Average Error: 63.0 → 0.0
Time: 7.0s
Precision: 64
Internal Precision: 1344
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[(n \cdot \left((\left(\frac{\frac{1}{3}}{n} - \frac{1}{2}\right) \cdot \left(\frac{1}{n \cdot n}\right) + \left(\frac{1}{n}\right))_*\right) + \left(\log_* (1 + n) - 1\right))_*\]

Error

Bits error versus n

Target

Original63.0
Target0.0
Herbie0.0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

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

  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. Simplified0.0

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

  5. Final simplification0.0

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

Runtime

Time bar (total: 7.0s)Debug logProfile

herbie shell --seed 2018217 +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))