Average Error: 15.6 → 0.4
Time: 11.8s
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 r123334 = N;
        double r123335 = 1.0;
        double r123336 = r123334 + r123335;
        double r123337 = atan(r123336);
        double r123338 = atan(r123334);
        double r123339 = r123337 - r123338;
        return r123339;
}
↓
double f(double N) {
        double r123340 = 1.0;
        double r123341 = N;
        double r123342 = r123341 + r123340;
        double r123343 = 1.0;
        double r123344 = fma(r123341, r123342, r123343);
        double r123345 = atan2(r123340, r123344);
        return r123345;
}

Error

Bits error versus N

Target

Original15.6
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.6

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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. Simplified0.4

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{0 + 1}}{1 + \left(N + 1\right) \cdot N}\]

  5. Simplified0.4

    \[\leadsto \tan^{-1}_* \frac{0 + 1}{\color{blue}{\mathsf{fma}\left(N, N + 1, 1\right)}}\]

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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