Average Error: 15.1 → 0.3
Time: 11.8s
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 r5063798 = N;
        double r5063799 = 1.0;
        double r5063800 = r5063798 + r5063799;
        double r5063801 = atan(r5063800);
        double r5063802 = atan(r5063798);
        double r5063803 = r5063801 - r5063802;
        return r5063803;
}
↓
double f(double N) {
        double r5063804 = 1.0;
        double r5063805 = N;
        double r5063806 = r5063805 * r5063805;
        double r5063807 = r5063805 + r5063806;
        double r5063808 = r5063807 + r5063804;
        double r5063809 = atan2(r5063804, r5063808);
        return r5063809;
}

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)))