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Derivation

1. Split input into 2 regimes

2. if N < 10502.73743544022

   1. Initial program 0.1

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

   3. Applied diff-log0.1

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

   5. Applied add-sqr-sqrt0.1

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

   6. Applied log-prod0.1

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

   8. Applied pow1/20.1

      \[\leadsto \log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{\frac{1}{2}}\right)}\]

   9. Applied log-pow0.1

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

   if 10502.73743544022 < N

   1. Initial program 59.4

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

   3. Applied diff-log59.2

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

   4. Taylor expanded around -inf 0.0

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

   5. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 10502.73743544022:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\frac{1 + N}{N}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1042149663 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 14.3s)Debug log