Average Error: 14.9 → 0.5
Time: 10.3s
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 r119307 = N;
        double r119308 = 1.0;
        double r119309 = r119307 + r119308;
        double r119310 = atan(r119309);
        double r119311 = atan(r119307);
        double r119312 = r119310 - r119311;
        return r119312;
}
↓
double f(double N) {
        double r119313 = 1.0;
        double r119314 = 1.0;
        double r119315 = N;
        double r119316 = r119315 + r119313;
        double r119317 = cbrt(r119316);
        double r119318 = r119317 * r119317;
        double r119319 = r119315 * r119315;
        double r119320 = r119313 * r119313;
        double r119321 = r119319 - r119320;
        double r119322 = cbrt(r119321);
        double r119323 = r119315 - r119313;
        double r119324 = cbrt(r119323);
        double r119325 = r119314 / r119324;
        double r119326 = r119322 * r119325;
        double r119327 = r119326 * r119315;
        double r119328 = r119318 * r119327;
        double r119329 = r119314 + r119328;
        double r119330 = atan2(r119313, r119329);
        return r119330;
}

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)))