Average Error: 13.7 → 1.0
Time: 7.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 7.575706194898737323694237493331283921083 \cdot 10^{-11}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} \cdot \frac{\frac{x}{\sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 7.575706194898737323694237493331283921083 \cdot 10^{-11}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} \cdot \frac{\frac{x}{\sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\

\end{array}
double f(double wj, double x) {
        double r262737 = wj;
        double r262738 = exp(r262737);
        double r262739 = r262737 * r262738;
        double r262740 = x;
        double r262741 = r262739 - r262740;
        double r262742 = r262738 + r262739;
        double r262743 = r262741 / r262742;
        double r262744 = r262737 - r262743;
        return r262744;
}
↓
double f(double wj, double x) {
        double r262745 = wj;
        double r262746 = 7.575706194898737e-11;
        bool r262747 = r262745 <= r262746;
        double r262748 = x;
        double r262749 = 2.0;
        double r262750 = pow(r262745, r262749);
        double r262751 = r262748 + r262750;
        double r262752 = r262745 * r262748;
        double r262753 = r262749 * r262752;
        double r262754 = r262751 - r262753;
        double r262755 = 1.0;
        double r262756 = r262745 + r262755;
        double r262757 = cbrt(r262756);
        double r262758 = r262757 * r262757;
        double r262759 = r262755 / r262758;
        double r262760 = exp(r262745);
        double r262761 = sqrt(r262760);
        double r262762 = r262759 / r262761;
        double r262763 = r262748 / r262757;
        double r262764 = r262763 / r262761;
        double r262765 = r262762 * r262764;
        double r262766 = r262765 + r262745;
        double r262767 = r262745 / r262756;
        double r262768 = r262766 - r262767;
        double r262769 = r262747 ? r262754 : r262768;
        return r262769;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
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 < 7.575706194898737e-11

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 0.9

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

    if 7.575706194898737e-11 < wj

    1. Initial program 25.5

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

    2. Simplified4.0

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

    4. Applied add-sqr-sqrt4.1

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

    5. Applied add-cube-cbrt4.3

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

    6. Applied *-un-lft-identity4.3

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

    7. Applied times-frac4.3

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

    8. Applied times-frac4.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.575706194898737323694237493331283921083 \cdot 10^{-11}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} \cdot \frac{\frac{x}{\sqrt[3]{wj + 1}}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))))