Average Error: 14.8 → 0.4
Time: 11.1s
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 r4673065 = N;
        double r4673066 = 1.0;
        double r4673067 = r4673065 + r4673066;
        double r4673068 = atan(r4673067);
        double r4673069 = atan(r4673065);
        double r4673070 = r4673068 - r4673069;
        return r4673070;
}
↓
double f(double N) {
        double r4673071 = 1.0;
        double r4673072 = N;
        double r4673073 = r4673072 * r4673072;
        double r4673074 = r4673072 + r4673071;
        double r4673075 = r4673073 + r4673074;
        double r4673076 = atan2(r4673071, r4673075);
        return r4673076;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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 add-cube-cbrt0.7

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{N}\right)}}\]

  7. Applied associate-*r*0.7

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\left(N + 1\right) \cdot \left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right)\right) \cdot \sqrt[3]{N}}}\]
  8. Using strategy rm

  9. Applied cbrt-unprod0.5

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\left(N + 1\right) \cdot \color{blue}{\sqrt[3]{N \cdot N}}\right) \cdot \sqrt[3]{N}}\]

  10. Taylor expanded around 0 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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