Average Error: 14.9 → 0.5
Time: 8.3s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right) \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right) \cdot N\right)}
double f(double N) {
        double r138440 = N;
        double r138441 = 1.0;
        double r138442 = r138440 + r138441;
        double r138443 = atan(r138442);
        double r138444 = atan(r138440);
        double r138445 = r138443 - r138444;
        return r138445;
}
↓
double f(double N) {
        double r138446 = 1.0;
        double r138447 = 1.0;
        double r138448 = N;
        double r138449 = r138448 + r138446;
        double r138450 = cbrt(r138449);
        double r138451 = r138450 * r138450;
        double r138452 = r138448 * r138448;
        double r138453 = r138446 * r138446;
        double r138454 = r138452 - r138453;
        double r138455 = cbrt(r138454);
        double r138456 = r138448 - r138446;
        double r138457 = cbrt(r138456);
        double r138458 = r138447 / r138457;
        double r138459 = r138455 * r138458;
        double r138460 = r138459 * r138448;
        double r138461 = r138451 * r138460;
        double r138462 = r138447 + r138461;
        double r138463 = atan2(r138446, r138462);
        return r138463;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.3
Herbie0.5
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.8

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.6

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

  7. Applied associate-*l*0.6

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

  9. Applied flip-+0.6

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

  10. Applied cbrt-div0.5

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

  12. Applied div-inv0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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