Error

Derivation

1. Split input into 2 regimes

2. if N < 9812.217374183212

   1. Initial program 0.1

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

   2. Simplified0.1

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
   3. Using strategy rm

   4. Applied log1p-udef0.1

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

   5. Applied diff-log0.1

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

   if 9812.217374183212 < N

   1. Initial program 59.4

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

   2. Simplified59.5

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]

   3. Taylor expanded around -inf 0.0

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

   4. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9812.217374183212:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))