Average Error: 14.6 → 0.4
Time: 2.9s
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 r109113 = N;
        double r109114 = 1.0;
        double r109115 = r109113 + r109114;
        double r109116 = atan(r109115);
        double r109117 = atan(r109113);
        double r109118 = r109116 - r109117;
        return r109118;
}
↓
double f(double N) {
        double r109119 = 1.0;
        double r109120 = 1.0;
        double r109121 = N;
        double r109122 = r109121 + r109119;
        double r109123 = r109122 * r109121;
        double r109124 = r109120 + r109123;
        double r109125 = atan2(r109119, r109124);
        return r109125;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.6

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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

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

Reproduce

herbie shell --seed 2020021 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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