Average Error: 14.9 → 0.4
Time: 15.9s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}
double f(double N) {
        double r5371607 = N;
        double r5371608 = 1.0;
        double r5371609 = r5371607 + r5371608;
        double r5371610 = atan(r5371609);
        double r5371611 = atan(r5371607);
        double r5371612 = r5371610 - r5371611;
        return r5371612;
}
↓
double f(double N) {
        double r5371613 = 1.0;
        double r5371614 = N;
        double r5371615 = r5371614 * r5371614;
        double r5371616 = r5371613 + r5371615;
        double r5371617 = r5371614 + r5371616;
        double r5371618 = atan2(r5371613, r5371617);
        return r5371618;
}

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. Using strategy rm

  8. Applied add-sqr-sqrt1.0

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

  9. Applied sqrt-prod0.6

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

  10. Taylor expanded around 0 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

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)))