Error

Target

Derivation

1. Initial program 63.0

   \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

2. Applied simplify44.3

   \[\leadsto \color{blue}{(n \cdot \left(\log_* (1 + n) - \log n\right) + \left(\log_* (1 + n) - 1\right))_*}\]

3. Taylor expanded around inf 44.3

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

4. Applied simplify0.0

   \[\leadsto \color{blue}{(\left(\frac{1}{n} - (\left(\frac{\frac{1}{2}}{n}\right) \cdot \left(\frac{1}{n}\right) + 0)_*\right) \cdot n + \left(\log_* (1 + n) - 1\right))_*}\]
5. Using strategy rm

6. Applied log1p-udef0

   \[\leadsto (\left(\frac{1}{n} - (\left(\frac{\frac{1}{2}}{n}\right) \cdot \left(\frac{1}{n}\right) + 0)_*\right) \cdot n + \left(\color{blue}{\log \left(1 + n\right)} - 1\right))_*\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))