Average Error: 13.7 → 1.2
Time: 7.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 5.3258678519809691 \cdot 10^{-17}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) + \left(1 \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj\right)}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 5.3258678519809691 \cdot 10^{-17}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\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 <= 5.325867851980969e-17)) {
		temp = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		temp = (((wj * ((((x / (wj + 1.0)) / exp(wj)) * ((x / (wj + 1.0)) / exp(wj))) - (wj * wj))) + ((1.0 * ((((x / (wj + 1.0)) / exp(wj)) * ((x / (wj + 1.0)) / exp(wj))) - (wj * wj))) - ((((x / (wj + 1.0)) / exp(wj)) - wj) * wj))) / ((((x / (wj + 1.0)) / exp(wj)) - wj) * (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

Original13.7
Target13.1
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 < 5.325867851980969e-17

    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 0.8

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

    if 5.325867851980969e-17 < wj

    1. Initial program 24.5

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

    2. Simplified7.3

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

    4. Applied flip-+16.8

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

    5. Applied frac-sub17.0

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

    7. Applied distribute-rgt-in16.8

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

    8. Applied associate--l+10.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 5.3258678519809691 \cdot 10^{-17}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) + \left(1 \cdot \left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj\right)}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))))))