Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 3.668446215776555e-05

   1. Initial program 59.7

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

   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. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}}\]
   4. Using strategy rm

   5. Applied add-log-exp0.0

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

   if 3.668446215776555e-05 < (- (log (+ N 1)) (log N))

   1. Initial program 0.1

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

   3. Applied flip3-+0.1

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

   4. Applied log-div0.1

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

   5. Applied simplify0.1

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

   6. Applied simplify0.1

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

Runtime

Time bar (total: 30.9s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))