Average Error: 13.9 → 0.5
Time: 4.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.58110533664 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{5}{2} \cdot {wj}^{2} - 2 \cdot wj, x\right) + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{wj + \frac{wj}{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.58110533664 \cdot 10^{-12}:\\
\;\;\;\;\mathsf{fma}\left(x, \frac{5}{2} \cdot {wj}^{2} - 2 \cdot wj, x\right) + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\

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

\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 - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))) <= 6.581105336643539e-12)) {
		VAR = (fma(x, ((2.5 * pow(wj, 2.0)) - (2.0 * wj)), x) + fma(wj, wj, (pow(wj, 4.0) - pow(wj, 3.0))));
	} else {
		VAR = (((x / (wj + 1.0)) / exp(wj)) + (((wj * wj) - ((wj / (wj + 1.0)) * (wj / (wj + 1.0)))) / (wj + (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

Original13.9
Target13.3
Herbie0.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 6.581105336643539e-12

    1. Initial program 18.4

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

    2. Simplified18.4

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

    4. Applied associate--l+10.0

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

    5. Taylor expanded around 0 0.2

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

    6. Simplified0.2

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

    7. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left({wj}^{2} \cdot x\right)\right) - 2 \cdot \left(wj \cdot x\right)\right)} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]

    8. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{5}{2} \cdot {wj}^{2} - 2 \cdot wj, x\right)} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]

    if 6.581105336643539e-12 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 2.3

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

    2. Simplified0.3

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

    4. Applied associate--l+0.3

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

    6. Applied flip--0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.58110533664 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{5}{2} \cdot {wj}^{2} - 2 \cdot wj, x\right) + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{wj + \frac{wj}{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))))