\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 28.9 s
Input Error: 31.0
Output Error: 29.4
Log: ⚲
Profile: 🕒
\((e^{\log_* (1 + {\left(\sqrt[3]{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N}\right)}^3)} - 1)^* - 1\)
  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 fma-udef to get
    \[(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{red}{(N * \left(\log N\right) + 1)_*} \leadsto (N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \color{blue}{\left(N \cdot \log N + 1\right)}\]
    30.1
  5. Applied associate--r+ to get
    \[\color{red}{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - \left(N \cdot \log N + 1\right)} \leadsto \color{blue}{\left((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N\right) - 1}\]
    30.1
  6. Using strategy rm
    30.1
  7. Applied expm1-log1p-u to get
    \[\color{red}{\left((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N\right)} - 1 \leadsto \color{blue}{(e^{\log_* (1 + \left((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N\right))} - 1)^*} - 1\]
    29.4
  8. Using strategy rm
    29.4
  9. Applied add-cube-cbrt to get
    \[(e^{\log_* (1 + \color{red}{\left((N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N\right)})} - 1)^* - 1 \leadsto (e^{\log_* (1 + \color{blue}{{\left(\sqrt[3]{(N * \left(\log_* (1 + N)\right) + \left(\log_* (1 + N)\right))_* - N \cdot \log N}\right)}^3})} - 1)^* - 1\]
    29.4
  10. Removed slow pow expressions

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))))))