Error

Derivation

1. Split input into 2 regimes

2. if N < 8059.32314534696

   1. Initial program 0.1

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

   3. Applied diff-log0.1

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

   if 8059.32314534696 < N

   1. Initial program 59.5

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

   3. Applied diff-log59.2

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

   5. Applied add-cbrt-cube59.2

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

   7. Applied pow1/359.2

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

   8. Taylor expanded around inf 0.0

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

   9. Simplified0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8059.32314534696:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 25.0s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))