\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 56.7 s
Input Error: 31.0
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\left(\frac{\left(N \cdot 2 + 1\right) \cdot \left(\log N \cdot \log N\right) + \left(1 + \frac{1}{N}\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + \log N \cdot N} - \frac{\log N \cdot \left(\frac{\frac{2}{3}}{N} + \left(3 + N \cdot 2\right)\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + \log N \cdot N}\right) - 1\)
  1. Started with
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    31.0
  2. Using strategy rm
    31.0
  3. Applied flip-+ to get
    \[\left(\color{red}{\left(N + 1\right)} \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{\frac{{N}^2 - {1}^2}{N - 1}} \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    30.1
  4. Applied associate-*l/ to get
    \[\left(\color{red}{\frac{{N}^2 - {1}^2}{N - 1} \cdot \log \left(N + 1\right)} - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{\frac{\left({N}^2 - {1}^2\right) \cdot \log \left(N + 1\right)}{N - 1}} - N \cdot \log N\right) - 1\]
    29.9
  5. Applied simplify to get
    \[\left(\frac{\color{red}{\left({N}^2 - {1}^2\right) \cdot \log \left(N + 1\right)}}{N - 1} - N \cdot \log N\right) - 1 \leadsto \left(\frac{\color{blue}{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{N - 1} - N \cdot \log N\right) - 1\]
    29.9
  6. Using strategy rm
    29.9
  7. Applied add-cube-cbrt to get
    \[\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{\color{red}{N - 1}} - N \cdot \log N\right) - 1 \leadsto \left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{\color{blue}{{\left(\sqrt[3]{N - 1}\right)}^3}} - N \cdot \log N\right) - 1\]
    29.9
  8. Applied add-cube-cbrt to get
    \[\left(\frac{\color{red}{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{{\left(\sqrt[3]{N - 1}\right)}^3} - N \cdot \log N\right) - 1 \leadsto \left(\frac{\color{blue}{{\left(\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}\right)}^3}}{{\left(\sqrt[3]{N - 1}\right)}^3} - N \cdot \log N\right) - 1\]
    29.9
  9. Applied cube-undiv to get
    \[\left(\color{red}{\frac{{\left(\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}\right)}^3}{{\left(\sqrt[3]{N - 1}\right)}^3}} - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{{\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3} - N \cdot \log N\right) - 1\]
    29.9
  10. Using strategy rm
    29.9
  11. Applied flip-- to get
    \[\color{red}{\left({\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3 - N \cdot \log N\right)} - 1 \leadsto \color{blue}{\frac{{\left({\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}{{\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3 + N \cdot \log N}} - 1\]
    29.9
  12. Applied simplify to get
    \[\frac{\color{red}{{\left({\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}}{{\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3 + N \cdot \log N} - 1 \leadsto \frac{\color{blue}{{\left(\frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}\right)}^2 - {\left(\log N \cdot N\right)}^2}}{{\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3 + N \cdot \log N} - 1\]
    30.9
  13. Applied simplify to get
    \[\frac{{\left(\frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}\right)}^2 - {\left(\log N \cdot N\right)}^2}{\color{red}{{\left(\frac{\sqrt[3]{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{\sqrt[3]{N - 1}}\right)}^3 + N \cdot \log N}} - 1 \leadsto \frac{{\left(\frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}\right)}^2 - {\left(\log N \cdot N\right)}^2}{\color{blue}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}}} - 1\]
    30.1
  14. Applied taylor to get
    \[\frac{{\left(\frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}\right)}^2 - {\left(\log N \cdot N\right)}^2}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}} - 1 \leadsto \frac{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}} - 1\]
    0.0
  15. Taylor expanded around inf to get
    \[\frac{\color{red}{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}} - 1 \leadsto \frac{\color{blue}{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}} - 1\]
    0.0
  16. Applied simplify to get
    \[\frac{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}{N \cdot \log N + \frac{\log \left(N + 1\right)}{\frac{1}{N + 1}}} - 1 \leadsto \frac{\left(\log N \cdot \log N + 1\right) + \left(\left(\left(N \cdot 2\right) \cdot \log N\right) \cdot \log N + \frac{1}{N}\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + N \cdot \log N} - \left(\frac{\frac{\log N}{\frac{N}{\frac{2}{3}}} + \left(\left(N \cdot 2\right) \cdot \log N + \log N \cdot 3\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + N \cdot \log N} + 1\right)\]
    0.4
  17. Applied final simplification
  18. Applied simplify to get
    \[\color{red}{\frac{\left(\log N \cdot \log N + 1\right) + \left(\left(\left(N \cdot 2\right) \cdot \log N\right) \cdot \log N + \frac{1}{N}\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + N \cdot \log N} - \left(\frac{\frac{\log N}{\frac{N}{\frac{2}{3}}} + \left(\left(N \cdot 2\right) \cdot \log N + \log N \cdot 3\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + N \cdot \log N} + 1\right)} \leadsto \color{blue}{\left(\frac{\left(N \cdot 2 + 1\right) \cdot \left(\log N \cdot \log N\right) + \left(1 + \frac{1}{N}\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + \log N \cdot N} - \frac{\log N \cdot \left(\frac{\frac{2}{3}}{N} + \left(3 + N \cdot 2\right)\right)}{\frac{\log \left(1 + N\right)}{\frac{1}{1 + N}} + \log N \cdot N}\right) - 1}\]
    0.4

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))))))