\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 21.2 s
Input Error: 63.3
Output Error: 0.3
Log: ⚲
Profile: 🕒
\((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (\left(\frac{1}{N}\right) * \left(-\log N\right) + 1)_*\)
  1. Started with
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    63.3
  2. Applied simplify to get
    \[\color{red}{\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1} \leadsto \color{blue}{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (N * \left(\log N\right) + 1)_*}\]
    62.6
  3. Applied taylor to get
    \[(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (N * \left(\log N\right) + 1)_* \leadsto (N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (\left(\frac{1}{N}\right) * \left(\log \left(\frac{1}{N}\right)\right) + 1)_*\]
    0.3
  4. Taylor expanded around inf to get
    \[(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{red}{(\left(\frac{1}{N}\right) * \left(\log \left(\frac{1}{N}\right)\right) + 1)_*} \leadsto (N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{blue}{(\left(\frac{1}{N}\right) * \left(\log \left(\frac{1}{N}\right)\right) + 1)_*}\]
    0.3
  5. Applied simplify to get
    \[\color{red}{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (\left(\frac{1}{N}\right) * \left(\log \left(\frac{1}{N}\right)\right) + 1)_*} \leadsto \color{blue}{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (\left(\frac{1}{N}\right) * \left(-\log N\right) + 1)_*}\]
    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))))))