Average Error: 14.9 → 0.4
Time: 10.6s
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 r5474371 = N;
        double r5474372 = 1.0;
        double r5474373 = r5474371 + r5474372;
        double r5474374 = atan(r5474373);
        double r5474375 = atan(r5474371);
        double r5474376 = r5474374 - r5474375;
        return r5474376;
}
↓
double f(double N) {
        double r5474377 = 1.0;
        double r5474378 = 1.0;
        double r5474379 = N;
        double r5474380 = r5474377 + r5474379;
        double r5474381 = r5474379 * r5474380;
        double r5474382 = r5474378 + r5474381;
        double r5474383 = atan2(r5474377, r5474382);
        return r5474383;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.8

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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