Average Error: 14.1 → 0.3
Time: 4.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.9673801197410912 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{wj + 1}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt{wj + 1}}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.9673801197410912 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{wj + 1}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt{wj + 1}}}{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 <= 9.967380119741091e-05)) {
		VAR = ((double) (((double) (((double) (x / ((double) (wj + 1.0)))) / ((double) exp(wj)))) + ((double) (((double) (((double) pow(wj, 4.0)) + ((double) pow(wj, 2.0)))) - ((double) pow(wj, 3.0))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) sqrt(((double) (wj + 1.0)))))) * ((double) (((double) (((double) cbrt(x)) / ((double) sqrt(((double) (wj + 1.0)))))) / ((double) exp(wj)))))) + wj)) - ((double) (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

Original14.1
Target13.5
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 < 9.967380119741091e-05

    1. Initial program 13.7

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

    2. Simplified13.7

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

    4. Applied associate--l+7.3

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

    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 9.967380119741091e-05 < wj

    1. Initial program 30.3

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

    2. Simplified1.1

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

    4. Applied *-un-lft-identity1.1

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

    5. Applied add-sqr-sqrt1.2

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

    6. Applied add-cube-cbrt1.3

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

    7. Applied times-frac1.3

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

    8. Applied times-frac1.3

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

    9. Simplified1.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.9673801197410912 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{wj + 1}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt{wj + 1}}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))))))