Average Error: 13.9 → 0.9
Time: 5.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.855396681840028 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}} \cdot \frac{\sqrt[3]{wj}}{\sqrt[3]{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.855396681840028 \cdot 10^{-8}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}} \cdot \frac{\sqrt[3]{wj}}{\sqrt[3]{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 temp;
	if ((wj <= 1.855396681840028e-08)) {
		temp = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		temp = ((((x / (wj + 1.0)) / exp(wj)) + wj) - (((cbrt(wj) * cbrt(wj)) / (cbrt((wj + 1.0)) * cbrt((wj + 1.0)))) * (cbrt(wj) / cbrt((wj + 1.0)))));
	}
	return temp;
}

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
Target13.4
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.855396681840028e-08

    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. Taylor expanded around 0 0.8

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

    if 1.855396681840028e-08 < wj

    1. Initial program 24.3

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

    2. Simplified3.3

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

    4. Applied add-cube-cbrt3.8

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

    5. Applied add-cube-cbrt4.2

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

    6. Applied times-frac4.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.855396681840028 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{\sqrt[3]{wj} \cdot \sqrt[3]{wj}}{\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}} \cdot \frac{\sqrt[3]{wj}}{\sqrt[3]{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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))))))