Average Error: 15.3 → 0.3
Time: 13.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}
double f(double N) {
        double r85357 = N;
        double r85358 = 1.0;
        double r85359 = r85357 + r85358;
        double r85360 = atan(r85359);
        double r85361 = atan(r85357);
        double r85362 = r85360 - r85361;
        return r85362;
}
↓
double f(double N) {
        double r85363 = 1.0;
        double r85364 = N;
        double r85365 = r85364 + r85363;
        double r85366 = 1.0;
        double r85367 = fma(r85364, r85365, r85366);
        double r85368 = atan2(r85363, r85367);
        return r85368;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.3

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

  2. Simplified15.3

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

  4. Applied diff-atan14.2

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))