\[\log \left(N + 1\right) - \log N\]
Test:
NMSE problem 3.3.6
Bits:
128 bits
Bits error versus N
Time: 5.9 s
Input Error: 19.2
Output Error: 8.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(\frac{N + 1}{N}\right) & \text{when } N \le 53.3103f0 \\ \frac{1}{N} + \left(\frac{\frac{1}{3}}{{N}^3} - \frac{\frac{1}{2}}{N \cdot N}\right) & \text{otherwise} \end{cases}\)

    if N < 53.3103f0

    1. Started with
      \[\log \left(N + 1\right) - \log N\]
      15.8
    2. Using strategy rm
      15.8
    3. Applied diff-log to get
      \[\color{red}{\log \left(N + 1\right) - \log N} \leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
      12.4

    if 53.3103f0 < N

    1. Started with
      \[\log \left(N + 1\right) - \log N\]
      26.7
    2. Applied taylor to get
      \[\log \left(N + 1\right) - \log N \leadsto \left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}\]
      0.0
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}}\]
      0.0
    4. Applied simplify to get
      \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\frac{1}{N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
      0.0
    5. Applied simplify to get
      \[\frac{1}{N} + \color{red}{\left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)} \leadsto \frac{1}{N} + \color{blue}{\left(\frac{\frac{1}{3}}{{N}^3} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
      0.0
  1. Removed slow pow expressions

Original test:


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