Average Error: 14.9 → 0.4
Time: 9.0s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}
double f(double N) {
        double r15284881 = N;
        double r15284882 = 1.0;
        double r15284883 = r15284881 + r15284882;
        double r15284884 = atan(r15284883);
        double r15284885 = atan(r15284881);
        double r15284886 = r15284884 - r15284885;
        return r15284886;
}
↓
double f(double N) {
        double r15284887 = 1.0;
        double r15284888 = N;
        double r15284889 = r15284888 + r15284887;
        double r15284890 = fma(r15284888, r15284889, r15284887);
        double r15284891 = atan2(r15284887, r15284890);
        return r15284891;
}

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-atan13.9

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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