Derivation

Split input into 2 regimes
if N < 8476.15339805449
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Initial simplification0.1
  \[\leadsto \log_* (1 + N) - \log N\]
3. Using strategy rm
4. Applied log1p-udef0.1
  \[\leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
5. Applied diff-log0.1
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
if 8476.15339805449 < N
1. Initial program 59.5
  \[\log \left(N + 1\right) - \log N\]
2. Initial simplification59.5
  \[\leadsto \log_* (1 + N) - \log N\]
3. Using strategy rm
4. Applied log1p-udef59.5
  \[\leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
5. Applied diff-log59.3
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
6. Taylor expanded around -inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]
7. Simplified0.0
  \[\leadsto \color{blue}{(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8476.15339805449:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

herbie shell --seed 2018234 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))