Average Error: 14.9 → 0.6
Time: 12.0s
Precision: 64
Internal Precision: 1408
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1 + 0}{1 + \left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}\]

Error

Bits error versus N

Target

Original14.9
Target0.4
Herbie0.6
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.7

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

  4. Applied simplify0.4

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

  6. Applied add-cube-cbrt0.6

    \[\leadsto \tan^{-1}_* \frac{1 + 0}{1 + \color{blue}{\left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}}\]

Runtime

Time bar (total: 12.0s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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