\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
Test:
NMSE example 3.5
Bits:
128 bits
Bits error versus N
Time: 2.7 s
Input Error: 6.9
Output Error: 0.6
Log: ⚲
Profile: 🕒
\(\tan^{-1}_* \frac{1 - 0}{(N * N + N)_* + 1}\)
  1. Started with
    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
    6.9
  2. Using strategy rm
    6.9
  3. Applied diff-atan to get
    \[\color{red}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}\]
    5.0
  4. Applied simplify to get
    \[\tan^{-1}_* \frac{\color{red}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \leadsto \tan^{-1}_* \frac{\color{blue}{1 - 0}}{1 + \left(N + 1\right) \cdot N}\]
    0.6
  5. Applied simplify to get
    \[\tan^{-1}_* \frac{1 - 0}{\color{red}{1 + \left(N + 1\right) \cdot N}} \leadsto \tan^{-1}_* \frac{1 - 0}{\color{blue}{(N * \left(1 + N\right) + 1)_*}}\]
    0.6
  6. Using strategy rm
    0.6
  7. Applied fma-udef to get
    \[\tan^{-1}_* \frac{1 - 0}{\color{red}{(N * \left(1 + N\right) + 1)_*}} \leadsto \tan^{-1}_* \frac{1 - 0}{\color{blue}{N \cdot \left(1 + N\right) + 1}}\]
    0.6
  8. Applied simplify to get
    \[\tan^{-1}_* \frac{1 - 0}{\color{red}{N \cdot \left(1 + N\right)} + 1} \leadsto \tan^{-1}_* \frac{1 - 0}{\color{blue}{(N * N + N)_*} + 1}\]
    0.6

Original test:


(lambda ((N default))
  #:name "NMSE example 3.5"
  (- (atan (+ N 1)) (atan N))
  #:target
  (atan (/ 1 (+ 1 (* N (+ N 1))))))