Error

Derivation

1. Split input into 2 regimes

2. if N < 5115.25510295185

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt0.1

      \[\leadsto \log \left(N + 1\right) - \log \color{blue}{\left(\sqrt{N} \cdot \sqrt{N}\right)}\]

   4. Applied log-prod0.1

      \[\leadsto \log \left(N + 1\right) - \color{blue}{\left(\log \left(\sqrt{N}\right) + \log \left(\sqrt{N}\right)\right)}\]

   5. Applied associate--r+0.1

      \[\leadsto \color{blue}{\left(\log \left(N + 1\right) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)}\]

   if 5115.25510295185 < N

   1. Initial program 59.4

      \[\log \left(N + 1\right) - \log N\]

   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

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5115.25510295185:\\ \;\;\;\;\left(\log \left(1 + N\right) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))