Average Error: 14.9 → 0.5
Time: 22.4s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\left(\sqrt{\sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}} \cdot \sqrt{\sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}}\right) \cdot \sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\left(\sqrt{\sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}} \cdot \sqrt{\sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}}\right) \cdot \sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}
double f(double N) {
        double r12275432 = N;
        double r12275433 = 1.0;
        double r12275434 = r12275432 + r12275433;
        double r12275435 = atan(r12275434);
        double r12275436 = atan(r12275432);
        double r12275437 = r12275435 - r12275436;
        return r12275437;
}
↓
double f(double N) {
        double r12275438 = 1.0;
        double r12275439 = N;
        double r12275440 = r12275439 + r12275438;
        double r12275441 = fma(r12275439, r12275440, r12275438);
        double r12275442 = atan2(r12275438, r12275441);
        double r12275443 = sqrt(r12275442);
        double r12275444 = sqrt(r12275443);
        double r12275445 = r12275444 * r12275444;
        double r12275446 = r12275445 * r12275443;
        return r12275446;
}

Error

Bits error versus N

Target

Original14.9
Target0.3
Herbie0.5
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

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. Simplified0.3

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

  5. Simplified0.3

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.9

    \[\leadsto \color{blue}{\sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}} \cdot \sqrt{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.9

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

  10. Applied sqrt-prod0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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