Average Error: 63.0 → 60.8

Time: 56.9s

Precision: 64

Internal Precision: 1408

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

↓

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

Error

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

Target

Original	63.0
Target	0
Herbie	60.8

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

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 60.8
\[\leadsto \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - \color{blue}{-1 \cdot \left(n \cdot \log n\right)}\right) - 1\]
Applied simplify60.8
\[\leadsto \color{blue}{\left(1 + n\right) \cdot \log \left(1 + n\right) - \left(1 + \left(-n\right) \cdot \log n\right)}\]

Runtime

herbie shell --seed '#(212267722 3993171362 1093346726 3605783651 2106536041 3335990851)' 
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)
  :herbie-expected #f

  :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))