Average Error: 14.9 → 0.4
Time: 16.5s
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 r4347042 = N;
        double r4347043 = 1.0;
        double r4347044 = r4347042 + r4347043;
        double r4347045 = atan(r4347044);
        double r4347046 = atan(r4347042);
        double r4347047 = r4347045 - r4347046;
        return r4347047;
}
↓
double f(double N) {
        double r4347048 = 1.0;
        double r4347049 = N;
        double r4347050 = fma(r4347049, r4347049, r4347049);
        double r4347051 = r4347048 + r4347050;
        double r4347052 = atan2(r4347048, r4347051);
        return r4347052;
}

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-atan13.7

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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