Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 4.798456013914258e-05

   1. Initial program 59.6

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

   2. Applied simplify59.6

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

   3. Taylor expanded around inf 0.0

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

   4. Applied simplify0.0

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

   if 4.798456013914258e-05 < (- (log (+ N 1)) (log N))

   1. Initial program 0.1

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

   2. Applied simplify0.1

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

   4. Applied add-sqr-sqrt0.1

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

   5. Applied fma-neg0.1

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))