Average Error: 14.8 → 0.3
Time: 9.5s
Precision: 64
Internal Precision: 1344
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \left({N}^{2} + 1\right)}\]

Error

Bits error versus N

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Your Program's Arguments

Results

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Target

Original14.8
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]

  2. Initial simplification14.8

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

  4. Applied diff-atan13.8

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

  5. Simplified0.3

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

  7. Applied add-sqr-sqrt0.9

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

  8. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{N + \left({N}^{2} + 1\right)}}\]

  9. Final simplification0.3

    \[\leadsto \tan^{-1}_* \frac{1}{N + \left({N}^{2} + 1\right)}\]

Runtime

Time bar (total: 9.5s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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