Average Error: 14.4 → 0.6
Time: 4.0s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq -6.170330481140069 \cdot 10^{-09}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 1.6940097534747777 \cdot 10^{-08}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq -6.170330481140069 \cdot 10^{-09}:\\
\;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{elif}\;wj \leq 1.6940097534747777 \cdot 10^{-08}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

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

\end{array}
double code(double wj, double x) {
	return ((double) (wj - (((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 <= -6.170330481140069e-09)) {
		VAR = ((double) (wj + (((double) (x - ((double) (wj * ((double) exp(wj)))))) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj)))))))));
	} else {
		double VAR_1;
		if ((wj <= 1.6940097534747777e-08)) {
			VAR_1 = ((double) (x + ((double) (wj * ((double) (wj + ((double) (x * -2.0))))))));
		} else {
			VAR_1 = (((double) (((double) (wj * wj)) + ((double) ((((double) ((x / ((double) exp(wj))) - wj)) / ((double) (wj + 1.0))) * (((double) (wj - (x / ((double) exp(wj))))) / ((double) (wj + 1.0))))))) / ((double) (wj + (((double) (wj - (x / ((double) exp(wj))))) / ((double) (wj + 1.0))))));
		}
		VAR = VAR_1;
	}
	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

Original14.4
Target13.6
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < -6.17033048114006892e-9

    1. Initial program 5.7

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

    if -6.17033048114006892e-9 < wj < 1.69400975347477775e-8

    1. Initial program 14.1

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

    2. Simplified14.1

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

    if 1.69400975347477775e-8 < wj

    1. Initial program 27.9

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

    2. Simplified2.5

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

    4. Applied flip-+9.9

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

    5. Simplified9.9

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

    6. Simplified9.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.170330481140069 \cdot 10^{-09}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 1.6940097534747777 \cdot 10^{-08}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))))