Average Error: 15.3 → 0.4
Time: 34.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, 1 + N, 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, 1 + N, 1\right)}
double f(double N) {
        double r7577971 = N;
        double r7577972 = 1.0;
        double r7577973 = r7577971 + r7577972;
        double r7577974 = atan(r7577973);
        double r7577975 = atan(r7577971);
        double r7577976 = r7577974 - r7577975;
        return r7577976;
}
↓
double f(double N) {
        double r7577977 = 1.0;
        double r7577978 = N;
        double r7577979 = r7577977 + r7577978;
        double r7577980 = 1.0;
        double r7577981 = fma(r7577978, r7577979, r7577980);
        double r7577982 = atan2(r7577977, r7577981);
        return r7577982;
}

Error

Bits error versus N

Target

Original15.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.3

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

  3. Applied diff-atan14.1

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +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)))