Average Error: 14.8 → 0.4
Time: 9.0s
Precision: 64
Ground Truth: 128
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
⬇
\[\tan^{-1}_* \frac{1 + 0}{\left({N}^2 + N\right) + 1}\]

Error

Bits error versus N

Target

Original14.8
Comparison0.4
Herbie0.4
\[ \tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 14.8

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

  3. Applied diff-atan 13.7

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

  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. Removed slow pow expressions

Runtime

Total time: 9.0s Debug log

Please report a bug with the following info:

herbie --seed '#(2673858385 3039539534 1461792205 3524151131 933919278 2324521657)'
(FPCore (N)
  :name "NMSE example 3.5"

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

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