Average Error: 14.1 → 0.3
Time: 4.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.3145565373278193 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - e^{\log \left(\frac{wj}{wj + 1}\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.3145565373278193 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

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

\end{array}
double code(double wj, double x) {
	return (wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj)))));
}
↓
double code(double wj, double x) {
	double VAR;
	if ((wj <= 0.00013145565373278193)) {
		VAR = (((x / (wj + 1.0)) / exp(wj)) + ((pow(wj, 4.0) + pow(wj, 2.0)) - pow(wj, 3.0)));
	} else {
		VAR = ((((x / (wj + 1.0)) / exp(wj)) + wj) - exp(log((wj / (wj + 1.0)))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.4
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 0.00013145565373278193

    1. Initial program 13.7

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

    2. Simplified13.6

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

    4. Applied associate--l+7.2

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

    5. Taylor expanded around 0 0.3

      \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

    if 0.00013145565373278193 < wj

    1. Initial program 33.1

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

    2. Simplified1.0

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

    4. Applied add-exp-log1.1

      \[\leadsto \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{\color{blue}{e^{\log \left(wj + 1\right)}}}\]

    5. Applied add-exp-log1.3

      \[\leadsto \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{\color{blue}{e^{\log wj}}}{e^{\log \left(wj + 1\right)}}\]

    6. Applied div-exp1.4

      \[\leadsto \left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \color{blue}{e^{\log wj - \log \left(wj + 1\right)}}\]

    7. Simplified1.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.3145565373278193 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - e^{\log \left(\frac{wj}{wj + 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))