\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 19.9 s
Input Error: 31.0
Output Error: 30.1
Log: ⚲
Profile: 🕒
\((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - {\left(\sqrt{(N * \left(\log N\right) + 1)_*}\right)}^2\)
  1. Started with
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    31.0
  2. Applied simplify to get
    \[\color{red}{\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1} \leadsto \color{blue}{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - (N * \left(\log N\right) + 1)_*}\]
    30.0
  3. Using strategy rm
    30.0
  4. Applied add-sqr-sqrt to get
    \[(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{red}{(N * \left(\log N\right) + 1)_*} \leadsto (N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{blue}{{\left(\sqrt{(N * \left(\log N\right) + 1)_*}\right)}^2}\]
    30.1

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))))))