Average Error: 13.2 → 0.8
Time: 34.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.783506544229532 \cdot 10^{-19}:\\ \;\;\;\;x + \left(wj + -2 \cdot x\right) \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{1 + wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.783506544229532 \cdot 10^{-19}:\\
\;\;\;\;x + \left(wj + -2 \cdot x\right) \cdot wj\\

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

\end{array}
double f(double wj, double x) {
        double r55892276 = wj;
        double r55892277 = exp(r55892276);
        double r55892278 = r55892276 * r55892277;
        double r55892279 = x;
        double r55892280 = r55892278 - r55892279;
        double r55892281 = r55892277 + r55892278;
        double r55892282 = r55892280 / r55892281;
        double r55892283 = r55892276 - r55892282;
        return r55892283;
}
↓
double f(double wj, double x) {
        double r55892284 = wj;
        double r55892285 = exp(r55892284);
        double r55892286 = r55892284 * r55892285;
        double r55892287 = x;
        double r55892288 = r55892286 - r55892287;
        double r55892289 = r55892285 + r55892286;
        double r55892290 = r55892288 / r55892289;
        double r55892291 = r55892284 - r55892290;
        double r55892292 = 3.783506544229532e-19;
        bool r55892293 = r55892291 <= r55892292;
        double r55892294 = -2.0;
        double r55892295 = r55892294 * r55892287;
        double r55892296 = r55892284 + r55892295;
        double r55892297 = r55892296 * r55892284;
        double r55892298 = r55892287 + r55892297;
        double r55892299 = 1.0;
        double r55892300 = r55892299 + r55892284;
        double r55892301 = r55892299 / r55892300;
        double r55892302 = r55892287 / r55892285;
        double r55892303 = r55892284 - r55892302;
        double r55892304 = r55892301 * r55892303;
        double r55892305 = r55892284 - r55892304;
        double r55892306 = r55892293 ? r55892298 : r55892305;
        return r55892306;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target12.7
Herbie0.8
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 3.783506544229532e-19

    1. Initial program 17.4

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.8

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

    if 3.783506544229532e-19 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 2.5

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied distribute-rgt1-in2.5

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

    4. Applied *-un-lft-identity2.5

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

    5. Applied times-frac2.5

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

    6. Simplified0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.783506544229532 \cdot 10^{-19}:\\ \;\;\;\;x + \left(wj + -2 \cdot x\right) \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{1 + wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))