Average Error: 15.3 → 0.3
Time: 11.3s
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 r145605 = N;
        double r145606 = 1.0;
        double r145607 = r145605 + r145606;
        double r145608 = atan(r145607);
        double r145609 = atan(r145605);
        double r145610 = r145608 - r145609;
        return r145610;
}
↓
double f(double N) {
        double r145611 = 1.0;
        double r145612 = N;
        double r145613 = r145612 + r145611;
        double r145614 = r145613 * r145612;
        double r145615 = 1.0;
        double r145616 = r145614 + r145615;
        double r145617 = atan2(r145611, r145616);
        return r145617;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 15.3

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

  3. Applied diff-atan14.2

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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