Average Error: 15.2 → 0.3
Time: 5.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}
double f(double N) {
        double r14744718 = N;
        double r14744719 = 1.0;
        double r14744720 = r14744718 + r14744719;
        double r14744721 = atan(r14744720);
        double r14744722 = atan(r14744718);
        double r14744723 = r14744721 - r14744722;
        return r14744723;
}
↓
double f(double N) {
        double r14744724 = 1.0;
        double r14744725 = N;
        double r14744726 = r14744725 * r14744725;
        double r14744727 = r14744724 + r14744726;
        double r14744728 = r14744725 + r14744727;
        double r14744729 = atan2(r14744724, r14744728);
        return r14744729;
}

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. Using strategy rm

  3. Applied diff-atan14.1

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

  4. Simplified0.3

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

  5. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{N + \left({N}^{2} + 1\right)}}\]

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019124 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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