Average Error: 14.7 → 0.3
Time: 11.6s
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 r99249 = N;
        double r99250 = 1.0;
        double r99251 = r99249 + r99250;
        double r99252 = atan(r99251);
        double r99253 = atan(r99249);
        double r99254 = r99252 - r99253;
        return r99254;
}
↓
double f(double N) {
        double r99255 = 1.0;
        double r99256 = N;
        double r99257 = r99256 + r99255;
        double r99258 = r99257 * r99256;
        double r99259 = 1.0;
        double r99260 = r99258 + r99259;
        double r99261 = atan2(r99255, r99260);
        return r99261;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

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

  2. Simplified14.7

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

  4. Applied diff-atan13.7

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

  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 2019195 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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