Average Error: 14.8 → 0.6
Time: 14.4s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\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}}
double f(double N) {
        double r114319 = N;
        double r114320 = 1.0;
        double r114321 = r114319 + r114320;
        double r114322 = atan(r114321);
        double r114323 = atan(r114319);
        double r114324 = r114322 - r114323;
        return r114324;
}
↓
double f(double N) {
        double r114325 = 1.0;
        double r114326 = 1.0;
        double r114327 = N;
        double r114328 = r114327 + r114325;
        double r114329 = cbrt(r114328);
        double r114330 = cbrt(r114327);
        double r114331 = r114329 * r114330;
        double r114332 = r114328 * r114327;
        double r114333 = cbrt(r114332);
        double r114334 = r114331 * r114333;
        double r114335 = r114334 * r114333;
        double r114336 = r114326 + r114335;
        double r114337 = atan2(r114325, r114336);
        return r114337;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.4
Herbie0.6
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

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

  3. Applied diff-atan13.7

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.7

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}}\]
  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019306 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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