Error

Derivation

1. Split input into 2 regimes

2. if (- (log (+ N 1)) (log N)) < 3.887471677899157e-09

   1. Initial program 60.1

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

   2. Taylor expanded around inf 60.1

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

   3. Applied simplify0.0

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

   if 3.887471677899157e-09 < (- (log (+ N 1)) (log N))

   1. Initial program 0.4

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

   3. Applied add-cube-cbrt0.5

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

   4. Applied fma-neg0.5

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

Runtime

Time bar (total: 40.9s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))