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 simplify44.2

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

3. Taylor expanded around inf 0.0

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

4. Applied simplify0.0

   \[\leadsto \color{blue}{\left((\left(\frac{\frac{1}{3}}{n}\right) \cdot \left(\frac{1}{n}\right) + 1)_* + (\left(\frac{1}{n}\right) \cdot \left(-\frac{1}{2}\right) + \left(\log_* (1 + n)\right))_*\right) - 1}\]
5. Using strategy rm

6. Applied log1p-expm1-u0.0

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

7. Applied simplify0.0

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

Runtime

Time bar (total: 31.3s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +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))