Average Error: 15.1 → 0.4
Time: 14.5s
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 r65007 = N;
        double r65008 = 1.0;
        double r65009 = r65007 + r65008;
        double r65010 = atan(r65009);
        double r65011 = atan(r65007);
        double r65012 = r65010 - r65011;
        return r65012;
}
↓
double f(double N) {
        double r65013 = 1.0;
        double r65014 = N;
        double r65015 = r65014 + r65013;
        double r65016 = 1.0;
        double r65017 = fma(r65014, r65015, r65016);
        double r65018 = atan2(r65013, r65017);
        return r65018;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.1

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +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)))