Average Error: 13.4 → 1.0
Time: 5.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.305370283524307 \cdot 10^{-10}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{wj}}} \cdot \frac{\frac{1}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.305370283524307 \cdot 10^{-10}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{\sqrt{e^{wj}}} \cdot \frac{\frac{1}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \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 <= 1.305370283524307e-10)) {
		VAR = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		VAR = ((((x / sqrt(exp(wj))) * ((1.0 / (wj + 1.0)) / sqrt(exp(wj)))) + 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

Original13.4
Target12.7
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.305370283524307e-10

    1. Initial program 13.0

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

    2. Simplified13.0

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

    3. Taylor expanded around 0 0.9

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

    if 1.305370283524307e-10 < wj

    1. Initial program 26.0

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

    2. Simplified3.8

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

    4. Applied add-sqr-sqrt3.9

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

    5. Applied div-inv3.9

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

    6. Applied times-frac3.9

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.305370283524307 \cdot 10^{-10}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{wj}}} \cdot \frac{\frac{1}{wj + 1}}{\sqrt{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(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))))))