Average Error: 13.7 → 2.1
Time: 3.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[x + wj \cdot \left(wj - x \cdot 2\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
x + wj \cdot \left(wj - x \cdot 2\right)
double code(double wj, double x) {
	return (wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj)))));
}
↓
double code(double wj, double x) {
	return (x + (wj * (wj - (x * 2.0))));
}

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
Target13.2
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

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

  4. Final simplification2.1

    \[\leadsto x + wj \cdot \left(wj - x \cdot 2\right)\]

Reproduce

herbie shell --seed 2020106 
(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))))))