Average Error: 14.9 → 0.4
Time: 14.9s
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 r5307038 = N;
        double r5307039 = 1.0;
        double r5307040 = r5307038 + r5307039;
        double r5307041 = atan(r5307040);
        double r5307042 = atan(r5307038);
        double r5307043 = r5307041 - r5307042;
        return r5307043;
}
↓
double f(double N) {
        double r5307044 = 1.0;
        double r5307045 = N;
        double r5307046 = r5307045 * r5307045;
        double r5307047 = r5307045 + r5307044;
        double r5307048 = r5307046 + r5307047;
        double r5307049 = atan2(r5307044, r5307048);
        return r5307049;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.7

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

  4. Simplified0.4

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

  6. Applied add-sqr-sqrt1.0

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

  7. Taylor expanded around 0 0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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