Average Error: 13.4 → 1.5
Time: 28.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{else}:\\
\;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r468297 = wj;
        double r468298 = exp(r468297);
        double r468299 = r468297 * r468298;
        double r468300 = x;
        double r468301 = r468299 - r468300;
        double r468302 = r468298 + r468299;
        double r468303 = r468301 / r468302;
        double r468304 = r468297 - r468303;
        return r468304;
}
↓
double f(double wj, double x) {
        double r468305 = wj;
        double r468306 = -5.00771993997508e-09;
        bool r468307 = r468305 <= r468306;
        double r468308 = exp(r468305);
        double r468309 = r468305 * r468308;
        double r468310 = x;
        double r468311 = r468309 - r468310;
        double r468312 = r468308 + r468309;
        double r468313 = r468311 / r468312;
        double r468314 = r468305 - r468313;
        double r468315 = 2.0;
        double r468316 = r468315 * r468310;
        double r468317 = r468305 - r468316;
        double r468318 = r468305 * r468317;
        double r468319 = r468310 + r468318;
        double r468320 = r468307 ? r468314 : r468319;
        return r468320;
}

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
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -5.00771993997508e-09

    1. Initial program 5.8

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

    if -5.00771993997508e-09 < wj

    1. Initial program 13.6

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

    2. Simplified12.9

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

    3. Taylor expanded around 0 1.4

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

    4. Simplified1.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))