\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 1.0 m
Input Error: 31.0
Output Error: 29.9
Log: ⚲
Profile: 🕒
\(\left(\sqrt{N \cdot \log N} + \frac{\frac{1}{N} - \left(\frac{\frac{\frac{1}{8}}{N}}{{N}^{4}} + \frac{\frac{1}{2}}{{N}^3}\right)}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}}\right) \cdot \left(\frac{\sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} \cdot \sqrt{{N}^2 - 1} - \sqrt{N \cdot \log N}\right) - 1\)
  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-sqr-sqrt to get
    \[\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1} - \color{red}{N \cdot \log N}\right) - 1 \leadsto \left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1} - \color{blue}{{\left(\sqrt{N \cdot \log N}\right)}^2}\right) - 1\]
    29.9
  8. Applied add-sqr-sqrt to get
    \[\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{\color{red}{N - 1}} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1 \leadsto \left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{\color{blue}{{\left(\sqrt{N - 1}\right)}^2}} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1\]
    29.9
  9. Applied add-sqr-sqrt to get
    \[\left(\frac{\left({N}^2 - 1\right) \cdot \color{red}{\log \left(N + 1\right)}}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1 \leadsto \left(\frac{\left({N}^2 - 1\right) \cdot \color{blue}{{\left(\sqrt{\log \left(N + 1\right)}\right)}^2}}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1\]
    29.9
  10. Applied add-sqr-sqrt to get
    \[\left(\frac{\color{red}{\left({N}^2 - 1\right)} \cdot {\left(\sqrt{\log \left(N + 1\right)}\right)}^2}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1 \leadsto \left(\frac{\color{blue}{{\left(\sqrt{{N}^2 - 1}\right)}^2} \cdot {\left(\sqrt{\log \left(N + 1\right)}\right)}^2}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1\]
    29.9
  11. Applied square-unprod to get
    \[\left(\frac{\color{red}{{\left(\sqrt{{N}^2 - 1}\right)}^2 \cdot {\left(\sqrt{\log \left(N + 1\right)}\right)}^2}}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1 \leadsto \left(\frac{\color{blue}{{\left(\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}\right)}^2}}{{\left(\sqrt{N - 1}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1\]
    29.9
  12. Applied square-undiv to get
    \[\left(\color{red}{\frac{{\left(\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}\right)}^2}{{\left(\sqrt{N - 1}\right)}^2}} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1 \leadsto \left(\color{blue}{{\left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}}\right)}^2} - {\left(\sqrt{N \cdot \log N}\right)}^2\right) - 1\]
    29.9
  13. Applied difference-of-squares to get
    \[\color{red}{\left({\left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}}\right)}^2 - {\left(\sqrt{N \cdot \log N}\right)}^2\right)} - 1 \leadsto \color{blue}{\left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right)} - 1\]
    29.9
  14. Using strategy rm
    29.9
  15. Applied add-exp-log to get
    \[\left(\frac{\color{red}{\sqrt{{N}^2 - 1}} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1 \leadsto \left(\frac{\color{blue}{e^{\log \left(\sqrt{{N}^2 - 1}\right)}} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1\]
    29.9
  16. Applied taylor to get
    \[\left(\frac{e^{\log \left(\sqrt{{N}^2 - 1}\right)} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1 \leadsto \left(\frac{\left(e^{-\log N} - \left(\frac{1}{8} \cdot \frac{e^{-\log N}}{{N}^{4}} + \frac{1}{2} \cdot \frac{e^{-\log N}}{{N}^2}\right)\right) \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1\]
    29.9
  17. Taylor expanded around inf to get
    \[\left(\frac{\color{red}{\left(e^{-\log N} - \left(\frac{1}{8} \cdot \frac{e^{-\log N}}{{N}^{4}} + \frac{1}{2} \cdot \frac{e^{-\log N}}{{N}^2}\right)\right)} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1 \leadsto \left(\frac{\color{blue}{\left(e^{-\log N} - \left(\frac{1}{8} \cdot \frac{e^{-\log N}}{{N}^{4}} + \frac{1}{2} \cdot \frac{e^{-\log N}}{{N}^2}\right)\right)} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1\]
    29.9
  18. Applied simplify to get
    \[\left(\frac{\left(e^{-\log N} - \left(\frac{1}{8} \cdot \frac{e^{-\log N}}{{N}^{4}} + \frac{1}{2} \cdot \frac{e^{-\log N}}{{N}^2}\right)\right) \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} + \sqrt{N \cdot \log N}\right) \cdot \left(\frac{\sqrt{{N}^2 - 1} \cdot \sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} - \sqrt{N \cdot \log N}\right) - 1 \leadsto \left(\frac{\sqrt{{N}^2 - 1}}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}} - \sqrt{\log N \cdot N}\right) \cdot \left(\frac{\frac{1}{N} - \left(\frac{\frac{\frac{1}{8}}{N}}{{N}^{4}} + \frac{\frac{\frac{1}{2}}{N}}{{N}^2}\right)}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}} + \sqrt{\log N \cdot N}\right) - 1\]
    29.9
  19. Applied final simplification
  20. Applied simplify to get
    \[\color{red}{\left(\frac{\sqrt{{N}^2 - 1}}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}} - \sqrt{\log N \cdot N}\right) \cdot \left(\frac{\frac{1}{N} - \left(\frac{\frac{\frac{1}{8}}{N}}{{N}^{4}} + \frac{\frac{\frac{1}{2}}{N}}{{N}^2}\right)}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}} + \sqrt{\log N \cdot N}\right) - 1} \leadsto \color{blue}{\left(\sqrt{N \cdot \log N} + \frac{\frac{1}{N} - \left(\frac{\frac{\frac{1}{8}}{N}}{{N}^{4}} + \frac{\frac{1}{2}}{{N}^3}\right)}{\frac{\sqrt{N - 1}}{\sqrt{\log \left(N + 1\right)}}}\right) \cdot \left(\frac{\sqrt{\log \left(N + 1\right)}}{\sqrt{N - 1}} \cdot \sqrt{{N}^2 - 1} - \sqrt{N \cdot \log N}\right) - 1}\]
    29.9
  21. 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))))))