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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. Initial simplification44.2

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

3. Taylor expanded around -inf 0.0

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

4. Simplified0.0

   \[\leadsto \left(\log \left(1 + n\right) + \color{blue}{\left(\left(1 + \frac{\frac{\frac{1}{3}}{n}}{n}\right) - \frac{\frac{1}{2}}{n}\right)}\right) - 1\]

5. Final simplification0.0

   \[\leadsto \left(\left(\left(1 + \frac{\frac{\frac{1}{3}}{n}}{n}\right) - \frac{\frac{1}{2}}{n}\right) + \log \left(n + 1\right)\right) - 1\]

Runtime

Time bar (total: 16.4s)Debug logProfile

herbie shell --seed 2018234 
(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))