Average Error: 63.0 → 61.5
Time: 12.2s
Precision: 64
Internal precision: 1408
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
⬇
\[\frac{{\left({\left({\left(\sqrt[3]{\sqrt[3]{\left(n + 1\right) \cdot \log \left(n + 1\right)}}\right)}^3\right)}^3\right)}^2 - {\left(n \cdot \log n\right)}^2}{\left(1 + n\right) \cdot \log \left(1 + n\right) + \log n \cdot n} - 1\]

Error

Bits error versus n

Target

Original63.0
Comparison0.0
Herbie61.5
\[ \log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot {n}^2} - \frac{4}{{n}^{3}}\right)\right) \]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
  2. Using strategy rm

  3. Applied add-cube-cbrt 61.8

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

  5. Applied add-cube-cbrt 61.6

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

  7. Applied flip-- 61.5

    \[\leadsto \color{blue}{\frac{{\left({\left({\left(\sqrt[3]{\sqrt[3]{\left(n + 1\right) \cdot \log \left(n + 1\right)}}\right)}^3\right)}^3\right)}^2 - {\left(n \cdot \log n\right)}^2}{{\left({\left(\sqrt[3]{\sqrt[3]{\left(n + 1\right) \cdot \log \left(n + 1\right)}}\right)}^3\right)}^3 + n \cdot \log n}} - 1\]

  8. Applied simplify 61.5

    \[\leadsto \frac{{\left({\left({\left(\sqrt[3]{\sqrt[3]{\left(n + 1\right) \cdot \log \left(n + 1\right)}}\right)}^3\right)}^3\right)}^2 - {\left(n \cdot \log n\right)}^2}{\color{blue}{\left(1 + n\right) \cdot \log \left(1 + n\right) + \log n \cdot n}} - 1\]
  9. Removed slow pow expressions

Runtime

Time bar (total: 12.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065003094 2074156664 2352254222 753858891 3745550101 3374585842)'
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (sqr n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))