Average Error: 15.0 → 0.4
Time: 10.7s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}
double f(double N) {
        double r2311441 = N;
        double r2311442 = 1.0;
        double r2311443 = r2311441 + r2311442;
        double r2311444 = atan(r2311443);
        double r2311445 = atan(r2311441);
        double r2311446 = r2311444 - r2311445;
        return r2311446;
}
↓
double f(double N) {
        double r2311447 = 1.0;
        double r2311448 = N;
        double r2311449 = r2311448 + r2311447;
        double r2311450 = fma(r2311448, r2311449, r2311447);
        double r2311451 = atan2(r2311447, r2311450);
        return r2311451;
}

Error

Bits error versus N

Target

Original15.0
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

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

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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