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Derivation

1. Split input into 2 regimes

2. if N < 7728.08331971911

   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 7728.08331971911 < 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

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

   8. Applied flip3--0.0

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

   9. Applied associate-*r/0.0

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

   10. Applied frac-sub0.0

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

   11. Simplified0.0

       \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{1}{6}}{N} + \frac{1}{4}\right) + \frac{\frac{\frac{1}{9}}{N}}{N}\right) - \frac{\frac{1}{8} - \frac{\frac{\frac{1}{27}}{N}}{N \cdot N}}{N}}}{N \cdot \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{3}}{N} \cdot \frac{\frac{1}{3}}{N} + \frac{1}{2} \cdot \frac{\frac{1}{3}}{N}\right)\right)}\]

   12. Simplified0.0

       \[\leadsto \frac{\left(\left(\frac{\frac{1}{6}}{N} + \frac{1}{4}\right) + \frac{\frac{\frac{1}{9}}{N}}{N}\right) - \frac{\frac{1}{8} - \frac{\frac{\frac{1}{27}}{N}}{N \cdot N}}{N}}{\color{blue}{\frac{1}{6} + \left(\frac{\frac{1}{9}}{N} + N \cdot \frac{1}{4}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7728.08331971911:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{\frac{1}{9}}{N}}{N} + \left(\frac{\frac{1}{6}}{N} + \frac{1}{4}\right)\right) - \frac{\frac{1}{8} - \frac{\frac{\frac{1}{27}}{N}}{N \cdot N}}{N}}{\frac{1}{6} + \left(\frac{1}{4} \cdot N + \frac{\frac{1}{9}}{N}\right)}\\ \end{array}\]

Runtime

Time bar (total: 19.8s)Debug logProfile

herbie shell --seed 2018352 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))