Error

Derivation

1. Split input into 2 regimes

2. if N < 6703.766594082657

   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 add-log-exp0.1

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

   5. Applied diff-log0.1

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

   7. Applied add-sqr-sqrt0.1

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

   8. Applied log-prod0.1

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

   10. Applied sqrt-div0.1

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

   11. Applied log-div0.1

       \[\leadsto \log \left(\sqrt{\frac{e^{\log_* (1 + N)}}{N}}\right) + \color{blue}{\left(\log \left(\sqrt{e^{\log_* (1 + N)}}\right) - \log \left(\sqrt{N}\right)\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt0.1

       \[\leadsto \log \left(\sqrt{\color{blue}{\sqrt{\frac{e^{\log_* (1 + N)}}{N}} \cdot \sqrt{\frac{e^{\log_* (1 + N)}}{N}}}}\right) + \left(\log \left(\sqrt{e^{\log_* (1 + N)}}\right) - \log \left(\sqrt{N}\right)\right)\]

   14. Applied sqrt-prod0.1

       \[\leadsto \log \color{blue}{\left(\sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}} \cdot \sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}}\right)} + \left(\log \left(\sqrt{e^{\log_* (1 + N)}}\right) - \log \left(\sqrt{N}\right)\right)\]

   15. Applied log-prod0.1

       \[\leadsto \color{blue}{\left(\log \left(\sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}}\right)\right)} + \left(\log \left(\sqrt{e^{\log_* (1 + N)}}\right) - \log \left(\sqrt{N}\right)\right)\]

   if 6703.766594082657 < N

   1. Initial program 59.6

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

   2. Initial simplification59.6

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

   4. Applied add-log-exp59.6

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

   5. Applied diff-log60.5

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6703.766594082657:\\ \;\;\;\;\left(\log \left(\sqrt{e^{\log_* (1 + N)}}\right) - \log \left(\sqrt{N}\right)\right) + \left(\log \left(\sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{e^{\log_* (1 + N)}}{N}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 17.5s)Debug logProfile

herbie shell --seed 2018258 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))