Error

Derivation

1. Split input into 2 regimes

2. if N < 6197.589456444199

   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-log0.1

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

   7. Applied add-cube-cbrt0.1

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{1 + N}{N}} \cdot \sqrt[3]{\frac{1 + N}{N}}\right) \cdot \sqrt[3]{\frac{1 + N}{N}}\right)}\]

   8. Applied log-prod0.4

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{1 + N}{N}} \cdot \sqrt[3]{\frac{1 + N}{N}}\right) + \log \left(\sqrt[3]{\frac{1 + N}{N}}\right)}\]
   9. Using strategy rm

   10. Applied pow1/30.3

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

   11. Applied log-pow0.3

       \[\leadsto \log \left(\sqrt[3]{\frac{1 + N}{N}} \cdot \sqrt[3]{\frac{1 + N}{N}}\right) + \color{blue}{\frac{1}{3} \cdot \log \left(\frac{1 + N}{N}\right)}\]

   12. Applied pow1/30.3

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

   13. Applied pow1/30.4

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

   14. Applied pow-prod-up0.4

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

   15. Applied log-pow0.4

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

   16. Applied distribute-rgt-out0.1

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

   17. Simplified0.1

       \[\leadsto \color{blue}{\left(\log_* (1 + N) - \log N\right)} \cdot \left(\left(\frac{1}{3} + \frac{1}{3}\right) + \frac{1}{3}\right)\]

   18. Simplified0.1

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

   if 6197.589456444199 < 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. Using strategy rm

   7. Applied add-cube-cbrt59.4

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{1 + N}{N}} \cdot \sqrt[3]{\frac{1 + N}{N}}\right) \cdot \sqrt[3]{\frac{1 + N}{N}}\right)}\]

   8. Applied log-prod59.4

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{1 + N}{N}} \cdot \sqrt[3]{\frac{1 + N}{N}}\right) + \log \left(\sqrt[3]{\frac{1 + N}{N}}\right)}\]

   9. Taylor expanded around inf 0.0

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

   10. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6197.589456444199:\\ \;\;\;\;\log_* (1 + N) - \log N\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 24.7s)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))