Average Error: 63.0 → 0

Time: 14.5s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `n`

Target

Original	63.0
Target	0
Herbie	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

Initial program 63.0
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
Taylor expanded around -inf 62.0
\[\leadsto \color{blue}{\left(\left(\log -1 + \left(1 + \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{-1}{n}\right)\right)\right)} - 1\]
Simplified0.0
\[\leadsto \color{blue}{\left(\left(1 + \left(\log n - \frac{\frac{1}{6}}{n \cdot n}\right)\right) + \frac{\frac{1}{2}}{n}\right)} - 1\]
Taylor expanded around inf 0.0
\[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]
Simplified0
\[\leadsto \color{blue}{\left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}}\]
Final simplification0
\[\leadsto \left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}\]

Reproduce

herbie shell --seed 2019050 
(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))