Derivation

Split input into 2 regimes
if (- (log (+ N 1)) (log N)) < 3.654607774921912e-08
1. Initial program 59.9
  \[\log \left(N + 1\right) - \log N\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}}\]
if 3.654607774921912e-08 < (- (log (+ N 1)) (log N))
1. Initial program 0.3
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied add-cube-cbrt0.4
  \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \sqrt[3]{N + 1}\right)} - \log N\]
4. Applied log-prod0.4
  \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) + \log \left(\sqrt[3]{N + 1}\right)\right)} - \log N\]
5. Applied associate--l+0.4
  \[\leadsto \color{blue}{\log \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) + \left(\log \left(\sqrt[3]{N + 1}\right) - \log N\right)}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 3.654607774921912 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{1 + N} \cdot \sqrt[3]{1 + N}\right) + \left(\log \left(\sqrt[3]{1 + N}\right) - \log N\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018215 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

In
Out