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Derivation

1. Split input into 2 regimes

2. if N < 5250.505171858608

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.1

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

   if 5250.505171858608 < N

   1. Initial program 59.5

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

   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 flip--0.0

      \[\leadsto \frac{1}{N} - \frac{1}{N \cdot N} \cdot \color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{1}{3}}{N} \cdot \frac{\frac{1}{3}}{N}}{\frac{1}{2} + \frac{\frac{1}{3}}{N}}}\]

   6. Applied associate-*r/0.0

      \[\leadsto \frac{1}{N} - \color{blue}{\frac{\frac{1}{N \cdot N} \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{1}{3}}{N} \cdot \frac{\frac{1}{3}}{N}\right)}{\frac{1}{2} + \frac{\frac{1}{3}}{N}}}\]

   7. Applied frac-sub0.0

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{2} + \frac{\frac{1}{3}}{N}\right) - N \cdot \left(\frac{1}{N \cdot N} \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{1}{3}}{N} \cdot \frac{\frac{1}{3}}{N}\right)\right)}{N \cdot \left(\frac{1}{2} + \frac{\frac{1}{3}}{N}\right)}}\]

   8. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}}{N \cdot \left(\frac{1}{2} + \frac{\frac{1}{3}}{N}\right)}\]

   9. Simplified0.1

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}{\color{blue}{\frac{1}{3} + N \cdot \frac{1}{2}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5250.505171858608:\\ \;\;\;\;\sqrt{\log \left(1 + N\right)} \cdot \sqrt{\log \left(1 + N\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{2} + \frac{\frac{1}{3}}{N}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}{\frac{1}{2} \cdot N + \frac{1}{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 18.7s)Debug log