Average Error: 14.9 → 0.4
Time: 2.1s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}
double f(double N) {
        double r192397 = N;
        double r192398 = 1.0;
        double r192399 = r192397 + r192398;
        double r192400 = atan(r192399);
        double r192401 = atan(r192397);
        double r192402 = r192400 - r192401;
        return r192402;
}
↓
double f(double N) {
        double r192403 = 1.0;
        double r192404 = N;
        double r192405 = 2.0;
        double r192406 = pow(r192404, r192405);
        double r192407 = r192403 * r192404;
        double r192408 = 1.0;
        double r192409 = r192407 + r192408;
        double r192410 = r192406 + r192409;
        double r192411 = atan2(r192403, r192410);
        return r192411;
}

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-atan13.8

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

  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

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

  7. Applied associate-*l*0.7

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

  8. Taylor expanded around 0 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019354 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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