Average Error: 14.9 → 0.4
Time: 8.5s
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 r2070166 = N;
        double r2070167 = 1.0;
        double r2070168 = r2070166 + r2070167;
        double r2070169 = atan(r2070168);
        double r2070170 = atan(r2070166);
        double r2070171 = r2070169 - r2070170;
        return r2070171;
}
↓
double f(double N) {
        double r2070172 = 1.0;
        double r2070173 = N;
        double r2070174 = fma(r2070173, r2070173, r2070172);
        double r2070175 = r2070173 + r2070174;
        double r2070176 = atan2(r2070172, r2070175);
        return r2070176;
}

Error

Bits error versus N

Target

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

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. Simplified0.4

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

  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 2019154 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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