Average Error: 63.0 → 61.1
Time: 25.5s
Precision: 64
Internal Precision: 1408
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\sqrt{(\left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*} \cdot \sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) \cdot \left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) + \left(-(n \cdot \left(\log n\right) + 1)_*\right))_*} \cdot \sqrt{(\left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*} \cdot \sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) \cdot \left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) + \left(-(n \cdot \left(\log n\right) + 1)_*\right))_*}\]

Error

Bits error versus n

Target

Original63.0
Target0.0
Herbie61.1
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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. Using strategy rm

  4. Applied add-cube-cbrt61.6

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

  5. Applied fma-neg61.6

    \[\leadsto \color{blue}{(\left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*} \cdot \sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) \cdot \left(\sqrt[3]{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_*}\right) + \left(-(n \cdot \left(\log n\right) + 1)_*\right))_*}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt61.1

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

Runtime

Time bar (total: 25.5s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))