Average Error: 63.0 → 0

Time: 13.7s

Precision: 64

Internal Precision: 1344

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

↓

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

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
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
Initial simplification44.2
\[\leadsto (n \cdot \left(\log_* (1 + n) - \log n\right) + \left(\log_* (1 + n) - 1\right))_*\]
Taylor expanded around inf 0.0
\[\leadsto (n \cdot \color{blue}{\left(\left(\frac{1}{3} \cdot \frac{1}{{n}^{3}} + \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)} + \left(\log_* (1 + n) - 1\right))_*\]
Simplified0.0
\[\leadsto (n \cdot \color{blue}{\left((\left(\frac{\frac{1}{3}}{n} - \frac{1}{2}\right) \cdot \left(\frac{1}{n \cdot n}\right) + \left(\frac{1}{n}\right))_*\right)} + \left(\log_* (1 + n) - 1\right))_*\]
Using strategy rm
Applied log1p-udef0
\[\leadsto (n \cdot \left((\left(\frac{\frac{1}{3}}{n} - \frac{1}{2}\right) \cdot \left(\frac{1}{n \cdot n}\right) + \left(\frac{1}{n}\right))_*\right) + \left(\color{blue}{\log \left(1 + n\right)} - 1\right))_*\]
Final simplification0
\[\leadsto (n \cdot \left((\left(\frac{\frac{1}{3}}{n} - \frac{1}{2}\right) \cdot \left(\frac{1}{n \cdot n}\right) + \left(\frac{1}{n}\right))_*\right) + \left(\log \left(1 + n\right) - 1\right))_*\]

Runtime

herbie shell --seed 2018216 +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))