Average Error: 15.2 → 0.3
Time: 16.9s
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 r84462 = N;
        double r84463 = 1.0;
        double r84464 = r84462 + r84463;
        double r84465 = atan(r84464);
        double r84466 = atan(r84462);
        double r84467 = r84465 - r84466;
        return r84467;
}
↓
double f(double N) {
        double r84468 = 1.0;
        double r84469 = N;
        double r84470 = r84469 + r84468;
        double r84471 = r84470 * r84469;
        double r84472 = 1.0;
        double r84473 = r84471 + r84472;
        double r84474 = atan2(r84468, r84473);
        return r84474;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.2
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  2. Simplified15.2

    \[\leadsto \color{blue}{\tan^{-1} \left(1 + N\right) - \tan^{-1} N}\]
  3. Using strategy rm

  4. Applied diff-atan14.0

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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