Average Error: 15.2 → 0.4
Time: 7.3s
Precision: 64
Internal Precision: 1344
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]

Error

Bits error versus N

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Your Program's Arguments

Results

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Target

Original15.2
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  2. Initial simplification15.2

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

  4. Applied diff-atan14.0

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

  5. Simplified0.4

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

  7. Applied add-sqr-sqrt1.0

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

  9. Applied rem-square-sqrt0.4

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

  10. Final simplification0.4

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

Runtime

Time bar (total: 7.3s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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