Average Error: 15.4 → 0.3
Time: 11.0s
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 r3978586 = N;
        double r3978587 = 1.0;
        double r3978588 = r3978586 + r3978587;
        double r3978589 = atan(r3978588);
        double r3978590 = atan(r3978586);
        double r3978591 = r3978589 - r3978590;
        return r3978591;
}
↓
double f(double N) {
        double r3978592 = 1.0;
        double r3978593 = N;
        double r3978594 = r3978593 + r3978592;
        double r3978595 = r3978594 * r3978593;
        double r3978596 = r3978595 + r3978592;
        double r3978597 = atan2(r3978592, r3978596);
        return r3978597;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.4
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.4

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

  3. Applied diff-atan14.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019132 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))