Average Error: 13.4 → 0.4
Time: 4.0s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq -4.5385141887408526 \cdot 10^{-09} \lor \neg \left(wj \leq 4.002066009806822 \cdot 10^{-09}\right):\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq -4.5385141887408526 \cdot 10^{-09} \lor \neg \left(wj \leq 4.002066009806822 \cdot 10^{-09}\right):\\
\;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\

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

\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 <= -4.5385141887408526e-09) || !(wj <= 4.002066009806822e-09))) {
		VAR = ((double) (((double) (wj + (x / ((double) (((double) exp(wj)) * ((double) (wj + 1.0))))))) - (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

Original13.4
Target12.8
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -4.53851418874085257e-9 or 4.00206600980682213e-9 < wj

    1. Initial program 18.3

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

    2. Simplified4.2

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

    4. Applied div-sub4.2

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

    5. Applied associate-+r-4.2

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

    6. Simplified4.2

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

    if -4.53851418874085257e-9 < wj < 4.00206600980682213e-9

    1. Initial program 13.2

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

    2. Simplified13.2

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

    3. Taylor expanded around 0 0.1

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

    4. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5385141887408526 \cdot 10^{-09} \lor \neg \left(wj \leq 4.002066009806822 \cdot 10^{-09}\right):\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))))