Average Error: 13.5 → 1.4
Time: 4.4s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -6.0276563278676428 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \left(\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -6.0276563278676428 \cdot 10^{-9}:\\
\;\;\;\;\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \left(\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right)\\

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

\end{array}
double code(double wj, double x) {
	return ((double) (wj - ((double) (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj))))))))));
}
↓
double code(double wj, double x) {
	double VAR;
	if ((wj <= -6.027656327867643e-09)) {
		VAR = ((double) (((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0)))))))) * ((double) (((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0)))))))) * ((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0))))))))))));
	} else {
		VAR = ((double) (x + ((double) (wj * ((double) (wj - ((double) (x * 2.0))))))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target13.0
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -6.0276563278676428e-9

    1. Initial program 5.6

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

    2. Simplified5.6

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

    4. Applied add-cube-cbrt6.2

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

    if -6.0276563278676428e-9 < wj

    1. Initial program 13.6

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

    2. Simplified13.2

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -6.0276563278676428 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \left(\sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj - x \cdot 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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