Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 1.9574390677234987e-06

   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

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

   if 1.9574390677234987e-06 < (- (log (+ N 1)) (log N))

   1. Initial program 0.2

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

   3. Applied diff-log0.2

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

   5. Applied add-sqr-sqrt0.2

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

   6. Applied log-prod0.2

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

   8. Applied add-sqr-sqrt0.2

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

   9. Applied sqrt-prod0.2

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

   10. Applied log-prod0.2

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

Runtime

Time bar (total: 25.6s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))