Average Error: 15.1 → 0.3
Time: 10.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + N \cdot N\right) + 1}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + N \cdot N\right) + 1}
double f(double N) {
        double r5130267 = N;
        double r5130268 = 1.0;
        double r5130269 = r5130267 + r5130268;
        double r5130270 = atan(r5130269);
        double r5130271 = atan(r5130267);
        double r5130272 = r5130270 - r5130271;
        return r5130272;
}
↓
double f(double N) {
        double r5130273 = 1.0;
        double r5130274 = N;
        double r5130275 = r5130274 * r5130274;
        double r5130276 = r5130274 + r5130275;
        double r5130277 = r5130276 + r5130273;
        double r5130278 = atan2(r5130273, r5130277);
        return r5130278;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.0

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

  4. Simplified0.3

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

  5. Taylor expanded around 0 0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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