Average Error: 13.8 → 1.0
Time: 3.6s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 6.379665546272698 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}} - wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 6.379665546272698 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

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

\end{array}
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.379665546272698e-09)
   (+ x (* wj (+ wj (* x -2.0))))
   (+ wj (/ (- (/ (/ x (sqrt (exp wj))) (sqrt (exp wj))) wj) (+ wj 1.0)))))
double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}
↓
double code(double wj, double x) {
	double tmp;
	if (wj <= 6.379665546272698e-09) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + ((((x / sqrt(exp(wj))) / sqrt(exp(wj))) - wj) / (wj + 1.0));
	}
	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

Original13.8
Target13.2
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 6.3796655462726977e-9

    1. Initial program 13.5

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 6.3796655462726977e-9 < wj

    1. Initial program 28.3

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

    2. Simplified2.4

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

    4. Applied add-sqr-sqrt_binary642.6

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

    5. Applied associate-/r*_binary642.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.379665546272698 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}} - wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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))))))