Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 1.5458009983149168e-08

   1. Initial program 60.1

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

   2. Taylor expanded around inf 60.1

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

   3. Applied simplify0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} - \log 1\right) - \frac{\frac{\frac{1}{2}}{N}}{N}}\]

   if 1.5458009983149168e-08 < (- (log (+ N 1)) (log N))

   1. Initial program 0.3

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

   3. Applied flip3-+0.3

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

   4. Applied log-div0.3

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

   5. Applied simplify0.3

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

   6. Applied simplify0.3

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

Runtime

Time bar (total: 47.2s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))