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Derivation

1. Split input into 2 regimes

2. if N < 8142.718291694942

   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

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

   6. Applied add-sqr-sqrt0.1

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

   7. Applied log-prod0.1

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

   9. Applied pow1/20.1

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

   10. Applied log-pow0.1

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

   12. Applied pow1/20.1

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

   13. Applied log-pow0.1

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

   if 8142.718291694942 < 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. Taylor expanded around -inf 0.0

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

   6. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8142.718291694942:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{1}{N \cdot N}\\ \end{array}\]

Runtime

Time bar (total: 13.1s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))