Average Error: 13.4 → 0.9
Time: 5.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 2.1380709412627531 \cdot 10^{-13}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(\frac{wj}{wj + 1} + wj\right) + {wj}^{2}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 2.1380709412627531 \cdot 10^{-13}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(\frac{wj}{wj + 1} + wj\right) + {wj}^{2}}\\

\end{array}
double f(double wj, double x) {
        double r128908 = wj;
        double r128909 = exp(r128908);
        double r128910 = r128908 * r128909;
        double r128911 = x;
        double r128912 = r128910 - r128911;
        double r128913 = r128909 + r128910;
        double r128914 = r128912 / r128913;
        double r128915 = r128908 - r128914;
        return r128915;
}
↓
double f(double wj, double x) {
        double r128916 = wj;
        double r128917 = 2.138070941262753e-13;
        bool r128918 = r128916 <= r128917;
        double r128919 = x;
        double r128920 = 2.0;
        double r128921 = pow(r128916, r128920);
        double r128922 = r128919 + r128921;
        double r128923 = r128916 * r128919;
        double r128924 = r128920 * r128923;
        double r128925 = r128922 - r128924;
        double r128926 = 1.0;
        double r128927 = r128916 + r128926;
        double r128928 = r128919 / r128927;
        double r128929 = exp(r128916);
        double r128930 = r128928 / r128929;
        double r128931 = 3.0;
        double r128932 = pow(r128916, r128931);
        double r128933 = r128916 / r128927;
        double r128934 = pow(r128933, r128931);
        double r128935 = r128932 - r128934;
        double r128936 = r128933 + r128916;
        double r128937 = r128933 * r128936;
        double r128938 = r128937 + r128921;
        double r128939 = r128935 / r128938;
        double r128940 = r128930 + r128939;
        double r128941 = r128918 ? r128925 : r128940;
        return r128941;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.7
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 2.138070941262753e-13

    1. Initial program 13.0

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

    2. Simplified13.0

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

    3. Taylor expanded around 0 0.8

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

    if 2.138070941262753e-13 < wj

    1. Initial program 23.1

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

    2. Simplified5.0

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

    4. Applied associate--l+5.0

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

    6. Applied flip3--5.2

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

    7. Simplified5.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 2.1380709412627531 \cdot 10^{-13}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(\frac{wj}{wj + 1} + wj\right) + {wj}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))))))