Average Error: 63.0 → 60.8
Time: 22.4s
Precision: 64
Internal Precision: 1408
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\sqrt{(\left(\log n\right) \cdot n + \left((n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_* - 1\right))_*} \cdot \sqrt{(\left(\log n\right) \cdot n + \left((n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_* - 1\right))_*}\]

Error

Bits error versus n

Target

Original63.0
Target0
Herbie60.8
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

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

  2. Applied simplify62.0

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

  3. Taylor expanded around inf 60.8

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

  4. Applied simplify60.8

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

  6. Applied add-sqr-sqrt60.8

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

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(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))