\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 24.1 s
Input Error: 63.3
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left(N + 1\right) \cdot \log \left(N + 1\right) - \left(\log N \cdot \left(-N\right) + 1\right)\)
  1. Started with
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    63.3
  2. Applied taylor to get
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1 \leadsto \left(\left(N + 1\right) \cdot \log \left(N + 1\right) - -1 \cdot \left(\log N \cdot N\right)\right) - 1\]
    0.3
  3. Taylor expanded around inf to get
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - \color{red}{-1 \cdot \left(\log N \cdot N\right)}\right) - 1 \leadsto \left(\left(N + 1\right) \cdot \log \left(N + 1\right) - \color{blue}{-1 \cdot \left(\log N \cdot N\right)}\right) - 1\]
    0.3
  4. Applied simplify to get
    \[\color{red}{\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - -1 \cdot \left(\log N \cdot N\right)\right) - 1} \leadsto \color{blue}{\left(N + 1\right) \cdot \log \left(N + 1\right) - \left(\log N \cdot \left(-N\right) + 1\right)}\]
    0.3

Original test:


(lambda ((N default))
  #:name "NMSE example 3.8"
  (- (- (* (+ N 1) (log (+ N 1))) (* N (log N))) 1)
  #:target
  (- (log (+ N 1)) (- (/ 1 (* 2 N)) (- (/ 1 (* 3 (sqr N))) (/ 4 (pow N 3))))))