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 r157027 = wj;
        double r157028 = exp(r157027);
        double r157029 = r157027 * r157028;
        double r157030 = x;
        double r157031 = r157029 - r157030;
        double r157032 = r157028 + r157029;
        double r157033 = r157031 / r157032;
        double r157034 = r157027 - r157033;
        return r157034;
}
↓
double f(double wj, double x) {
        double r157035 = wj;
        double r157036 = 7.575706194898737e-11;
        bool r157037 = r157035 <= r157036;
        double r157038 = x;
        double r157039 = 2.0;
        double r157040 = pow(r157035, r157039);
        double r157041 = r157038 + r157040;
        double r157042 = r157035 * r157038;
        double r157043 = r157039 * r157042;
        double r157044 = r157041 - r157043;
        double r157045 = 1.0;
        double r157046 = r157035 + r157045;
        double r157047 = cbrt(r157046);
        double r157048 = r157047 * r157047;
        double r157049 = r157045 / r157048;
        double r157050 = exp(r157035);
        double r157051 = sqrt(r157050);
        double r157052 = r157049 / r157051;
        double r157053 = r157038 / r157047;
        double r157054 = r157053 / r157051;
        double r157055 = r157052 * r157054;
        double r157056 = r157055 + r157035;
        double r157057 = r157035 / r157046;
        double r157058 = r157056 - r157057;
        double r157059 = r157037 ? r157044 : r157058;
        return r157059;
}

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))))))