Average Error: 14.5 → 0.4
Time: 2.5s
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 r191018 = N;
        double r191019 = 1.0;
        double r191020 = r191018 + r191019;
        double r191021 = atan(r191020);
        double r191022 = atan(r191018);
        double r191023 = r191021 - r191022;
        return r191023;
}
↓
double f(double N) {
        double r191024 = 1.0;
        double r191025 = N;
        double r191026 = r191025 + r191024;
        double r191027 = 1.0;
        double r191028 = fma(r191025, r191026, r191027);
        double r191029 = atan2(r191024, r191028);
        return r191029;
}

Error

Bits error versus N

Target

Original14.5
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.5

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

  3. Applied diff-atan13.4

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +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)))