\[\log \left(N + 1\right) - \log N\]
Test:
NMSE problem 3.3.6
Bits:
128 bits
Bits error versus N
Time: 2.7 s
Input Error: 40.7
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\log_* (1 + \frac{1}{N})\)
  1. Started with
    \[\log \left(N + 1\right) - \log N\]
    40.7
  2. Applied simplify to get
    \[\color{red}{\log \left(N + 1\right) - \log N} \leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    40.7
  3. Using strategy rm
    40.7
  4. Applied log1p-udef to get
    \[\color{red}{\log_* (1 + N)} - \log N \leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
    40.7
  5. Applied diff-log to get
    \[\color{red}{\log \left(1 + N\right) - \log N} \leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
    39.1
  6. Applied taylor to get
    \[\log \left(\frac{1 + N}{N}\right) \leadsto \log \left(1 + \frac{1}{N}\right)\]
    39.1
  7. Taylor expanded around 0 to get
    \[\log \color{red}{\left(1 + \frac{1}{N}\right)} \leadsto \log \color{blue}{\left(1 + \frac{1}{N}\right)}\]
    39.1
  8. Applied simplify to get
    \[\color{red}{\log \left(1 + \frac{1}{N}\right)} \leadsto \color{blue}{\log_* (1 + \frac{1}{N})}\]
    0.0

Original test:


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