Average Error: 14.9 → 0.4
Time: 12.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}
double f(double N) {
        double r11379469 = N;
        double r11379470 = 1.0;
        double r11379471 = r11379469 + r11379470;
        double r11379472 = atan(r11379471);
        double r11379473 = atan(r11379469);
        double r11379474 = r11379472 - r11379473;
        return r11379474;
}
↓
double f(double N) {
        double r11379475 = 1.0;
        double r11379476 = N;
        double r11379477 = r11379476 + r11379475;
        double r11379478 = fma(r11379476, r11379477, r11379475);
        double r11379479 = atan2(r11379475, r11379478);
        return r11379479;
}

Error

Bits error versus N

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-atan13.7

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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