Average Error: 15.1 → 0.3
Time: 14.0s
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 r5151928 = N;
        double r5151929 = 1.0;
        double r5151930 = r5151928 + r5151929;
        double r5151931 = atan(r5151930);
        double r5151932 = atan(r5151928);
        double r5151933 = r5151931 - r5151932;
        return r5151933;
}
↓
double f(double N) {
        double r5151934 = 1.0;
        double r5151935 = N;
        double r5151936 = r5151935 + r5151934;
        double r5151937 = 1.0;
        double r5151938 = fma(r5151935, r5151936, r5151937);
        double r5151939 = atan2(r5151934, r5151938);
        return r5151939;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.1

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

  4. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{1 \cdot N + \left({N}^{2} + 1\right)}}\]

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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)))