Average Error: 14.9 → 0.8
Time: 9.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[(e^{\log_* (1 + \tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*})} - 1)^*\]

Error

Bits error versus N

Target

Original14.9
Target0.3
Herbie0.8
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.8

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  7. Applied expm1-log1p-u0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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