Average Error: 15.1 → 0.3
Time: 12.5s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + N \cdot \left(1 + N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + N \cdot \left(1 + N\right)}
double f(double N) {
        double r5409512 = N;
        double r5409513 = 1.0;
        double r5409514 = r5409512 + r5409513;
        double r5409515 = atan(r5409514);
        double r5409516 = atan(r5409512);
        double r5409517 = r5409515 - r5409516;
        return r5409517;
}
↓
double f(double N) {
        double r5409518 = 1.0;
        double r5409519 = 1.0;
        double r5409520 = N;
        double r5409521 = r5409518 + r5409520;
        double r5409522 = r5409520 * r5409521;
        double r5409523 = r5409519 + r5409522;
        double r5409524 = atan2(r5409518, r5409523);
        return r5409524;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.1
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.1

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))