Average Error: 13.6 → 0.3
Time: 6.2s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -3.7212948846946436 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \le 6.4820602777102907 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{e^{wj}}} \cdot \frac{\frac{x}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -3.7212948846946436 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{elif}\;wj \le 6.4820602777102907 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double code(double wj, double x) {
	return ((double) (wj - ((double) (((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 <= -3.7212948846946436e-09)) {
		VAR = ((double) (wj - ((double) (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj))))))))));
	} else {
		double VAR_1;
		if ((wj <= 6.482060277710291e-09)) {
			VAR_1 = ((double) (((double) (x + ((double) pow(wj, 2.0)))) - ((double) (2.0 * ((double) (wj * x))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / ((double) sqrt(((double) exp(wj)))))) * ((double) (((double) (x / ((double) (wj + 1.0)))) / ((double) sqrt(((double) exp(wj)))))))) + wj)) - ((double) (wj / ((double) (wj + 1.0))))));
		}
		VAR = VAR_1;
	}
	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

Original13.6
Target12.8
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < -3.7212948846946436e-9

    1. Initial program 4.4

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

    if -3.7212948846946436e-9 < wj < 6.4820602777102907e-9

    1. Initial program 13.3

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

    2. Simplified13.3

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

    3. Taylor expanded around 0 0.2

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

    if 6.4820602777102907e-9 < wj

    1. Initial program 26.7

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

    2. Simplified2.8

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

    4. Applied add-sqr-sqrt2.9

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

    5. Applied *-un-lft-identity2.9

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

    6. Applied *-un-lft-identity2.9

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

    7. Applied times-frac2.9

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

    8. Applied times-frac2.9

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

    9. Simplified2.9

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -3.7212948846946436 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \le 6.4820602777102907 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{e^{wj}}} \cdot \frac{\frac{x}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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))))))