Average Error: 14.9 → 0.3
Time: 2.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}
double f(double N) {
        double r177850 = N;
        double r177851 = 1.0;
        double r177852 = r177850 + r177851;
        double r177853 = atan(r177852);
        double r177854 = atan(r177850);
        double r177855 = r177853 - r177854;
        return r177855;
}
↓
double f(double N) {
        double r177856 = 1.0;
        double r177857 = 1.0;
        double r177858 = N;
        double r177859 = r177858 + r177856;
        double r177860 = r177859 * r177858;
        double r177861 = r177857 + r177860;
        double r177862 = atan2(r177856, r177861);
        return r177862;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-atan13.7

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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