Average Error: 13.7 → 0.6
Time: 5.5s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 2.941308039666007 \cdot 10^{-06}:\\ \;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right) + 2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 2.941308039666007 \cdot 10^{-06}:\\
\;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right) + 2 \cdot \left(wj \cdot x\right)\right)\right)\\

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

\end{array}
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.941308039666007e-06)
   (-
    (+ (pow wj 2.0) (+ x (* 2.5 (* (pow wj 2.0) x))))
    (+
     (pow wj 3.0)
     (+ (* 2.6666666666666665 (* x (pow wj 3.0))) (* 2.0 (* wj x)))))
   (+ wj (* (/ (- (/ x (exp wj)) wj) (- (* wj wj) 1.0)) (- 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 <= 2.941308039666007e-06) {
		tmp = (pow(wj, 2.0) + (x + (2.5 * (pow(wj, 2.0) * x)))) - (pow(wj, 3.0) + ((2.6666666666666665 * (x * pow(wj, 3.0))) + (2.0 * (wj * x))));
	} else {
		tmp = wj + ((((x / exp(wj)) - wj) / ((wj * wj) - 1.0)) * (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

Original13.7
Target13.0
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 2.9413080396660069e-6

    1. Initial program 13.3

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

    2. Simplified13.3

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

    3. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2.6666666666666665 \cdot \left({wj}^{3} \cdot x\right) + 2 \cdot \left(wj \cdot x\right)\right)\right)}\]

    if 2.9413080396660069e-6 < wj

    1. Initial program 29.6

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

    2. Simplified1.5

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

    4. Applied flip-+_binary641.5

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

    5. Applied associate-/r/_binary641.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.941308039666007 \cdot 10^{-06}:\\ \;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right) + 2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))))))