Average Error: 13.8 → 0.3
Time: 5.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

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

\end{array}
double f(double wj, double x) {
        double r196283 = wj;
        double r196284 = exp(r196283);
        double r196285 = r196283 * r196284;
        double r196286 = x;
        double r196287 = r196285 - r196286;
        double r196288 = r196284 + r196285;
        double r196289 = r196287 / r196288;
        double r196290 = r196283 - r196289;
        return r196290;
}
↓
double f(double wj, double x) {
        double r196291 = wj;
        double r196292 = 4.811043021165421e-08;
        bool r196293 = r196291 <= r196292;
        double r196294 = x;
        double r196295 = 1.0;
        double r196296 = r196291 + r196295;
        double r196297 = r196294 / r196296;
        double r196298 = exp(r196291);
        double r196299 = r196297 / r196298;
        double r196300 = 4.0;
        double r196301 = pow(r196291, r196300);
        double r196302 = 2.0;
        double r196303 = pow(r196291, r196302);
        double r196304 = r196301 + r196303;
        double r196305 = 3.0;
        double r196306 = pow(r196291, r196305);
        double r196307 = r196304 - r196306;
        double r196308 = r196299 + r196307;
        double r196309 = r196298 * r196296;
        double r196310 = r196294 / r196309;
        double r196311 = r196291 / r196296;
        double r196312 = r196291 - r196311;
        double r196313 = r196310 + r196312;
        double r196314 = r196293 ? r196308 : r196313;
        return r196314;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.1
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 4.811043021165421e-08

    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. 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 4.811043021165421e-08 < wj

    1. Initial program 28.7

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

    2. Simplified2.3

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

    4. Applied associate--l+2.3

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

    6. Applied div-inv2.3

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

    7. Applied associate-/l*2.3

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

    8. Simplified2.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(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))))))