Average Error: 14.0 → 1.2
Time: 4.1s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 8.746783178569327 \cdot 10^{-06}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} - wj}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 8.746783178569327 \cdot 10^{-06}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} - 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 <= 8.746783178569327e-06)) {
		VAR = ((double) (x + ((double) (wj * ((double) (wj + ((double) (x * -2.0))))))));
	} else {
		VAR = (((double) (((double) pow(wj, 3.0)) + ((double) pow((((double) ((x / ((double) exp(wj))) - wj)) / ((double) (wj + 1.0))), 3.0)))) / ((double) (((double) (wj * wj)) + ((double) (((double) ((x / ((double) exp(wj))) - wj)) * (((double) ((((double) ((x / ((double) exp(wj))) - wj)) / ((double) (wj + 1.0))) - 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

Original14.0
Target13.5
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.7467831785693269e-6

    1. Initial program Error: 13.7 bits

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

    2. SimplifiedError: 13.7 bits

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

    3. Taylor expanded around 0 Error: 0.9 bits

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

    4. SimplifiedError: 0.9 bits

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

    if 8.7467831785693269e-6 < wj

    1. Initial program Error: 26.5 bits

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

    2. SimplifiedError: 1.7 bits

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

    4. Applied flip3-+Error: 14.4 bits

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

    5. SimplifiedError: 14.4 bits

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

  4. Final simplificationError: 1.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.746783178569327 \cdot 10^{-06}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} - wj}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))))