Average Error: 14.7 → 0.4
Time: 7.2s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \mathsf{fma}\left(N, N, 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N + \mathsf{fma}\left(N, N, 1\right)}
double f(double N) {
        double r2091650 = N;
        double r2091651 = 1.0;
        double r2091652 = r2091650 + r2091651;
        double r2091653 = atan(r2091652);
        double r2091654 = atan(r2091650);
        double r2091655 = r2091653 - r2091654;
        return r2091655;
}
↓
double f(double N) {
        double r2091656 = 1.0;
        double r2091657 = N;
        double r2091658 = fma(r2091657, r2091657, r2091656);
        double r2091659 = r2091657 + r2091658;
        double r2091660 = atan2(r2091656, r2091659);
        return r2091660;
}

Error

Bits error versus N

Target

Original14.7
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

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

  3. Applied diff-atan13.5

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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