Average Error: 13.3 → 0.3
Time: 18.3s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 6.1360703267965559 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(wj + \frac{wj}{wj + 1}\right) + wj \cdot wj} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 6.1360703267965559 \cdot 10^{-5}:\\
\;\;\;\;\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(wj + \frac{wj}{wj + 1}\right) + wj \cdot wj} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\

\end{array}
double code(double wj, double x) {
	return ((double) (wj - (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj)))))))));
}
↓
double code(double wj, double x) {
	double VAR;
	if ((wj <= 6.136070326796556e-05)) {
		VAR = ((double) (((double) (((double) (((double) pow(wj, 2.0)) + ((double) pow(wj, 4.0)))) - ((double) pow(wj, 3.0)))) + ((x / ((double) exp(wj))) / ((double) (wj + 1.0)))));
	} else {
		VAR = ((double) ((((double) (((double) pow(wj, 3.0)) - ((double) pow((wj / ((double) (wj + 1.0))), 3.0)))) / ((double) (((double) ((wj / ((double) (wj + 1.0))) * ((double) (wj + (wj / ((double) (wj + 1.0))))))) + ((double) (wj * wj))))) + ((x / ((double) exp(wj))) / ((double) (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

Original13.3
Target12.6
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 < 6.1360703267965559e-5

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied div-sub12.9

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

    5. Applied associate--r-6.4

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

    6. Taylor expanded around 0 0.2

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

    if 6.1360703267965559e-5 < wj

    1. Initial program 33.5

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

    2. Simplified0.9

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

    4. Applied div-sub0.9

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

    5. Applied associate--r-0.9

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

    7. Applied flip3--1.2

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

    8. Simplified1.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.1360703267965559 \cdot 10^{-5}:\\ \;\;\;\;\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} - {\left(\frac{wj}{wj + 1}\right)}^{3}}{\frac{wj}{wj + 1} \cdot \left(wj + \frac{wj}{wj + 1}\right) + wj \cdot wj} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))))