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Derivation

1. Split input into 2 regimes

2. if N < 5102.476402697487

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.1

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

   4. Applied log-prod0.1

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

   6. Applied add-cbrt-cube0.1

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

   if 5102.476402697487 < N

   1. Initial program 59.6

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

   2. Taylor expanded around inf 0.0

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

   3. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5102.476402697487:\\ \;\;\;\;\left(\log \left(\sqrt[3]{\left(\sqrt{1 + N} \cdot \sqrt{1 + N}\right) \cdot \sqrt{1 + N}}\right) + \log \left(\sqrt{1 + N}\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed 2018336 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))