Error

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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. Simplified62.0

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

3. Taylor expanded around -inf 62.0

   \[\leadsto \color{blue}{\left(\log -1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{-1}{n}\right)\right)}\]

4. Simplified0

   \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{n} + \log n\right) - \frac{\frac{1}{6}}{n \cdot n}}\]

5. Final simplification0

   \[\leadsto \left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{1}{6}}{n \cdot n}\]

Reproduce

herbie shell --seed 2019005 +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))

Details

Time bar (total: 12.6s)Debug log