Average Error: 15.2 → 0.3
Time: 9.9s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}
double f(double N) {
        double r12706979 = N;
        double r12706980 = 1.0;
        double r12706981 = r12706979 + r12706980;
        double r12706982 = atan(r12706981);
        double r12706983 = atan(r12706979);
        double r12706984 = r12706982 - r12706983;
        return r12706984;
}
↓
double f(double N) {
        double r12706985 = 1.0;
        double r12706986 = N;
        double r12706987 = r12706986 + r12706985;
        double r12706988 = fma(r12706986, r12706987, r12706985);
        double r12706989 = atan2(r12706985, r12706988);
        return r12706989;
}

Error

Bits error versus N

Target

Original15.2
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  3. Applied diff-atan14.1

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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