Average Error: 14.9 → 0.6
Time: 3.1s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \left(\sqrt[3]{N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right)\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \left(\sqrt[3]{N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right)\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}
double f(double N) {
        double r129507 = N;
        double r129508 = 1.0;
        double r129509 = r129507 + r129508;
        double r129510 = atan(r129509);
        double r129511 = atan(r129507);
        double r129512 = r129510 - r129511;
        return r129512;
}
↓
double f(double N) {
        double r129513 = 1.0;
        double r129514 = 1.0;
        double r129515 = N;
        double r129516 = r129515 + r129513;
        double r129517 = cbrt(r129516);
        double r129518 = cbrt(r129515);
        double r129519 = r129516 * r129515;
        double r129520 = cbrt(r129519);
        double r129521 = r129518 * r129520;
        double r129522 = r129517 * r129521;
        double r129523 = r129522 * r129520;
        double r129524 = r129514 + r129523;
        double r129525 = atan2(r129513, r129524);
        return r129525;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.4
Herbie0.6
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.9

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

  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-cbrt0.6

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

  8. Applied cbrt-prod0.6

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

  9. Applied associate-*l*0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020001 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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