Average Error: 14.9 → 0.4
Time: 9.0s
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 r6076389 = N;
        double r6076390 = 1.0;
        double r6076391 = r6076389 + r6076390;
        double r6076392 = atan(r6076391);
        double r6076393 = atan(r6076389);
        double r6076394 = r6076392 - r6076393;
        return r6076394;
}
↓
double f(double N) {
        double r6076395 = 1.0;
        double r6076396 = 1.0;
        double r6076397 = N;
        double r6076398 = r6076395 + r6076397;
        double r6076399 = r6076397 * r6076398;
        double r6076400 = r6076396 + r6076399;
        double r6076401 = atan2(r6076395, r6076400);
        return r6076401;
}

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)))