Derivation

Split input into 2 regimes
if N < 11879.940791136825
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Initial simplification0.1
  \[\leadsto \log \left(1 + N\right) - \log N\]
3. Using strategy rm
4. Applied diff-log0.1
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
if 11879.940791136825 < N
1. Initial program 59.6
  \[\log \left(N + 1\right) - \log N\]
2. Initial simplification59.6
  \[\leadsto \log \left(1 + N\right) - \log N\]
3. Using strategy rm
4. Applied diff-log59.4
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
5. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 11879.940791136825:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018249 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

In
Out