Average Error: 14.9 → 0.3
Time: 11.2s
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 r3617164 = N;
        double r3617165 = 1.0;
        double r3617166 = r3617164 + r3617165;
        double r3617167 = atan(r3617166);
        double r3617168 = atan(r3617164);
        double r3617169 = r3617167 - r3617168;
        return r3617169;
}
↓
double f(double N) {
        double r3617170 = 1.0;
        double r3617171 = N;
        double r3617172 = r3617171 + r3617170;
        double r3617173 = fma(r3617171, r3617172, r3617170);
        double r3617174 = atan2(r3617170, r3617173);
        return r3617174;
}

Error

Bits error versus N

Target

Original14.9
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.9

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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