Average Error: 13.7 → 0.6
Time: 20.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.48959693341686037244700827053556224655 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \left(\left(wj \cdot \left(\frac{5}{2} \cdot wj - 2\right)\right) \cdot x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.48959693341686037244700827053556224655 \cdot 10^{-5}:\\
\;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \left(\left(wj \cdot \left(\frac{5}{2} \cdot wj - 2\right)\right) \cdot x + x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r189947 = wj;
        double r189948 = exp(r189947);
        double r189949 = r189947 * r189948;
        double r189950 = x;
        double r189951 = r189949 - r189950;
        double r189952 = r189948 + r189949;
        double r189953 = r189951 / r189952;
        double r189954 = r189947 - r189953;
        return r189954;
}
↓
double f(double wj, double x) {
        double r189955 = wj;
        double r189956 = 1.4895969334168604e-05;
        bool r189957 = r189955 <= r189956;
        double r189958 = 4.0;
        double r189959 = pow(r189955, r189958);
        double r189960 = 2.0;
        double r189961 = pow(r189955, r189960);
        double r189962 = r189959 + r189961;
        double r189963 = 3.0;
        double r189964 = pow(r189955, r189963);
        double r189965 = r189962 - r189964;
        double r189966 = 2.5;
        double r189967 = r189966 * r189955;
        double r189968 = r189967 - r189960;
        double r189969 = r189955 * r189968;
        double r189970 = x;
        double r189971 = r189969 * r189970;
        double r189972 = r189971 + r189970;
        double r189973 = r189965 + r189972;
        double r189974 = 1.0;
        double r189975 = r189955 + r189974;
        double r189976 = r189955 / r189975;
        double r189977 = r189955 - r189976;
        double r189978 = r189974 + r189955;
        double r189979 = exp(r189955);
        double r189980 = r189978 * r189979;
        double r189981 = r189970 / r189980;
        double r189982 = r189977 + r189981;
        double r189983 = r189957 ? r189973 : r189982;
        return r189983;
}

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
Target13.0
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.4895969334168604e-05

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied div-sub13.3

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

    5. Applied associate--r-6.9

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

    6. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

    7. Taylor expanded around 0 0.6

      \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left({wj}^{2} \cdot x\right)\right) - 2 \cdot \left(wj \cdot x\right)\right)}\]

    8. Simplified0.6

      \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(wj \cdot \left(\frac{5}{2} \cdot wj - 2\right)\right) \cdot x + x\right)}\]

    if 1.4895969334168604e-05 < wj

    1. Initial program 30.1

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

    2. Simplified1.2

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

    4. Applied div-sub1.2

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

    5. Applied associate--r-1.2

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

    7. Applied div-inv1.2

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

    8. Applied associate-/l*1.2

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

    9. Simplified1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.48959693341686037244700827053556224655 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \left(\left(wj \cdot \left(\frac{5}{2} \cdot wj - 2\right)\right) \cdot x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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))))))