Average Error: 14.7 → 0.4
Time: 15.4s
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 r116626 = N;
        double r116627 = 1.0;
        double r116628 = r116626 + r116627;
        double r116629 = atan(r116628);
        double r116630 = atan(r116626);
        double r116631 = r116629 - r116630;
        return r116631;
}
↓
double f(double N) {
        double r116632 = 1.0;
        double r116633 = N;
        double r116634 = r116633 + r116632;
        double r116635 = 1.0;
        double r116636 = fma(r116633, r116634, r116635);
        double r116637 = atan2(r116632, r116636);
        return r116637;
}

Error

Bits error versus N

Target

Original14.7
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

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

  3. Applied diff-atan13.6

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

  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 2019325 +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)))