Average Error: 13.8 → 0.3
Time: 45.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -4.580237846772234 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{\frac{e^{wj} \cdot wj - x}{wj + 1}}{e^{wj}}\\ \mathbf{elif}\;wj \le 6.696494124738262 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -4.580237846772234 \cdot 10^{-09}:\\
\;\;\;\;wj - \frac{\frac{e^{wj} \cdot wj - x}{wj + 1}}{e^{wj}}\\

\mathbf{elif}\;wj \le 6.696494124738262 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\

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

\end{array}
double f(double wj, double x) {
        double r50761319 = wj;
        double r50761320 = exp(r50761319);
        double r50761321 = r50761319 * r50761320;
        double r50761322 = x;
        double r50761323 = r50761321 - r50761322;
        double r50761324 = r50761320 + r50761321;
        double r50761325 = r50761323 / r50761324;
        double r50761326 = r50761319 - r50761325;
        return r50761326;
}
↓
double f(double wj, double x) {
        double r50761327 = wj;
        double r50761328 = -4.580237846772234e-09;
        bool r50761329 = r50761327 <= r50761328;
        double r50761330 = exp(r50761327);
        double r50761331 = r50761330 * r50761327;
        double r50761332 = x;
        double r50761333 = r50761331 - r50761332;
        double r50761334 = 1.0;
        double r50761335 = r50761327 + r50761334;
        double r50761336 = r50761333 / r50761335;
        double r50761337 = r50761336 / r50761330;
        double r50761338 = r50761327 - r50761337;
        double r50761339 = 6.696494124738262e-09;
        bool r50761340 = r50761327 <= r50761339;
        double r50761341 = -2.0;
        double r50761342 = r50761341 * r50761332;
        double r50761343 = r50761342 + r50761327;
        double r50761344 = r50761327 * r50761343;
        double r50761345 = r50761332 + r50761344;
        double r50761346 = r50761332 / r50761330;
        double r50761347 = r50761327 - r50761346;
        double r50761348 = r50761334 / r50761335;
        double r50761349 = r50761347 * r50761348;
        double r50761350 = r50761327 - r50761349;
        double r50761351 = r50761340 ? r50761345 : r50761350;
        double r50761352 = r50761329 ? r50761338 : r50761351;
        return r50761352;
}

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
Target13.0
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < -4.580237846772234e-09

    1. Initial program 4.7

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

    3. Applied distribute-rgt1-in4.6

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

    4. Applied associate-/r*4.6

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

    if -4.580237846772234e-09 < wj < 6.696494124738262e-09

    1. Initial program 13.5

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.2

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

    if 6.696494124738262e-09 < wj

    1. Initial program 29.5

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

    3. Applied distribute-rgt1-in29.6

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

    4. Applied *-un-lft-identity29.6

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

    5. Applied times-frac29.5

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

    6. Simplified3.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -4.580237846772234 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{\frac{e^{wj} \cdot wj - x}{wj + 1}}{e^{wj}}\\ \mathbf{elif}\;wj \le 6.696494124738262 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(-2 \cdot x + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(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))))))