Average Error: 13.4 → 1.2
Time: 8.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r285911 = wj;
        double r285912 = exp(r285911);
        double r285913 = r285911 * r285912;
        double r285914 = x;
        double r285915 = r285913 - r285914;
        double r285916 = r285912 + r285913;
        double r285917 = r285915 / r285916;
        double r285918 = r285911 - r285917;
        return r285918;
}
↓
double f(double wj, double x) {
        double r285919 = wj;
        double r285920 = 3.1088434258944978e-06;
        bool r285921 = r285919 <= r285920;
        double r285922 = x;
        double r285923 = 2.0;
        double r285924 = pow(r285919, r285923);
        double r285925 = r285922 + r285924;
        double r285926 = r285919 * r285922;
        double r285927 = r285923 * r285926;
        double r285928 = r285925 - r285927;
        double r285929 = 1.0;
        double r285930 = r285919 + r285929;
        double r285931 = r285922 / r285930;
        double r285932 = exp(r285919);
        double r285933 = r285931 / r285932;
        double r285934 = r285933 * r285933;
        double r285935 = r285919 * r285919;
        double r285936 = r285934 - r285935;
        double r285937 = r285936 * r285930;
        double r285938 = r285933 - r285919;
        double r285939 = r285938 * r285919;
        double r285940 = r285937 - r285939;
        double r285941 = r285938 * r285930;
        double r285942 = r285940 / r285941;
        double r285943 = r285921 ? r285928 : r285942;
        return r285943;
}

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
Target12.8
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 3.1088434258944978e-06

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 1.0

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

    if 3.1088434258944978e-06 < wj

    1. Initial program 27.7

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

    2. Simplified1.7

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

    4. Applied flip-+10.6

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

    5. Applied frac-sub10.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.10884342589449778 \cdot 10^{-6}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(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))))))