Average Error: 14.7 → 0.3
Time: 15.2s
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 r5146512 = N;
        double r5146513 = 1.0;
        double r5146514 = r5146512 + r5146513;
        double r5146515 = atan(r5146514);
        double r5146516 = atan(r5146512);
        double r5146517 = r5146515 - r5146516;
        return r5146517;
}
↓
double f(double N) {
        double r5146518 = 1.0;
        double r5146519 = N;
        double r5146520 = r5146519 + r5146518;
        double r5146521 = 1.0;
        double r5146522 = fma(r5146519, r5146520, r5146521);
        double r5146523 = atan2(r5146518, r5146522);
        return r5146523;
}

Error

Bits error versus N

Target

Original14.7
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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. 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 2019192 +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)))