Average Error: 14.8 → 0.4
Time: 14.9s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}
double f(double N) {
        double r9456737 = N;
        double r9456738 = 1.0;
        double r9456739 = r9456737 + r9456738;
        double r9456740 = atan(r9456739);
        double r9456741 = atan(r9456737);
        double r9456742 = r9456740 - r9456741;
        return r9456742;
}
↓
double f(double N) {
        double r9456743 = 1.0;
        double r9456744 = N;
        double r9456745 = r9456744 + r9456743;
        double r9456746 = r9456745 * r9456744;
        double r9456747 = r9456746 + r9456743;
        double r9456748 = atan2(r9456743, r9456747);
        return r9456748;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

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

  3. Applied diff-atan13.7

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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