Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 0.0046142372679678426

   1. Initial program 59.2

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

   2. Applied simplify59.2

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]

   3. Taylor expanded around inf 0.2

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

   4. Applied simplify0.2

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

   if 0.0046142372679678426 < (- (log (+ N 1)) (log N))

   1. Initial program 0.0

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

   2. Applied simplify0.0

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
   3. Using strategy rm

   4. Applied log1p-udef0.0

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

   5. Applied diff-log0.0

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

Runtime

Time bar (total: 16.4s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))