Average Error: 14.9 → 0.4
Time: 11.7s
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 r5343832 = N;
        double r5343833 = 1.0;
        double r5343834 = r5343832 + r5343833;
        double r5343835 = atan(r5343834);
        double r5343836 = atan(r5343832);
        double r5343837 = r5343835 - r5343836;
        return r5343837;
}
↓
double f(double N) {
        double r5343838 = 1.0;
        double r5343839 = N;
        double r5343840 = fma(r5343839, r5343839, r5343839);
        double r5343841 = r5343838 + r5343840;
        double r5343842 = atan2(r5343838, r5343841);
        return r5343842;
}

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-atan13.7

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied fma-udef0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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