Average Error: 14.9 → 0.6
Time: 3.2s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{N + 1}\right) \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{N + 1}\right) \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}
double f(double N) {
        double r125401 = N;
        double r125402 = 1.0;
        double r125403 = r125401 + r125402;
        double r125404 = atan(r125403);
        double r125405 = atan(r125401);
        double r125406 = r125404 - r125405;
        return r125406;
}
↓
double f(double N) {
        double r125407 = 1.0;
        double r125408 = 1.0;
        double r125409 = N;
        double r125410 = r125409 + r125407;
        double r125411 = r125410 * r125409;
        double r125412 = cbrt(r125411);
        double r125413 = cbrt(r125410);
        double r125414 = r125412 * r125413;
        double r125415 = cbrt(r125409);
        double r125416 = r125414 * r125415;
        double r125417 = r125416 * r125412;
        double r125418 = r125408 + r125417;
        double r125419 = atan2(r125407, r125418);
        return r125419;
}

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

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

  9. Applied associate-*r*0.6

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

  10. Final simplification0.6

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

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)))