Average Error: 13.7 → 0.2
Time: 5.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.293516541346027667620105505719152461097 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{wj \cdot wj - 1} \cdot \frac{wj - 1}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.293516541346027667620105505719152461097 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

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

\end{array}
double f(double wj, double x) {
        double r490003 = wj;
        double r490004 = exp(r490003);
        double r490005 = r490003 * r490004;
        double r490006 = x;
        double r490007 = r490005 - r490006;
        double r490008 = r490004 + r490005;
        double r490009 = r490007 / r490008;
        double r490010 = r490003 - r490009;
        return r490010;
}
↓
double f(double wj, double x) {
        double r490011 = wj;
        double r490012 = 9.293516541346028e-05;
        bool r490013 = r490011 <= r490012;
        double r490014 = x;
        double r490015 = 1.0;
        double r490016 = r490011 + r490015;
        double r490017 = r490014 / r490016;
        double r490018 = exp(r490011);
        double r490019 = r490017 / r490018;
        double r490020 = 4.0;
        double r490021 = pow(r490011, r490020);
        double r490022 = 2.0;
        double r490023 = pow(r490011, r490022);
        double r490024 = r490021 + r490023;
        double r490025 = 3.0;
        double r490026 = pow(r490011, r490025);
        double r490027 = r490024 - r490026;
        double r490028 = r490019 + r490027;
        double r490029 = r490011 * r490011;
        double r490030 = r490029 - r490015;
        double r490031 = r490014 / r490030;
        double r490032 = r490011 - r490015;
        double r490033 = r490032 / r490018;
        double r490034 = r490031 * r490033;
        double r490035 = r490011 / r490016;
        double r490036 = r490011 - r490035;
        double r490037 = r490034 + r490036;
        double r490038 = r490013 ? r490028 : r490037;
        return r490038;
}

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
Herbie0.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.293516541346028e-05

    1. Initial program 13.4

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

    2. Simplified13.3

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

    4. Applied associate--l+6.9

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

    5. Taylor expanded around 0 0.2

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

    if 9.293516541346028e-05 < wj

    1. Initial program 30.9

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

    2. Simplified1.0

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

    4. Applied associate--l+1.0

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

    6. Applied *-un-lft-identity1.0

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

    7. Applied flip-+1.0

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

    8. Applied associate-/r/1.0

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

    9. Applied times-frac1.0

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

    10. Simplified1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.293516541346027667620105505719152461097 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{wj \cdot wj - 1} \cdot \frac{wj - 1}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))))