Average Error: 15.4 → 0.3
Time: 19.4s
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 r4321726 = N;
        double r4321727 = 1.0;
        double r4321728 = r4321726 + r4321727;
        double r4321729 = atan(r4321728);
        double r4321730 = atan(r4321726);
        double r4321731 = r4321729 - r4321730;
        return r4321731;
}
↓
double f(double N) {
        double r4321732 = 1.0;
        double r4321733 = N;
        double r4321734 = fma(r4321733, r4321733, r4321733);
        double r4321735 = r4321732 + r4321734;
        double r4321736 = atan2(r4321732, r4321735);
        return r4321736;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.4

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

  3. Applied diff-atan14.3

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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