Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 3.654607774921912e-08

   1. Initial program 59.9

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified0.0

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

   if 3.654607774921912e-08 < (- (log (+ N 1)) (log N))

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt0.4

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

   4. Applied log-prod0.4

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

   5. Applied associate--l+0.4

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 3.654607774921912 \cdot 10^{-08}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{1 + N} \cdot \sqrt[3]{1 + N}\right) + \left(\log \left(\sqrt[3]{1 + N}\right) - \log N\right)\\ \end{array}\]

Runtime

Time bar (total: 21.6s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))