\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Test:
Jmat.Real.lambertw, newton loop step
Bits:
128 bits
Bits error versus wj
Bits error versus x
Time: 7.7 s
Input Error: 3.1
Output Error: 3.1
Log: ⚲
Profile: 🕒
\(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\)
  1. Started with
    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    3.1

Original test:


(lambda ((wj default) (x default))
  #:name "Jmat.Real.lambertw, newton loop step"
  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
  #:target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj)))))))