Error

Derivation

1. Split input into 2 regimes

2. if N < 6332.576051332236

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   4. Applied log1p-udef0.1

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

   5. Applied diff-log0.0

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

   7. Applied add-sqr-sqrt0.1

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

   8. Applied log-prod0.1

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

   10. Applied sqrt-div0.1

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

   12. Applied add-sqr-sqrt0.1

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

   13. Applied sqrt-prod0.1

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

   14. Applied log-prod0.1

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

   if 6332.576051332236 < N

   1. Initial program 59.5

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

   2. Initial simplification59.5

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

   4. Applied log1p-udef59.5

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

   5. Applied diff-log59.3

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

   6. Taylor expanded around -inf 0.0

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

   7. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6332.576051332236:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right)\right) + \log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 16.5s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))