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Derivation

1. Split input into 2 regimes

2. if N < 10143.115174419936

   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 10143.115174419936 < 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. Using strategy rm

   6. Applied add-cbrt-cube59.4

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(\frac{1 + N}{N}\right) \cdot \log \left(\frac{1 + N}{N}\right)\right) \cdot \log \left(\frac{1 + N}{N}\right)}}\]
   7. Using strategy rm

   8. Applied add-exp-log59.4

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

   9. Taylor expanded around inf 0.0

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

   10. Simplified0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 10143.115174419936:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 8.3s)Debug log