / r115765;
        double r115771 = r115769 - r115770;
        double r115772 = r115753 ? r115760 : r115771;
        return r115772;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.6
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 < 7.48741129916649e-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 7.48741129916649e-09 < wj

    1. Initial program 23.0

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

    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 add-sqr-sqrt3.2

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

    5. Applied div-inv3.2

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

    6. Applied times-frac3.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le \frac{2262931210409171}{302231454903657293676544}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{wj}}} \cdot \frac{\frac{1}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))))