Average Error: 15.4 → 0.4
Time: 29.7s
Precision: 64
\[\tan^{-1} \left(N + 1.0\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1.0}{\mathsf{fma}\left(N, N + 1.0, 1\right)}\]
\tan^{-1} \left(N + 1.0\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1.0}{\mathsf{fma}\left(N, N + 1.0, 1\right)}
double f(double N) {
        double r6708733 = N;
        double r6708734 = 1.0;
        double r6708735 = r6708733 + r6708734;
        double r6708736 = atan(r6708735);
        double r6708737 = atan(r6708733);
        double r6708738 = r6708736 - r6708737;
        return r6708738;
}
↓
double f(double N) {
        double r6708739 = 1.0;
        double r6708740 = N;
        double r6708741 = r6708740 + r6708739;
        double r6708742 = 1.0;
        double r6708743 = fma(r6708740, r6708741, r6708742);
        double r6708744 = atan2(r6708739, r6708743);
        return r6708744;
}

Error

Bits error versus N

Target

Original15.4
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1.0}{1.0 + N \cdot \left(N + 1.0\right)}\right)\]

Derivation

  1. Initial program 15.4

    \[\tan^{-1} \left(N + 1.0\right) - \tan^{-1} N\]
  2. Using strategy rm

  3. Applied diff-atan14.3

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1.0\right) - N}{1 + \left(N + 1.0\right) \cdot N}}\]

  4. Simplified0.4

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{1.0}}{1 + \left(N + 1.0\right) \cdot N}\]

  5. Simplified0.4

    \[\leadsto \tan^{-1}_* \frac{1.0}{\color{blue}{\mathsf{fma}\left(N, 1.0 + N, 1\right)}}\]

  6. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{1.0}{\mathsf{fma}\left(N, N + 1.0, 1\right)}\]

Reproduce

herbie shell --seed 2019165 +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)))