Average Error: 14.3 → 0.9
Time: 5.1s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 7.795004187076639 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{wj + 1}{\frac{x}{e^{wj}} - wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 7.795004187076639 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

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

\end{array}
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.795004187076639e-09)
   (+ x (* wj (+ wj (* x -2.0))))
   (+ wj (/ 1.0 (/ (+ wj 1.0) (- (/ x (exp wj)) wj))))))
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 tmp;
	if ((wj <= 7.795004187076639e-09)) {
		tmp = ((double) (x + ((double) (wj * ((double) (wj + ((double) (x * -2.0))))))));
	} else {
		tmp = ((double) (wj + (1.0 / (((double) (wj + 1.0)) / ((double) ((x / ((double) exp(wj))) - wj))))));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.3
Target13.7
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 7.795004187076639e-9

    1. Initial program Error: 14.0 bits

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

    2. SimplifiedError: 14.0 bits

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

    3. Taylor expanded around 0 Error: 0.8 bits

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

    4. SimplifiedError: 0.9 bits

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

    if 7.795004187076639e-9 < wj

    1. Initial program Error: 24.2 bits

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

    2. SimplifiedError: 2.8 bits

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

    4. Applied clear-numError: 2.9 bits

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

  4. Final simplificationError: 0.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.795004187076639 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{wj + 1}{\frac{x}{e^{wj}} - wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))))