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Derivation

1. Split input into 2 regimes

2. if N < 12668.320172312919

   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 clear-num0.1

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

   7. Applied add-exp-log0.1

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

   8. Applied rec-exp0.1

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

   9. Applied rem-log-exp0.1

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

   if 12668.320172312919 < N

   1. Initial program 59.5

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

   3. Applied diff-log59.3

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

   5. Applied clear-num59.3

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

   7. Applied add-exp-log59.3

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

   8. Applied rec-exp59.3

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

   9. Applied rem-log-exp59.3

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

   10. Taylor expanded around -inf 0.0

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

   11. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 12668.320172312919:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\frac{1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{-1}{3}}{N}}{N \cdot N} - \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 8.0s)Debug log