Error

Try it out

Derivation

1. Split input into 2 regimes

2. if N < 14302.326937808648

   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. Taylor expanded around 0 0.1

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

   if 14302.326937808648 < 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. Taylor expanded around -inf 0.0

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

   5. Simplified0.0

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

   6. Taylor expanded around 0 0.0

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

   7. Simplified0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 14302.326937808648:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \left(\frac{\frac{1}{2}}{N \cdot N} + \frac{-1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 13.9s)Debug log