Error

Derivation

1. Initial program 31.3

   \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
2. Using strategy rm

3. Applied flip3-+30.9

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

4. Applied log-div30.7

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

5. Applied simplify30.7

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

6. Applied simplify30.7

   \[\leadsto \left(\left(N + 1\right) \cdot \left(\log \left(1 + {N}^{3}\right) - \color{blue}{\log \left(N \cdot N + \left(1 - N\right)\right)}\right) - N \cdot \log N\right) - 1\]
7. Using strategy rm

8. Applied flip--30.7

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

10. Applied add-cube-cbrt30.8

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

12. Applied add-cube-cbrt30.8

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

13. Applied log-prod30.8

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

Runtime

Time bar (total: 2.1m)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (N)
  :name "NMSE example 3.8"
  :pre (> N 0)
  (- (- (* (+ N 1) (log (+ N 1))) (* N (log N))) 1))