Average Error: 15.1 → 0.3
Time: 7.7s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]
double f(double N) {
        double r6327615 = N;
        double r6327616 = 1.0;
        double r6327617 = r6327615 + r6327616;
        double r6327618 = atan(r6327617);
        double r6327619 = atan(r6327615);
        double r6327620 = r6327618 - r6327619;
        return r6327620;
}
↓
double f(double N) {
        double r6327621 = 1.0;
        double r6327622 = N;
        double r6327623 = r6327622 + r6327621;
        double r6327624 = r6327623 * r6327622;
        double r6327625 = r6327624 + r6327621;
        double r6327626 = atan2(r6327621, r6327625);
        return r6327626;
}


\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}

Error

Bits error versus N

Target

Original15.1
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.1

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019102 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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