Average Error: 14.9 → 0.3
Time: 12.4s
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 r4919051 = N;
        double r4919052 = 1.0;
        double r4919053 = r4919051 + r4919052;
        double r4919054 = atan(r4919053);
        double r4919055 = atan(r4919051);
        double r4919056 = r4919054 - r4919055;
        return r4919056;
}
↓
double f(double N) {
        double r4919057 = 1.0;
        double r4919058 = N;
        double r4919059 = r4919058 + r4919057;
        double r4919060 = r4919059 * r4919058;
        double r4919061 = r4919060 + r4919057;
        double r4919062 = atan2(r4919057, r4919061);
        return r4919062;
}

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-atan13.9

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

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

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

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