\[\log \left(N + 1\right) - \log N\]
Test:
NMSE problem 3.3.6
Bits:
128 bits
Bits error versus N
Time: 5.6 s
Input Error: 61.4
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{1}{N} - \frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}\)
  1. Started with
    \[\log \left(N + 1\right) - \log N\]
    61.4
  2. Applied simplify to get
    \[\color{red}{\log \left(N + 1\right) - \log N} \leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    61.4
  3. Using strategy rm
    61.4
  4. Applied log1p-udef to get
    \[\color{red}{\log_* (1 + N)} - \log N \leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
    61.4
  5. Applied diff-log to get
    \[\color{red}{\log \left(1 + N\right) - \log N} \leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
    59.6
  6. Applied taylor to get
    \[\log \left(\frac{1 + N}{N}\right) \leadsto \left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}\]
    0.1
  7. Taylor expanded around inf to get
    \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}}\]
    0.1
  8. Applied simplify to get
    \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)}\]
    0.1
  9. Applied simplify to get
    \[\frac{1}{N} - \color{red}{\frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)} \leadsto \frac{1}{N} - \color{blue}{\frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}}\]
    0.1

Original test:


(lambda ((N default))
  #:name "NMSE problem 3.3.6"
  (- (log (+ N 1)) (log N)))