Average Error: 15.0 → 0.4
Time: 13.3s
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 r5842243 = N;
        double r5842244 = 1.0;
        double r5842245 = r5842243 + r5842244;
        double r5842246 = atan(r5842245);
        double r5842247 = atan(r5842243);
        double r5842248 = r5842246 - r5842247;
        return r5842248;
}
↓
double f(double N) {
        double r5842249 = 1.0;
        double r5842250 = N;
        double r5842251 = r5842250 + r5842249;
        double r5842252 = r5842251 * r5842250;
        double r5842253 = r5842252 + r5842249;
        double r5842254 = atan2(r5842249, r5842253);
        return r5842254;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

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

  6. Applied *-commutative0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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