Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 3.542537891792339e-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. Simplified0.0

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

   if 3.542537891792339e-05 < (- (log (+ N 1)) (log N))

   1. Initial program 0.1

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

   3. Applied diff-log0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 3.542537891792339 \cdot 10^{-05}:\\ \;\;\;\;(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}\]

Runtime

Time bar (total: 27.3s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))