Average Error: 13.5 → 1.1
Time: 3.9s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 5.067437279036206 \cdot 10^{-16}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 5.067437279036206 \cdot 10^{-16}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{\frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}}{\sqrt{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 5.067437279036206e-16)
   (+ x (* wj (+ wj (* x -2.0))))
   (+ wj (/ (/ (- (/ x (exp wj)) wj) (sqrt (+ wj 1.0))) (sqrt (+ 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 <= 5.067437279036206e-16) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + ((((x / exp(wj)) - wj) / sqrt(wj + 1.0)) / sqrt(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

Original13.5
Target12.9
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 5.0674372790362063e-16

    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 0.9

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

    4. Simplified0.9

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

    if 5.0674372790362063e-16 < wj

    1. Initial program 20.6

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

    2. Simplified5.5

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

    4. Applied add-sqr-sqrt_binary645.8

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

    5. Applied associate-/r*_binary645.8

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.067437279036206 \cdot 10^{-16}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))))))