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Derivation

1. Split input into 2 regimes

2. if N < 5454.710586921827

   1. Initial program 0.1

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

   3. Applied flip3-+0.1

      \[\leadsto \log \color{blue}{\left(\frac{{N}^{3} + {1}^{3}}{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}\right)} - \log N\]

   4. Applied log-div0.1

      \[\leadsto \color{blue}{\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right)} - \log N\]

   5. Simplified0.1

      \[\leadsto \left(\color{blue}{\log \left(1 + {N}^{3}\right)} - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right) - \log N\]

   if 5454.710586921827 < 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

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)}\]
   4. Using strategy rm

   5. Applied add-log-exp0.6

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5454.710586921827:\\ \;\;\;\;\left(\log \left(1 + {N}^{3}\right) - \log \left(N \cdot N + \left(1 - N\right)\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \log \left(e^{\left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{1}{N \cdot N}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

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