Error: 14.3 → 0.4
Time: 7.3s
Precision: 64
Ground Truth: 128
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
⬇
\[\tan^{-1}_* \frac{1 + 0}{\left(N + 1\right) \cdot N + 1}\]

Error

Bits error versus N

Derivation

  1. Initial program 14.3

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

  3. Applied diff-atan 13.5

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

  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

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

  7. Applied square-mult 0.4

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

  8. Applied distribute-lft1-in 0.4

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

Runtime

Total time: 7.3s Debug log

Please report a bug with the following info:

herbie --seed '#(3279971195 628103085 3858102579 1384554080 810072436 2288626801)'
(FPCore (N)
  :name "NMSE example 3.5"
  
  :target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))(- (atan (+ N 1)) (atan N)))