Average Error: 15.8 → 0.3
Time: 7.5s
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 r11528493 = N;
        double r11528494 = 1.0;
        double r11528495 = r11528493 + r11528494;
        double r11528496 = atan(r11528495);
        double r11528497 = atan(r11528493);
        double r11528498 = r11528496 - r11528497;
        return r11528498;
}
↓
double f(double N) {
        double r11528499 = 1.0;
        double r11528500 = N;
        double r11528501 = r11528500 + r11528499;
        double r11528502 = r11528501 * r11528500;
        double r11528503 = r11528502 + r11528499;
        double r11528504 = atan2(r11528499, r11528503);
        return r11528504;
}


\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

Original15.8
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.8

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

  3. Applied diff-atan14.6

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

  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 2019101 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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