Average Error: 15.6 → 0.4
Time: 16.0s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N \cdot N + \left(N + 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N \cdot N + \left(N + 1\right)}
double f(double N) {
        double r4538258 = N;
        double r4538259 = 1.0;
        double r4538260 = r4538258 + r4538259;
        double r4538261 = atan(r4538260);
        double r4538262 = atan(r4538258);
        double r4538263 = r4538261 - r4538262;
        return r4538263;
}
↓
double f(double N) {
        double r4538264 = 1.0;
        double r4538265 = N;
        double r4538266 = r4538265 * r4538265;
        double r4538267 = r4538265 + r4538264;
        double r4538268 = r4538266 + r4538267;
        double r4538269 = atan2(r4538264, r4538268);
        return r4538269;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.6
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.6

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

  3. Applied diff-atan14.5

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

  4. Simplified0.4

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

  5. Taylor expanded around -inf 0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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