Average Error: 15.6 → 0.4
Time: 11.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}
double f(double N) {
        double r4743102 = N;
        double r4743103 = 1.0;
        double r4743104 = r4743102 + r4743103;
        double r4743105 = atan(r4743104);
        double r4743106 = atan(r4743102);
        double r4743107 = r4743105 - r4743106;
        return r4743107;
}
↓
double f(double N) {
        double r4743108 = 1.0;
        double r4743109 = N;
        double r4743110 = fma(r4743109, r4743109, r4743109);
        double r4743111 = r4743108 + r4743110;
        double r4743112 = atan2(r4743108, r4743111);
        return r4743112;
}

Error

Bits error versus N

Target

Original15.6
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.6

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

  3. Applied diff-atan14.5

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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