Average Error: 15.1 → 0.3
Time: 12.9s
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 r4771336 = N;
        double r4771337 = 1.0;
        double r4771338 = r4771336 + r4771337;
        double r4771339 = atan(r4771338);
        double r4771340 = atan(r4771336);
        double r4771341 = r4771339 - r4771340;
        return r4771341;
}
↓
double f(double N) {
        double r4771342 = 1.0;
        double r4771343 = N;
        double r4771344 = fma(r4771343, r4771343, r4771343);
        double r4771345 = r4771342 + r4771344;
        double r4771346 = atan2(r4771342, r4771345);
        return r4771346;
}

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-atan14.0

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  7. Applied fma-udef0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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