Average Error: 14.3 → 0.4
Time: 3.1s
Precision: 64
Internal precision: 1408
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
⬇
\[\tan^{-1}_* \frac{1}{\left(1 + N\right) + {N}^2}\]

Error

Bits error versus N

Target

Original14.3
Comparison0.4
Herbie0.4
\[ \tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 14.3

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  2. Using strategy rm

  3. Applied diff-atan 13.2

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}\]

  4. Applied simplify 0.4

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{1 + 0}}{1 + \left(N + 1\right) \cdot N}\]

  5. Applied simplify 0.4

    \[\leadsto \tan^{-1}_* \frac{1 + 0}{\color{blue}{1 + \left(N + {N}^2\right)}}\]

  6. Applied taylor 0.4

    \[\leadsto \tan^{-1}_* \frac{1 + 0}{1 + \left({N}^2 + N\right)}\]

  7. Taylor expanded around 0 0.4

    \[\leadsto \tan^{-1}_* \frac{1 + 0}{\color{blue}{1 + \left({N}^2 + N\right)}}\]

  8. Applied simplify 0.4

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{\left(1 + N\right) + {N}^2}}\]
  9. Removed slow pow expressions

Runtime

Time bar (total: 3.1s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065003094 2074156664 2352254222 753858891 3745550101 3374585842)'
(FPCore (N)
  :name "2atan (example 3.5)"

  :target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))