Average Error: 15.0 → 0.4
Time: 18.6s
Precision: 64
Internal Precision: 1408
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1 + 0}{1 + \left(N + 1\right) \cdot N}\]

Error

Bits error versus N

Target

Original15.0
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

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

  3. Applied diff-atan14.0

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

  4. Applied simplify0.4

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{1 + 0}}{1 + \left(N + 1\right) \cdot N}\]
  5. Removed slow pow expressions.

Runtime

Time bar (total: 18.6s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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