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Derivation

1. Split input into 2 regimes

2. if N < 5473.209040664214

   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-cube-cbrt0.1

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

   7. Applied associate-/l*0.1

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

   if 5473.209040664214 < N

   1. Initial program 59.4

      \[\log \left(N + 1\right) - \log N\]

   2. Initial simplification59.4

      \[\leadsto \log \left(1 + N\right) - \log N\]

   3. Taylor expanded around -inf 0.1

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

   4. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5473.209040664214:\\ \;\;\;\;\log \left(\frac{\sqrt[3]{1 + N} \cdot \sqrt[3]{1 + N}}{\frac{N}{\sqrt[3]{1 + N}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Runtime

Time bar (total: 17.2s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))