Error

Derivation

1. Split input into 2 regimes

2. if N < 5314.928572296327

   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

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

   7. Applied add-sqr-sqrt0.1

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

   8. Applied associate-/r*0.1

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

   if 5314.928572296327 < N

   1. Initial program 59.4

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

   2. Simplified59.4

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

   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. Simplified0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5314.928572296327:\\ \;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))