Average Error: 15.1 → 0.4
Time: 16.3s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}
double f(double N) {
        double r2357758 = N;
        double r2357759 = 1.0;
        double r2357760 = r2357758 + r2357759;
        double r2357761 = atan(r2357760);
        double r2357762 = atan(r2357758);
        double r2357763 = r2357761 - r2357762;
        return r2357763;
}
↓
double f(double N) {
        double r2357764 = 1.0;
        double r2357765 = N;
        double r2357766 = r2357765 + r2357764;
        double r2357767 = fma(r2357765, r2357766, r2357764);
        double r2357768 = atan2(r2357764, r2357767);
        return r2357768;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.0

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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