Average Error: 13.5 → 0.7
Time: 13.2s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 1.3775701032820146 \cdot 10^{-06}:\\ \;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2 \cdot \left(wj \cdot x\right) + 2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{1}{\frac{e^{wj}}{x}} - wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 1.3775701032820146 \cdot 10^{-06}:\\
\;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2 \cdot \left(wj \cdot x\right) + 2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{\frac{1}{\frac{e^{wj}}{x}} - 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 1.3775701032820146e-06)
   (-
    (+ (pow wj 2.0) (+ x (* 2.5 (* (pow wj 2.0) x))))
    (+
     (pow wj 3.0)
     (+ (* 2.0 (* wj x)) (* 2.6666666666666665 (* x (pow wj 3.0))))))
   (+ wj (/ (- (/ 1.0 (/ (exp wj) x)) 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 <= 1.3775701032820146e-06) {
		tmp = (pow(wj, 2.0) + (x + (2.5 * (pow(wj, 2.0) * x)))) - (pow(wj, 3.0) + ((2.0 * (wj * x)) + (2.6666666666666665 * (x * pow(wj, 3.0)))));
	} else {
		tmp = wj + (((1.0 / (exp(wj) / x)) - 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

Original13.5
Target12.8
Herbie0.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.37757010328201456e-6

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 0.6

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

    4. Simplified0.7

      \[\leadsto \color{blue}{x + \left(\left(2.5 \cdot x + 1\right) \cdot \left(wj \cdot wj\right) - \left({wj}^{3} + x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\right)}\]

    5. Taylor expanded around 0 0.6

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

    if 1.37757010328201456e-6 < wj

    1. Initial program 27.7

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

    2. Simplified1.7

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

    4. Applied clear-num_binary64_34871.7

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3775701032820146 \cdot 10^{-06}:\\ \;\;\;\;\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2 \cdot \left(wj \cdot x\right) + 2.6666666666666665 \cdot \left(x \cdot {wj}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{1}{\frac{e^{wj}}{x}} - wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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))))))